{"id":119,"date":"2021-04-16T14:10:05","date_gmt":"2021-04-16T06:10:05","guid":{"rendered":"http:\/\/18.141.9.105\/?p=119"},"modified":"2021-04-17T22:35:56","modified_gmt":"2021-04-17T14:35:56","slug":"optsum","status":"publish","type":"post","link":"https:\/\/yunpeng.site\/index.php\/2021\/04\/16\/optsum\/","title":{"rendered":"\u4f18\u5316\u603b\u7ed3"},"content":{"rendered":"\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t
\n\t\t\t\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\n\n\n\nOptimization Cheatsheet<\/title><link href='https:\/\/fonts.loli.net\/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text\/css' \/><style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida Console\",Consolas,\"Courier\",monospace; --title-bar-height: 20px; }\n.mac-os-11 { --title-bar-height: 28px; }\nhtml { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: \"Helvetica Neue\", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }\nbody { margin: 0px; padding: 0px; height: auto; bottom: 0px; top: 0px; left: 0px; right: 0px; font-size: 1rem; line-height: 1.42857143; overflow-x: hidden; background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; background-color: inherit; tab-size: 4; background-position: inherit inherit; background-repeat: inherit inherit; }\niframe { margin: auto; }\na.url { word-break: break-all; }\na:active, a:hover { outline: 0px; }\n.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }\n#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; word-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }\n#write.first-line-indent p { text-indent: 2em; }\n#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }\n#write.first-line-indent li { margin-left: 2em; }\n.for-image #write { padding-left: 8px; padding-right: 8px; }\nbody.typora-export { padding-left: 30px; padding-right: 30px; }\n.typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; }\n.typora-export .task-list-item input { pointer-events: none; }\n@media screen and (max-width: 500px) { \n body.typora-export { padding-left: 0px; padding-right: 0px; }\n #write { padding-left: 20px; padding-right: 20px; }\n .CodeMirror-sizer { margin-left: 0px !important; }\n .CodeMirror-gutters { display: none !important; }\n}\n#write li > figure:last-child { margin-bottom: 0.5rem; }\n#write ol, #write ul { position: relative; }\nimg { max-width: 100%; vertical-align: middle; image-orientation: from-image; }\nbutton, input, select, textarea { color: inherit; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; }\ninput[type=\"checkbox\"], input[type=\"radio\"] { line-height: normal; padding: 0px; }\n*, ::after, ::before { box-sizing: border-box; }\n#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }\n#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }\np { line-height: inherit; }\nh1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; }\np { orphans: 4; }\nh1 { font-size: 2rem; }\nh2 { font-size: 1.8rem; }\nh3 { font-size: 1.6rem; }\nh4 { font-size: 1.4rem; }\nh5 { font-size: 1.2rem; }\nh6 { font-size: 1rem; }\n.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }\n.hidden { display: none; }\n.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }\na { cursor: pointer; }\nsup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-top-left-radius: 4px; border-top-right-radius: 4px; border-bottom-right-radius: 4px; border-bottom-left-radius: 4px; cursor: pointer; }\nsup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; }\n#write input[type=\"checkbox\"] { cursor: pointer; width: inherit; height: inherit; }\nfigure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }\nfigure > table { margin: 0px; }\ntr { break-inside: avoid; break-after: auto; }\nthead { display: table-header-group; }\ntable { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }\ntable.md-table td { min-width: 32px; }\n.CodeMirror-gutters { border-right-width: 0px; background-color: inherit; }\n.CodeMirror-linenumber { }\n.CodeMirror { text-align: left; }\n.CodeMirror-placeholder { opacity: 0.3; }\n.CodeMirror pre { padding: 0px 4px; }\n.CodeMirror-lines { padding: 0px; }\ndiv.hr:focus { cursor: none; }\n#write pre { white-space: pre-wrap; }\n#write.fences-no-line-wrapping pre { white-space: pre; }\n#write pre.ty-contain-cm { white-space: normal; }\n.CodeMirror-gutters { margin-right: 4px; }\n.md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; background-color: inherit; position: relative !important; background-position: inherit inherit; background-repeat: inherit inherit; }\n.md-diagram-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }\n#write .md-fences.mock-cm { white-space: pre-wrap; }\n.md-fences.md-fences-with-lineno { padding-left: 0px; }\n#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }\n.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }\n.CodeMirror-line, twitterwidget { break-inside: avoid; }\n.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }\n.footnotes + .footnotes { margin-top: 0px; }\n.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; background-position: 0px 0px; background-repeat: initial initial; }\nli div { padding-top: 0px; }\nblockquote { margin: 1rem 0px; }\nli .mathjax-block, li p { margin: 0.5rem 0px; }\nli blockquote { margin: 1rem 0px; }\nli { margin: 0px; position: relative; }\nblockquote > :last-child { margin-bottom: 0px; }\nblockquote > :first-child, li > :first-child { margin-top: 0px; }\n.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }\n#write .footnote-line { white-space: pre-wrap; }\n@media print { \n body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; }\n #write { margin-top: 0px; padding-top: 0px; border-color: transparent !important; }\n .typora-export * { -webkit-print-color-adjust: exact; }\n .typora-export #write { break-after: avoid; }\n .typora-export #write::after { height: 0px; }\n .is-mac table { break-inside: avoid; }\n}\n.footnote-line { margin-top: 0.714em; font-size: 0.7em; }\na img, img a { cursor: pointer; }\npre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background-color: rgb(204, 204, 204); display: block; overflow-x: hidden; background-position: initial initial; background-repeat: initial initial; }\np > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }\n#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }\np > .md-image:only-child { display: inline-block; width: 100%; }\n#write .MathJax_Display { margin: 0.8em 0px 0px; }\n.md-math-block { width: 100%; }\n.md-math-block:not(:empty)::after { display: none; }\n.MathJax_ref { fill: currentcolor; }\n[contenteditable=\"true\"]:active, [contenteditable=\"true\"]:focus, [contenteditable=\"false\"]:active, [contenteditable=\"false\"]:focus { outline: 0px; box-shadow: none; }\n.md-task-list-item { position: relative; list-style-type: none; }\n.task-list-item.md-task-list-item { padding-left: 0px; }\n.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }\n.math { font-size: 1rem; }\n.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-top-left-radius: 10px; border-top-right-radius: 10px; border-bottom-right-radius: 10px; border-bottom-left-radius: 10px; }\n.md-toc-content { position: relative; margin-left: 0px; }\n.md-toc-content::after, .md-toc::after { display: none; }\n.md-toc-item { display: block; color: rgb(65, 131, 196); }\n.md-toc-item a { text-decoration: none; }\n.md-toc-inner:hover { text-decoration: underline; }\n.md-toc-inner { display: inline-block; cursor: pointer; }\n.md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; }\n.md-toc-h2 .md-toc-inner { margin-left: 2em; }\n.md-toc-h3 .md-toc-inner { margin-left: 4em; }\n.md-toc-h4 .md-toc-inner { margin-left: 6em; }\n.md-toc-h5 .md-toc-inner { margin-left: 8em; }\n.md-toc-h6 .md-toc-inner { margin-left: 10em; }\n@media screen and (max-width: 48em) { \n .md-toc-h3 .md-toc-inner { margin-left: 3.5em; }\n .md-toc-h4 .md-toc-inner { margin-left: 5em; }\n .md-toc-h5 .md-toc-inner { margin-left: 6.5em; }\n .md-toc-h6 .md-toc-inner { margin-left: 8em; }\n}\na.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }\n.footnote-line a:not(.reversefootnote) { color: inherit; }\n.md-attr { display: none; }\n.md-fn-count::after { content: \".\"; }\ncode, pre, samp, tt { font-family: var(--monospace); }\nkbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background-color: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; background-position: initial initial; background-repeat: initial initial; }\n.md-comment { color: rgb(162, 127, 3); opacity: 0.8; font-family: var(--monospace); }\ncode { text-align: left; }\na.md-print-anchor { white-space: pre !important; border: none !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; text-shadow: initial !important; background-position: 0px 0px !important; background-repeat: initial initial !important; }\n.md-inline-math .MathJax_SVG .noError { display: none !important; }\n.html-for-mac .inline-math-svg .MathJax_SVG { vertical-align: 0.2px; }\n.md-math-block .MathJax_SVG_Display { text-align: center; margin: 0px; position: relative; text-indent: 0px; max-width: none; max-height: none; min-height: 0px; min-width: 100%; width: auto; overflow-y: hidden; display: block !important; }\n.MathJax_SVG_Display, .md-inline-math .MathJax_SVG_Display { width: auto; margin: inherit; display: inline-block !important; }\n.MathJax_SVG .MJX-monospace { font-family: var(--monospace); }\n.MathJax_SVG .MJX-sans-serif { font-family: sans-serif; }\n.MathJax_SVG { display: inline; font-style: normal; font-weight: 400; line-height: normal; zoom: 90%; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; }\n.MathJax_SVG * { transition: none; }\n.MathJax_SVG_Display svg { vertical-align: middle !important; margin-bottom: 0px !important; margin-top: 0px !important; }\n.os-windows.monocolor-emoji .md-emoji { font-family: \"Segoe UI Symbol\", sans-serif; }\n.md-diagram-panel > svg { max-width: 100%; }\n[lang=\"flow\"] svg, [lang=\"mermaid\"] svg { max-width: 100%; height: auto; }\n[lang=\"mermaid\"] .node text { font-size: 1rem; }\ntable tr th { border-bottom-width: 0px; }\nvideo { max-width: 100%; display: block; margin: 0px auto; }\niframe { max-width: 100%; width: 100%; border: none; }\n.highlight td, .highlight tr { border: 0px; }\nmark { background-color: rgb(255, 255, 0); color: rgb(0, 0, 0); background-position: initial initial; background-repeat: initial initial; }\n.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }\nmark .md-meta { color: rgb(0, 0, 0); opacity: 0.3 !important; }\n@media print { \n .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; }\n}\n.md-diagram-panel .messageText { stroke: none !important; }\n.md-diagram-panel .start-state { fill: var(--node-fill); }\n.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }\n.md-require-zoom-fix foreignObject { font-size: var(--mermaid-font-zoom); }\n\n\n:root {\n --side-bar-bg-color: #fafafa;\n --control-text-color: #777;\n}\n\n@include-when-export url(https:\/\/fonts.loli.net\/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);\n\n\/* open-sans-regular - latin-ext_latin *\/\n \/* open-sans-italic - latin-ext_latin *\/\n \/* open-sans-700 - latin-ext_latin *\/\n \/* open-sans-700italic - latin-ext_latin *\/\n html {\n font-size: 16px;\n}\n\nbody {\n font-family: \"Open Sans\",\"Clear Sans\", \"Helvetica Neue\", Helvetica, Arial, sans-serif;\n color: rgb(51, 51, 51);\n line-height: 1.6;\n}\n\n#write {\n max-width: 860px;\n \tmargin: 0 auto;\n \tpadding: 30px;\n padding-bottom: 100px;\n}\n\n@media only screen and (min-width: 1400px) {\n\t#write {\n\t\tmax-width: 1024px;\n\t}\n}\n\n@media only screen and (min-width: 1800px) {\n\t#write {\n\t\tmax-width: 1200px;\n\t}\n}\n\n#write > ul:first-child,\n#write > ol:first-child{\n margin-top: 30px;\n}\n\na {\n color: #4183C4;\n}\nh1,\nh2,\nh3,\nh4,\nh5,\nh6 {\n position: relative;\n margin-top: 1rem;\n margin-bottom: 1rem;\n font-weight: bold;\n line-height: 1.4;\n cursor: text;\n}\nh1:hover a.anchor,\nh2:hover a.anchor,\nh3:hover a.anchor,\nh4:hover a.anchor,\nh5:hover a.anchor,\nh6:hover a.anchor {\n text-decoration: none;\n}\nh1 tt,\nh1 code {\n font-size: inherit;\n}\nh2 tt,\nh2 code {\n font-size: inherit;\n}\nh3 tt,\nh3 code {\n font-size: inherit;\n}\nh4 tt,\nh4 code {\n font-size: inherit;\n}\nh5 tt,\nh5 code {\n font-size: inherit;\n}\nh6 tt,\nh6 code {\n font-size: inherit;\n}\nh1 {\n font-size: 2.25em;\n line-height: 1.2;\n border-bottom: 1px solid #eee;\n}\nh2 {\n font-size: 1.75em;\n line-height: 1.225;\n border-bottom: 1px solid #eee;\n}\n\n\/*@media print {\n .typora-export h1,\n .typora-export h2 {\n border-bottom: none;\n padding-bottom: initial;\n }\n\n .typora-export h1::after,\n .typora-export h2::after {\n content: \"\";\n display: block;\n height: 100px;\n margin-top: -96px;\n border-top: 1px solid #eee;\n }\n}*\/\n\nh3 {\n font-size: 1.5em;\n line-height: 1.43;\n}\nh4 {\n font-size: 1.25em;\n}\nh5 {\n font-size: 1em;\n}\nh6 {\n font-size: 1em;\n color: #777;\n}\np,\nblockquote,\nul,\nol,\ndl,\ntable{\n margin: 0.8em 0;\n}\nli>ol,\nli>ul {\n margin: 0 0;\n}\nhr {\n height: 2px;\n padding: 0;\n margin: 16px 0;\n background-color: #e7e7e7;\n border: 0 none;\n overflow: hidden;\n box-sizing: content-box;\n}\n\nli p.first {\n display: inline-block;\n}\nul,\nol {\n padding-left: 30px;\n}\nul:first-child,\nol:first-child {\n margin-top: 0;\n}\nul:last-child,\nol:last-child {\n margin-bottom: 0;\n}\nblockquote {\n border-left: 4px solid #dfe2e5;\n padding: 0 15px;\n color: #777777;\n}\nblockquote blockquote {\n padding-right: 0;\n}\ntable {\n padding: 0;\n word-break: initial;\n}\ntable tr {\n border-top: 1px solid #dfe2e5;\n margin: 0;\n padding: 0;\n}\ntable tr:nth-child(2n),\nthead {\n background-color: #f8f8f8;\n}\ntable th {\n font-weight: bold;\n border: 1px solid #dfe2e5;\n border-bottom: 0;\n margin: 0;\n padding: 6px 13px;\n}\ntable td {\n border: 1px solid #dfe2e5;\n margin: 0;\n padding: 6px 13px;\n}\ntable th:first-child,\ntable td:first-child {\n margin-top: 0;\n}\ntable th:last-child,\ntable td:last-child {\n margin-bottom: 0;\n}\n\n.CodeMirror-lines {\n padding-left: 4px;\n}\n\n.code-tooltip {\n box-shadow: 0 1px 1px 0 rgba(0,28,36,.3);\n border-top: 1px solid #eef2f2;\n}\n\n.md-fences,\ncode,\ntt {\n border: 1px solid #e7eaed;\n background-color: #f8f8f8;\n border-radius: 3px;\n padding: 0;\n padding: 2px 4px 0px 4px;\n font-size: 0.9em;\n}\n\ncode {\n background-color: #f3f4f4;\n padding: 0 2px 0 2px;\n}\n\n.md-fences {\n margin-bottom: 15px;\n margin-top: 15px;\n padding-top: 8px;\n padding-bottom: 6px;\n}\n\n\n.md-task-list-item > input {\n margin-left: -1.3em;\n}\n\n@media print {\n html {\n font-size: 13px;\n }\n table,\n pre {\n page-break-inside: avoid;\n }\n pre {\n word-wrap: break-word;\n }\n}\n\n.md-fences {\n\tbackground-color: #f8f8f8;\n}\n#write pre.md-meta-block {\n\tpadding: 1rem;\n font-size: 85%;\n line-height: 1.45;\n background-color: #f7f7f7;\n border: 0;\n border-radius: 3px;\n color: #777777;\n margin-top: 0 !important;\n}\n\n.mathjax-block>.code-tooltip {\n\tbottom: .375rem;\n}\n\n.md-mathjax-midline {\n background: #fafafa;\n}\n\n#write>h3.md-focus:before{\n\tleft: -1.5625rem;\n\ttop: .375rem;\n}\n#write>h4.md-focus:before{\n\tleft: -1.5625rem;\n\ttop: .285714286rem;\n}\n#write>h5.md-focus:before{\n\tleft: -1.5625rem;\n\ttop: .285714286rem;\n}\n#write>h6.md-focus:before{\n\tleft: -1.5625rem;\n\ttop: .285714286rem;\n}\n.md-image>.md-meta {\n \/*border: 1px solid #ddd;*\/\n border-radius: 3px;\n padding: 2px 0px 0px 4px;\n font-size: 0.9em;\n color: inherit;\n}\n\n.md-tag {\n color: #a7a7a7;\n opacity: 1;\n}\n\n.md-toc { \n margin-top:20px;\n padding-bottom:20px;\n}\n\n.sidebar-tabs {\n border-bottom: none;\n}\n\n#typora-quick-open {\n border: 1px solid #ddd;\n background-color: #f8f8f8;\n}\n\n#typora-quick-open-item {\n background-color: #FAFAFA;\n border-color: #FEFEFE #e5e5e5 #e5e5e5 #eee;\n border-style: solid;\n border-width: 1px;\n}\n\n\/** focus mode *\/\n.on-focus-mode blockquote {\n border-left-color: rgba(85, 85, 85, 0.12);\n}\n\nheader, .context-menu, .megamenu-content, footer{\n font-family: \"Segoe UI\", \"Arial\", sans-serif;\n}\n\n.file-node-content:hover .file-node-icon,\n.file-node-content:hover .file-node-open-state{\n visibility: visible;\n}\n\n.mac-seamless-mode #typora-sidebar {\n background-color: #fafafa;\n background-color: var(--side-bar-bg-color);\n}\n\n.md-lang {\n color: #b4654d;\n}\n\n.html-for-mac .context-menu {\n --item-hover-bg-color: #E6F0FE;\n}\n\n#md-notification .btn {\n border: 0;\n}\n\n.dropdown-menu .divider {\n border-color: #e5e5e5;\n}\n\n.ty-preferences .window-content {\n background-color: #fafafa;\n}\n\n.ty-preferences .nav-group-item.active {\n color: white;\n background: #999;\n}\n\n :root {--mermaid-font-zoom:1em ;} .typora-export li, .typora-export p, .typora-export, .footnote-line {white-space: normal;} \n<\/style>\n<\/head>\n<body class='typora-export'>\n<div id='write' class=''><h1><a name=\"optimization-cheatsheet\" class=\"md-header-anchor\"><\/a><span>Optimization Cheatsheet<\/span><\/h1><h2><a name=\"handout-1\tintroduction\" class=\"md-header-anchor\"><\/a><span>Handout 1<\/span><span>\t<\/span><span>Introduction<\/span><\/h2><h3><a name=\"basic-notions\" class=\"md-header-anchor\"><\/a><span>Basic Notions<\/span><\/h3><p><strong><span>Optimal Value:<\/span><\/strong><span>\t<\/span><span>Is defined to be the greatest lower bound or infimum of the set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.818ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6380.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E1-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E1-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMATHI-66\" x=\"500\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-28\" x=\"1050\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMATHI-78\" x=\"1439\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-29\" x=\"2011\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-3A\" x=\"2677\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMATHI-78\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-2208\" x=\"4083\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMATHI-58\" x=\"5028\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-7D\" x=\"5880\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{f(x): x\\in X\\}<\/script><\/p><p><strong><span>Global Minimizer:<\/span><\/strong><span>\t<\/span><span>The <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.382ex\" height=\"1.994ex\" viewBox=\"0 -755.9 1025.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E2-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E2-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E2-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMAIN-2217\" x=\"808\" y=\"513\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x^*<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.977ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5587.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E3-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E3-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><g transform=\"translate(939,0)\"><use xlink:href=\"#E3-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMAIN-2217\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E3-MJMAIN-29\" x=\"1964\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-2264\" x=\"2631\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-66\" x=\"3687\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-28\" x=\"4237\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-78\" x=\"4626\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-29\" x=\"5198\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x^*) \\le f(x) <\/script><\/p><p><span>Note that optimal value can be finite even if global minimizer does not exist<\/span><\/p><h3><a name=\"different-problems\" class=\"md-header-anchor\"><\/a><span>Different Problems<\/span><\/h3><p><strong><span>LP Problems\uff1a<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.413ex\" height=\"2.577ex\" viewBox=\"0 -806.1 1900 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E386-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E386-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x)<\/script><span> is a linear function and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.979ex\" height=\"1.877ex\" viewBox=\"0 -755.9 852 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E372-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E372-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X<\/script><span> is a set defined by linear inequalities<\/span><\/p><p><span>minimize <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.723ex\" height=\"2.344ex\" viewBox=\"0 -906.7 1602.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E261-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E261-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E261-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><use xlink:href=\"#E261-MJMATHI-78\" x=\"1030\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">c^Tx<\/script><span>\t\t<\/span><span>subject to <\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.164ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3084.6 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E7-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E7-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E7-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E7-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E7-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E7-MJMATHI-78\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E7-MJMAIN-2264\" x=\"1599\" y=\"0\"><\/use><use xlink:href=\"#E7-MJMATHI-62\" x=\"2655\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Ax \\le b<\/script><\/p><p><span>Note that eualities can be transformed to inequalities by:<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.651ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2863.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E8-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E8-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMATHI-78\" x=\"529\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMAIN-3D\" x=\"1378\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMATHI-62\" x=\"2434\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">ax = b<\/script><span>\t<\/span><span>=> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"22.593ex\" height=\"2.577ex\" viewBox=\"0 -806.1 9727.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E9-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E9-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMATHI-61\" x=\"500\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMATHI-78\" x=\"1029\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-2264\" x=\"1878\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMATHI-62\" x=\"2934\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-2212\" x=\"4585\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMATHI-61\" x=\"5586\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMATHI-78\" x=\"6115\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-2264\" x=\"6964\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-2212\" x=\"8020\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMATHI-62\" x=\"8798\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-7D\" x=\"9227\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{ax \\le b \\quad -ax \\le -b\\}<\/script><\/p><p><strong><span>Quadratic Programming (QP) Problems:<\/span><\/strong><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.979ex\" height=\"1.877ex\" viewBox=\"0 -755.9 852 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E372-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E372-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X<\/script><span> is also a set defined by linear inequalities while <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.393ex\" height=\"2.811ex\" viewBox=\"0 -906.7 5766.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E11-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E11-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E11-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E11-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E11-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E11-MJMAIN-3D\" x=\"2177\" y=\"0\"><\/use><g transform=\"translate(3233,0)\"><use xlink:href=\"#E11-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E11-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E11-MJMATHI-51\" x=\"4403\" y=\"0\"><\/use><use xlink:href=\"#E11-MJMATHI-78\" x=\"5194\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) = x^TQx<\/script><span> (<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.628ex\" height=\"3.978ex\" viewBox=\"0 -1258.5 4576 1712.8\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.055ex;\"><defs><path stroke-width=\"0\" id=\"E12-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E12-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E12-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E12-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E12-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E12-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E12-MJMAIN-3D\" x=\"1068\" y=\"0\"><\/use><g transform=\"translate(1846,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"2211\" height=\"60\" x=\"0\" y=\"220\"><\/rect><g transform=\"translate(60,541)\"><use transform=\"scale(0.707)\" xlink:href=\"#E12-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E12-MJMAIN-2B\" x=\"791\" y=\"0\"><\/use><g transform=\"translate(1109,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E12-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E12-MJMATHI-54\" x=\"1118\" y=\"609\"><\/use><\/g><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E12-MJMAIN-32\" x=\"1313\" y=\"-531\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">Q = \\frac{Q+Q^T}{2}<\/script><span> is symmertric without loss)<\/span><\/p><p><span>Note that this is different from Quadratically Constrained Quadratic Programming Problems.<\/span><\/p><p><strong><span>Semidefinite Programming (SDP) Problems:<\/span><\/strong><span>\t<\/span><\/p><p><span>inf <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.539ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1523.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E271-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E271-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E271-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E271-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty<\/script><span>\t\t<\/span><span>subject to <\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"19.078ex\" height=\"2.811ex\" viewBox=\"0 -856.4 8214.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E14-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E14-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E14-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E14-MJMAIN-2212\" x=\"982\" y=\"0\"><\/use><g transform=\"translate(1982,0)\"><use xlink:href=\"#E14-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E14-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E14-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E14-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E14-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(4452,0)\"><use xlink:href=\"#E14-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E14-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(5286,0)\"><use xlink:href=\"#E14-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E14-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E14-MJMAIN-2265\" x=\"6658\" y=\"0\"><\/use><use xlink:href=\"#E14-MJMAIN-30\" x=\"7714\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C - \\sum ^m _{i=1}y_iA_i \\ge 0<\/script><span>\t<\/span><span>,<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.431ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3199.4 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E273-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E273-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E273-MJMAIN-2208\" x=\"774\" y=\"0\"><\/use><g transform=\"translate(1719,0)\"><use xlink:href=\"#E273-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E273-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y \\in R^m<\/script><\/p><p><span>Positive Semidefinte: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.096ex\" height=\"2.461ex\" viewBox=\"0 -806.1 2624.6 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E16-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E16-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E16-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E16-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E16-MJMAIN-2265\" x=\"1068\" y=\"0\"><\/use><use xlink:href=\"#E16-MJMAIN-30\" x=\"2124\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Q \\ge 0<\/script><span> or <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.61ex\" height=\"2.811ex\" viewBox=\"0 -806.1 3276.7 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E17-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E17-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E17-MJMAIN-2208\" x=\"1068\" y=\"0\"><\/use><g transform=\"translate(2013,0)\"><use xlink:href=\"#E17-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-6E\" x=\"925\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMAIN-2B\" x=\"866\" y=\"-351\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">Q \\in S^n _+<\/script><span> if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.141ex\" height=\"2.694ex\" viewBox=\"0 -906.7 4366.4 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E18-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E18-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><use xlink:href=\"#E18-MJMATHI-51\" x=\"1169\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMATHI-78\" x=\"1960\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2265\" x=\"2810\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-30\" x=\"3866\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x^TQx \\ge 0<\/script><span> for any <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E62-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E62-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E62-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E62-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E62-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in R^n<\/script><\/p><p><span>How to combine multiple constraints?<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.027ex\" height=\"2.811ex\" viewBox=\"0 -856.4 8622.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E20-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E20-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E20-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMAIN-31\" x=\"1011\" y=\"-213\"><\/use><use xlink:href=\"#E20-MJMAIN-2212\" x=\"1390\" y=\"0\"><\/use><g transform=\"translate(2390,0)\"><use xlink:href=\"#E20-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(4861,0)\"><use xlink:href=\"#E20-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(5695,0)\"><use xlink:href=\"#E20-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E20-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E20-MJMAIN-2265\" x=\"7066\" y=\"0\"><\/use><use xlink:href=\"#E20-MJMAIN-30\" x=\"8122\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C_1 - \\sum ^m _{i = 1} y_i A_i \\ge 0<\/script><span>\t\t\t<\/span><span>=><\/span><span>\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"36.723ex\" height=\"5.846ex\" viewBox=\"0 -1509.8 15811.1 2517\" role=\"img\" focusable=\"false\" style=\"vertical-align: -2.339ex;\"><defs><path stroke-width=\"0\" id=\"E21-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJSZ3-5B\" d=\"M247 -949V1450H516V1388H309V-887H516V-949H247Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJSZ3-5D\" d=\"M11 1388V1450H280V-949H11V-887H218V1388H11Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-42\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E21-MJSZ3-5B\"><\/use><g transform=\"translate(695,0)\"><g transform=\"translate(-15,0)\"><g transform=\"translate(0,650)\"><use xlink:href=\"#E21-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMAIN-31\" x=\"1011\" y=\"-213\"><\/use><\/g><\/g><g transform=\"translate(2153,0)\"><g transform=\"translate(0,-750)\"><use xlink:href=\"#E21-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMAIN-32\" x=\"1011\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E21-MJSZ3-5D\" x=\"4184\" y=\"-1\"><\/use><use xlink:href=\"#E21-MJMAIN-2212\" x=\"4934\" y=\"0\"><\/use><g transform=\"translate(5934,0)\"><use xlink:href=\"#E21-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(8404,0)\"><use xlink:href=\"#E21-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(9405,0)\"><use xlink:href=\"#E21-MJSZ3-5B\"><\/use><g transform=\"translate(695,0)\"><g transform=\"translate(-15,0)\"><g transform=\"translate(0,650)\"><use xlink:href=\"#E21-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><\/g><\/g><g transform=\"translate(2079,0)\"><g transform=\"translate(0,-750)\"><use xlink:href=\"#E21-MJMATHI-42\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"1073\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E21-MJSZ3-5D\" x=\"4043\" y=\"-1\"><\/use><\/g><use xlink:href=\"#E21-MJMAIN-2265\" x=\"14255\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMAIN-30\" x=\"15311\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\left [ \\begin{matrix}C_1 & \\\\ & C_2 \\end{matrix} \\right ] - \\sum^m _{i=1} y_i \\left [ \\begin{matrix}A_i & \\\\ & B_i \\end{matrix} \\right ] \\ge 0<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.048ex\" height=\"2.811ex\" viewBox=\"0 -856.4 8631.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E22-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMATHI-42\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E22-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMAIN-32\" x=\"1011\" y=\"-213\"><\/use><use xlink:href=\"#E22-MJMAIN-2212\" x=\"1390\" y=\"0\"><\/use><g transform=\"translate(2390,0)\"><use xlink:href=\"#E22-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(4861,0)\"><use xlink:href=\"#E22-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(5695,0)\"><use xlink:href=\"#E22-MJMATHI-42\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMATHI-69\" x=\"1073\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E22-MJMAIN-2265\" x=\"7075\" y=\"0\"><\/use><use xlink:href=\"#E22-MJMAIN-30\" x=\"8131\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C_2 - \\sum ^m _{i=1} y_iB_i \\ge 0<\/script><\/p><p> <\/p><h3><a name=\"properties-of-semidefinte-matrix\" class=\"md-header-anchor\"><\/a><span>Properties of Semidefinte Matrix<\/span><\/h3><p><strong><span>1)<\/span><\/strong><\/p><p><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.54ex\" height=\"6.313ex\" viewBox=\"0 -1610.3 8843.5 2718\" role=\"img\" focusable=\"false\" style=\"vertical-align: -2.573ex;\"><defs><path stroke-width=\"0\" id=\"E23-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJSZ3-5B\" d=\"M247 -949V1450H516V1388H309V-887H516V-949H247Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJSZ3-5D\" d=\"M11 1388V1450H280V-949H11V-887H218V1388H11Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E23-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E23-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E23-MJMAIN-3D\" x=\"1027\" y=\"0\"><\/use><g transform=\"translate(2083,0)\"><use xlink:href=\"#E23-MJSZ3-5B\"><\/use><g transform=\"translate(695,0)\"><g transform=\"translate(-15,0)\"><g transform=\"translate(72,714)\"><use xlink:href=\"#E23-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMAIN-31\" x=\"1060\" y=\"-213\"><\/use><\/g><g transform=\"translate(0,-709)\"><use xlink:href=\"#E23-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMATHI-54\" x=\"1060\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMAIN-32\" x=\"1060\" y=\"-434\"><\/use><\/g><\/g><g transform=\"translate(2332,0)\"><g transform=\"translate(0,714)\"><use xlink:href=\"#E23-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMAIN-32\" x=\"1060\" y=\"-213\"><\/use><\/g><g transform=\"translate(0,-709)\"><use xlink:href=\"#E23-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMAIN-33\" x=\"1060\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E23-MJSZ3-5D\" x=\"4398\" y=\"-1\"><\/use><\/g><use xlink:href=\"#E23-MJMAIN-2265\" x=\"7287\" y=\"0\"><\/use><use xlink:href=\"#E23-MJMAIN-30\" x=\"8343\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A = \\left [ \\begin{matrix}A_1 & A_2\\\\ A_2^T & A_3 \\end{matrix} \\right ] \\ge 0<\/script><span>, then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.054ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3037.1 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E24-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E24-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E24-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E24-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E24-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E24-MJMAIN-31\" x=\"1060\" y=\"-213\"><\/use><use xlink:href=\"#E24-MJMAIN-2265\" x=\"1481\" y=\"0\"><\/use><use xlink:href=\"#E24-MJMAIN-30\" x=\"2537\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A_1 \\ge 0<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.054ex\" height=\"2.461ex\" viewBox=\"0 -806.1 3037.1 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E25-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E25-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMAIN-33\" x=\"1060\" y=\"-213\"><\/use><use xlink:href=\"#E25-MJMAIN-2265\" x=\"1481\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-30\" x=\"2537\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A_3 \\ge 0<\/script><\/p><h2><a name=\"handout-2\telements-of-convex-analysis\" class=\"md-header-anchor\"><\/a><span>Handout 2<\/span><span>\t<\/span><span>Elements of Convex Analysis<\/span><\/h2><h3><a name=\"affine-and-convex-sets\" class=\"md-header-anchor\"><\/a><span>Affine and Convex Sets<\/span><\/h3><p><strong><span>Affine Set:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.44ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7939.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E26-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E26-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMATHI-78\" x=\"640\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-2B\" x=\"1434\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-28\" x=\"2434\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-31\" x=\"2823\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-2212\" x=\"3545\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMATHI-3B1\" x=\"4545\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-29\" x=\"5185\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMATHI-79\" x=\"5574\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-2208\" x=\"6349\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMATHI-53\" x=\"7294\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha x+(1-\\alpha)y \\in S<\/script><\/p><p><strong><span>Convex Set:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.44ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7939.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E27-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E27-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E27-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMATHI-78\" x=\"640\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMAIN-2B\" x=\"1434\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMAIN-28\" x=\"2434\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMAIN-31\" x=\"2823\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMAIN-2212\" x=\"3545\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMATHI-3B1\" x=\"4545\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMAIN-29\" x=\"5185\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMATHI-79\" x=\"5574\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMAIN-2208\" x=\"6349\" y=\"0\"><\/use><use xlink:href=\"#E27-MJMATHI-53\" x=\"7294\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha x + (1-\\alpha)y \\in S<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.973ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3863.2 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E28-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E28-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E28-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E28-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E28-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E28-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E28-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E28-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E28-MJMAIN-2208\" x=\"917\" y=\"0\"><\/use><use xlink:href=\"#E28-MJMAIN-5B\" x=\"1862\" y=\"0\"><\/use><use xlink:href=\"#E28-MJMAIN-30\" x=\"2140\" y=\"0\"><\/use><use xlink:href=\"#E28-MJMAIN-2C\" x=\"2640\" y=\"0\"><\/use><use xlink:href=\"#E28-MJMAIN-31\" x=\"3085\" y=\"0\"><\/use><use xlink:href=\"#E28-MJMAIN-5D\" x=\"3585\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha \\in [0,1]<\/script><\/p><p><span>Note that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.161ex\" height=\"2.344ex\" viewBox=\"0 -856.4 500 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E29-MJMAIN-2205\" d=\"M331 696Q335 708 339 722T345 744T350 759T357 769T367 772Q374 772 381 767T388 754Q388 746 377 712L366 673L378 661Q460 575 460 344Q460 281 456 234T432 126T373 27Q319 -22 250 -22Q214 -22 180 -7Q168 -3 168 -4L159 -33Q148 -71 142 -75Q138 -78 132 -78Q124 -78 118 -72T111 -60Q111 -52 122 -18L133 21L125 29Q39 111 39 344Q39 596 137 675Q187 716 251 716Q265 716 278 714T296 710T315 703T331 696ZM276 676Q264 679 246 679Q196 679 159 631Q134 597 128 536T121 356Q121 234 127 174T151 80L234 366Q253 430 275 506T308 618L318 654Q318 656 294 669L276 676ZM181 42Q207 16 250 16Q291 16 324 47Q354 78 366 136T378 356Q378 470 372 528T349 616L348 613Q348 611 264 326L181 42Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E29-MJMAIN-2205\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\empty<\/script><span> is convex.<\/span><\/p><p><strong><span>Proposition 1:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is not empty<\/span><\/p><p><span>1) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is affine<\/span><\/p><p><span>2) Any affine combination of finite points in <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> belongs to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><\/p><p><span>3) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is the translation of some linear subspace <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.864ex\" height=\"2.227ex\" viewBox=\"0 -755.9 3385.8 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E35-MJMATHI-56\" d=\"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z\"><\/path><path stroke-width=\"0\" id=\"E35-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E35-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E35-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E35-MJMATHI-56\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E35-MJMAIN-2286\" x=\"1046\" y=\"0\"><\/use><g transform=\"translate(2102,0)\"><use xlink:href=\"#E35-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E35-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">V \\subseteq R ^n<\/script><span>; <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S <\/script><span> is of the form <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.281ex\" height=\"2.577ex\" viewBox=\"0 -806.1 13468.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E37-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMATHI-56\" d=\"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E37-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E37-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-78\" x=\"500\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-7D\" x=\"1072\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-2B\" x=\"1794\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-56\" x=\"2794\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-3D\" x=\"3841\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-7B\" x=\"4897\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-78\" x=\"5397\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-2B\" x=\"6191\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-76\" x=\"7191\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-2208\" x=\"7954\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-52\" x=\"8899\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-3A\" x=\"9935\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-76\" x=\"10491\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-2208\" x=\"11254\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMATHI-56\" x=\"12199\" y=\"0\"><\/use><use xlink:href=\"#E37-MJMAIN-7D\" x=\"12968\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{x\\}+V = \\{x+v\\in R: v \\in V\\}<\/script><span> for some <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.931ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2553.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E38-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E38-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E38-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E38-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E38-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E38-MJMATHI-52\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in R<\/script><\/p><p><strong><span>Proposition 2:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.576ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3261.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E118-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E118-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E118-MJMAIN-2286\" x=\"922\" y=\"0\"><\/use><g transform=\"translate(1978,0)\"><use xlink:href=\"#E118-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E118-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">S \\subseteq R^n<\/script><span> is arbitary<\/span><\/p><p><span>1) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is convex<\/span><\/p><p><span>2) Any convex combination of points in <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> belongs to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><\/p><p><strong><span>Examples of Convex Sets:<\/span><\/strong><span>\t<\/span><span>Non-negative orthant, hyperplane, halfspaces, euclidean ball, ellipsoid, simplex, convex cone, positive semidefinte cone<\/span><\/p><p><strong><span>Cone:<\/span><\/strong><span>\t<\/span><span>if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.723ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7630.7 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E43-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMAIN-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E43-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E43-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMATHI-3B1\" x=\"500\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMATHI-78\" x=\"1140\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMAIN-3A\" x=\"1989\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMATHI-3B1\" x=\"2545\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMAIN-3E\" x=\"3463\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMAIN-30\" x=\"4519\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMAIN-7D\" x=\"5019\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMAIN-2208\" x=\"5796\" y=\"0\"><\/use><use xlink:href=\"#E43-MJMATHI-4B\" x=\"6741\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{\\alpha x: \\alpha \\gt 0\\} \\in K<\/script><span> for every <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.233ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2683.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E44-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E44-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E44-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E44-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E44-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E44-MJMATHI-4B\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in K<\/script><\/p><p><span>Note that not every cone is convex and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.934ex\" height=\"2.811ex\" viewBox=\"0 -806.1 1263.1 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E45-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E45-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E45-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E45-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E45-MJMATHI-6E\" x=\"925\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E45-MJMAIN-2B\" x=\"866\" y=\"-351\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S^n_+<\/script><span> is a convex cone.<\/span><\/p><p><strong><span>Affine Hull:<\/span><\/strong><span>\t<\/span><span>The intersection of all affine subspaces containing <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span>, denoted by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.089ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3052 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E52-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E52-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-66\" x=\"529\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-66\" x=\"1079\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-28\" x=\"1629\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-53\" x=\"2018\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-29\" x=\"2663\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">aff(S)<\/script><\/p><p><strong><span>Convex Hull:<\/span><\/strong><span>\t<\/span><span>The intersection of all convex sets containing <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span>, denoted by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.957ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3426 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E54-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E54-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-6F\" x=\"433\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-6E\" x=\"918\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-76\" x=\"1518\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"2003\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-53\" x=\"2392\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"3037\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">conv(S)<\/script><span>. <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.553ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5404.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E50-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E50-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMATHI-6F\" x=\"433\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMATHI-6E\" x=\"918\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMATHI-76\" x=\"1518\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-28\" x=\"2003\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMATHI-53\" x=\"2392\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-29\" x=\"3037\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-3D\" x=\"3703\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMATHI-53\" x=\"4759\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">conv(S) = S<\/script><span> if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is convex<\/span><\/p><p><strong><span>Proposition 3:<\/span><\/strong><\/p><p><span>1) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.089ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3052 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E52-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E52-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-66\" x=\"529\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-66\" x=\"1079\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-28\" x=\"1629\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-53\" x=\"2018\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-29\" x=\"2663\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">aff(S)<\/script><span> is the set of all affine combinations of points in <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><\/p><p><span>2) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.957ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3426 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E54-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E54-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-6F\" x=\"433\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-6E\" x=\"918\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-76\" x=\"1518\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"2003\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-53\" x=\"2392\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"3037\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">conv(S)<\/script><span> is the set of all convex combinations of points in <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><\/p><h3><a name=\"convexity-preserving-operations\" class=\"md-header-anchor\"><\/a><span>Convexity-Preserving Operations<\/span><\/h3><p><strong><span>Affine Functions:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"46.511ex\" height=\"2.577ex\" viewBox=\"0 -806.1 20025.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E56-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-28\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMATHI-3B1\" x=\"1139\" y=\"0\"><\/use><g transform=\"translate(1779,0)\"><use xlink:href=\"#E56-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E56-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E56-MJMAIN-2B\" x=\"3026\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-28\" x=\"4026\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-31\" x=\"4415\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-2212\" x=\"5138\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMATHI-3B1\" x=\"6138\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-29\" x=\"6778\" y=\"0\"><\/use><g transform=\"translate(7167,0)\"><use xlink:href=\"#E56-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E56-MJMAIN-32\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E56-MJMAIN-29\" x=\"8192\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-3D\" x=\"8859\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMATHI-3B1\" x=\"9915\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMATHI-41\" x=\"10555\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-28\" x=\"11305\" y=\"0\"><\/use><g transform=\"translate(11694,0)\"><use xlink:href=\"#E56-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E56-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E56-MJMAIN-29\" x=\"12720\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-2B\" x=\"13331\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-28\" x=\"14331\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-31\" x=\"14720\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-2212\" x=\"15442\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMATHI-3B1\" x=\"16442\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-29\" x=\"17082\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMATHI-41\" x=\"17471\" y=\"0\"><\/use><use xlink:href=\"#E56-MJMAIN-28\" x=\"18221\" y=\"0\"><\/use><g transform=\"translate(18610,0)\"><use xlink:href=\"#E56-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E56-MJMAIN-32\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E56-MJMAIN-29\" x=\"19636\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A(\\alpha x_1 + (1-\\alpha)x_2) = \\alpha A(x_1) + (1-\\alpha)A(x_2)<\/script><\/p><p><span>Note that translation (<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.297ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5725 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E57-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E57-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-28\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMATHI-78\" x=\"1139\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-29\" x=\"1711\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-3D\" x=\"2377\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMATHI-78\" x=\"3433\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-2B\" x=\"4227\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMATHI-79\" x=\"5228\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A(x) = x + y<\/script><span>), projection (<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.085ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4772.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E58-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-55\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E58-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-28\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMATHI-78\" x=\"1139\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-29\" x=\"1711\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-3D\" x=\"2377\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMATHI-55\" x=\"3433\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMATHI-78\" x=\"4200\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A(x) = Ux<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.385ex\" height=\"2.344ex\" viewBox=\"0 -906.7 7485.2 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E59-MJMATHI-55\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E59-MJMATHI-55\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E59-MJMATHI-54\" x=\"1120\" y=\"513\"><\/use><use xlink:href=\"#E59-MJMATHI-55\" x=\"1390\" y=\"0\"><\/use><use xlink:href=\"#E59-MJMAIN-3D\" x=\"2434\" y=\"0\"><\/use><use xlink:href=\"#E59-MJMATHI-55\" x=\"3490\" y=\"0\"><\/use><g transform=\"translate(4257,0)\"><use xlink:href=\"#E59-MJMATHI-55\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E59-MJMATHI-54\" x=\"1120\" y=\"513\"><\/use><\/g><use xlink:href=\"#E59-MJMAIN-3D\" x=\"5925\" y=\"0\"><\/use><use xlink:href=\"#E59-MJMATHI-49\" x=\"6981\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">U^TU = UU^T = I<\/script><span>) and rotation (<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.048ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4756.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E60-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E60-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-28\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMATHI-78\" x=\"1139\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-29\" x=\"1711\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-3D\" x=\"2377\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMATHI-50\" x=\"3433\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMATHI-78\" x=\"4184\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A(x) = Px<\/script><span>) are all affine operations<\/span><\/p><p><strong><span>Proposition 4:<\/span><\/strong><span>\t<\/span><span>Affine functions operated on a convex set remains its convexity<\/span><\/p><h3><a name=\"projection-onto-closed-convex-sets\" class=\"md-header-anchor\"><\/a><span>Projection onto Closed Convex Sets<\/span><\/h3><p><span>Note that Projection points do not always exist and are not always unique.<\/span><\/p><p><span>Note that every finite point set is closed since it has no limit points thus fulfill the conditon that every limit points belong to itself.<\/span><\/p><p><strong><span>Theorem 4:<\/span><\/strong><span>\t<\/span><span>If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is non-empty, closed and convex, then for every <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E62-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E62-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E62-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E62-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E62-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in R^n<\/script><span>, there exists a unique point <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.48ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2790.1 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E71-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E71-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E71-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E71-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E71-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E71-MJMAIN-2217\" x=\"663\" y=\"513\"><\/use><use xlink:href=\"#E71-MJMAIN-2208\" x=\"1200\" y=\"0\"><\/use><use xlink:href=\"#E71-MJMATHI-53\" x=\"2145\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">z^* \\in S<\/script><span> that s closest to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E168-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E168-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><\/p><p><strong><span>Projection:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.711ex\" height=\"3.161ex\" viewBox=\"0 -1007.2 13653.3 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E65-MJSZ1-220F\" d=\"M158 656Q147 684 131 694Q110 707 69 710H55V750H888V710H874Q840 708 820 698T795 678T786 656V-155Q798 -206 874 -210H888V-250H570V-210H584Q618 -208 638 -197T663 -178T673 -155V710H270V277L271 -155Q283 -206 359 -210H373V-250H55V-210H69Q103 -208 123 -197T148 -178T158 -155V656Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-72\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E65-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E65-MJSZ1-220F\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E65-MJMATHI-53\" x=\"1335\" y=\"-404\"><\/use><use xlink:href=\"#E65-MJMAIN-28\" x=\"1500\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-78\" x=\"1889\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMAIN-29\" x=\"2461\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMAIN-3D\" x=\"3127\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-61\" x=\"4183\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-72\" x=\"4712\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-67\" x=\"5163\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-6D\" x=\"6643\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-69\" x=\"7521\" y=\"0\"><\/use><g transform=\"translate(7866,0)\"><use xlink:href=\"#E65-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(600,-155)\"><use transform=\"scale(0.707)\" xlink:href=\"#E65-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E65-MJMAIN-2208\" x=\"467\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E65-MJMATHI-53\" x=\"1135\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E65-MJMAIN-7C\" x=\"9825\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMAIN-7C\" x=\"10103\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-78\" x=\"10381\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMAIN-2212\" x=\"11175\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMATHI-7A\" x=\"12175\" y=\"0\"><\/use><use xlink:href=\"#E65-MJMAIN-7C\" x=\"12643\" y=\"0\"><\/use><g transform=\"translate(12921,0)\"><use xlink:href=\"#E65-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E65-MJMAIN-32\" x=\"393\" y=\"674\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E65-MJMAIN-32\" x=\"393\" y=\"-403\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\prod _S (x) = arg \\quad min _{z\\in S} ||x - z||^2 _2<\/script><\/p><p><strong><span>Weierstrass's Theorem:<\/span><\/strong><span>\t<\/span><span>If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is continuous on a compact set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.635ex\" height=\"1.877ex\" viewBox=\"0 -755.9 704 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E67-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E67-MJMATHI-54\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">T\t<\/script><span>, then it attains its maximum and minimum on <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.635ex\" height=\"1.877ex\" viewBox=\"0 -755.9 704 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E68-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E68-MJMATHI-54\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">T<\/script><\/p><p><strong><span>Theorem 5:<\/span><\/strong><span>\t<\/span><span>If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is non-empty, closed and convex, we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.859ex\" height=\"2.694ex\" viewBox=\"0 -806.1 5106.1 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E70-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJSZ1-220F\" d=\"M158 656Q147 684 131 694Q110 707 69 710H55V750H888V710H874Q840 708 820 698T795 678T786 656V-155Q798 -206 874 -210H888V-250H570V-210H584Q618 -208 638 -197T663 -178T673 -155V710H270V277L271 -155Q283 -206 359 -210H373V-250H55V-210H69Q103 -208 123 -197T148 -178T158 -155V656Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E70-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E70-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E70-MJMAIN-2217\" x=\"663\" y=\"513\"><\/use><use xlink:href=\"#E70-MJMAIN-3D\" x=\"1200\" y=\"0\"><\/use><g transform=\"translate(2256,0)\"><use xlink:href=\"#E70-MJSZ1-220F\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E70-MJMATHI-53\" x=\"1335\" y=\"-404\"><\/use><\/g><use xlink:href=\"#E70-MJMAIN-28\" x=\"3756\" y=\"0\"><\/use><use xlink:href=\"#E70-MJMATHI-78\" x=\"4145\" y=\"0\"><\/use><use xlink:href=\"#E70-MJMAIN-29\" x=\"4717\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">z^* = \\prod _S (x)<\/script><span> iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.48ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2790.1 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E71-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E71-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E71-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E71-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E71-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E71-MJMAIN-2217\" x=\"663\" y=\"513\"><\/use><use xlink:href=\"#E71-MJMAIN-2208\" x=\"1200\" y=\"0\"><\/use><use xlink:href=\"#E71-MJMATHI-53\" x=\"2145\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">z^* \\in S<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"21.64ex\" height=\"2.811ex\" viewBox=\"0 -906.7 9317.3 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E72-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E72-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E72-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E72-MJMATHI-7A\" x=\"389\" y=\"0\"><\/use><use xlink:href=\"#E72-MJMAIN-2212\" x=\"1079\" y=\"0\"><\/use><g transform=\"translate(2079,0)\"><use xlink:href=\"#E72-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E72-MJMAIN-2217\" x=\"663\" y=\"513\"><\/use><\/g><g transform=\"translate(3001,0)\"><use xlink:href=\"#E72-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E72-MJMATHI-54\" x=\"550\" y=\"513\"><\/use><\/g><use xlink:href=\"#E72-MJMAIN-28\" x=\"3988\" y=\"0\"><\/use><use xlink:href=\"#E72-MJMATHI-78\" x=\"4377\" y=\"0\"><\/use><use xlink:href=\"#E72-MJMAIN-2212\" x=\"5171\" y=\"0\"><\/use><g transform=\"translate(6172,0)\"><use xlink:href=\"#E72-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E72-MJMAIN-2217\" x=\"663\" y=\"513\"><\/use><\/g><use xlink:href=\"#E72-MJMAIN-29\" x=\"7094\" y=\"0\"><\/use><use xlink:href=\"#E72-MJMAIN-2264\" x=\"7761\" y=\"0\"><\/use><use xlink:href=\"#E72-MJMAIN-30\" x=\"8817\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(z-z^*)^T(x-z^*)\\le0<\/script><span> for all <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.425ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2335.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E73-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E73-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E73-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E73-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E73-MJMAIN-2208\" x=\"745\" y=\"0\"><\/use><use xlink:href=\"#E73-MJMATHI-53\" x=\"1690\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">z \\in S<\/script><\/p><h3><a name=\"separation-theorems\" class=\"md-header-anchor\"><\/a><span>Separation Theorems<\/span><\/h3><p><strong><span>Theorem 6 (Point-Set Separation):<\/span><\/strong><span>\t<\/span><span>If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> is non-empty, closed and convex, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.808ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4222.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E75-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E75-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E75-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E75-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E75-MJMAIN-2216\" d=\"M56 731Q56 740 62 745T75 750Q85 750 92 740Q96 733 270 255T444 -231Q444 -239 438 -244T424 -250Q414 -250 407 -240Q404 -236 230 242T56 731Z\"><\/path><path stroke-width=\"0\" id=\"E75-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E75-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E75-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E75-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E75-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E75-MJMAIN-2216\" x=\"3077\" y=\"0\"><\/use><use xlink:href=\"#E75-MJMATHI-53\" x=\"3577\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in R^n \\backslash S<\/script><span> is arbitary. Then there exists a <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.974ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3002.8 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E76-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E76-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E76-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E76-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E76-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E76-MJMAIN-2208\" x=\"774\" y=\"0\"><\/use><g transform=\"translate(1719,0)\"><use xlink:href=\"#E76-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E76-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y \\in R^n<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.36ex\" height=\"2.811ex\" viewBox=\"0 -906.7 7905.1 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E77-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E77-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E77-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E77-MJMATHI-61\" x=\"878\" y=\"0\"><\/use><g transform=\"translate(1407,0)\"><use xlink:href=\"#E77-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(572,-155)\"><use transform=\"scale(0.707)\" xlink:href=\"#E77-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E77-MJMAIN-2208\" x=\"467\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E77-MJMATHI-53\" x=\"1135\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(3337,0)\"><use xlink:href=\"#E77-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E77-MJMATHI-54\" x=\"705\" y=\"513\"><\/use><\/g><use xlink:href=\"#E77-MJMATHI-7A\" x=\"4434\" y=\"0\"><\/use><use xlink:href=\"#E77-MJMAIN-3C\" x=\"5180\" y=\"0\"><\/use><g transform=\"translate(6236,0)\"><use xlink:href=\"#E77-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E77-MJMATHI-54\" x=\"705\" y=\"513\"><\/use><\/g><use xlink:href=\"#E77-MJMATHI-78\" x=\"7333\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">max _{z\\in S} y^Tz \\lt y^T x<\/script><\/p><p><strong><span>Theorem 7:<\/span><\/strong><span>\t<\/span><span>A closed convex set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.576ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3261.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E118-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E118-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E118-MJMAIN-2286\" x=\"922\" y=\"0\"><\/use><g transform=\"translate(1978,0)\"><use xlink:href=\"#E118-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E118-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">S \\subseteq R^n<\/script><span>is the intersection of all the halfspaces containing <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><\/p><p><span>Note that set-set separation <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"26.893ex\" height=\"2.928ex\" viewBox=\"0 -906.7 11578.9 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E80-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E80-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E80-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E80-MJMATHI-61\" x=\"878\" y=\"0\"><\/use><g transform=\"translate(1407,0)\"><use xlink:href=\"#E80-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(572,-155)\"><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMAIN-2208\" x=\"467\" y=\"0\"><\/use><g transform=\"translate(802,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E80-MJMAIN-31\" x=\"866\" y=\"-213\"><\/use><\/g><\/g><\/g><g transform=\"translate(3635,0)\"><use xlink:href=\"#E80-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMATHI-54\" x=\"705\" y=\"513\"><\/use><\/g><use xlink:href=\"#E80-MJMATHI-7A\" x=\"4732\" y=\"0\"><\/use><use xlink:href=\"#E80-MJMAIN-3C\" x=\"5478\" y=\"0\"><\/use><use xlink:href=\"#E80-MJMATHI-6D\" x=\"6534\" y=\"0\"><\/use><use xlink:href=\"#E80-MJMATHI-69\" x=\"7412\" y=\"0\"><\/use><g transform=\"translate(7757,0)\"><use xlink:href=\"#E80-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(600,-155)\"><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMAIN-2208\" x=\"467\" y=\"0\"><\/use><g transform=\"translate(802,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E80-MJMAIN-32\" x=\"866\" y=\"-213\"><\/use><\/g><\/g><\/g><g transform=\"translate(10013,0)\"><use xlink:href=\"#E80-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E80-MJMATHI-54\" x=\"705\" y=\"513\"><\/use><\/g><use xlink:href=\"#E80-MJMATHI-7A\" x=\"11110\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">max _{z \\in S_1} y^T z \\lt min _{z \\in S_2} y^Tz<\/script><span> does not always holds, example can be <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"19.862ex\" height=\"3.395ex\" viewBox=\"0 -956.9 8551.6 1461.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E81-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E81-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E81-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E81-MJMAIN-28\" x=\"500\" y=\"0\"><\/use><g transform=\"translate(889,0)\"><use xlink:href=\"#E81-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E81-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E81-MJMAIN-2C\" x=\"1914\" y=\"0\"><\/use><g transform=\"translate(2359,0)\"><use xlink:href=\"#E81-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E81-MJMAIN-32\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E81-MJMAIN-29\" x=\"3384\" y=\"0\"><\/use><use xlink:href=\"#E81-MJMAIN-3A\" x=\"4051\" y=\"0\"><\/use><g transform=\"translate(4607,0)\"><use xlink:href=\"#E81-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E81-MJMAIN-32\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E81-MJMAIN-2265\" x=\"5910\" y=\"0\"><\/use><g transform=\"translate(6688,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"845\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E81-MJMAIN-31\" x=\"347\" y=\"571\"><\/use><g transform=\"translate(60,-345)\"><use transform=\"scale(0.707)\" xlink:href=\"#E81-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E81-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E81-MJMAIN-7D\" x=\"8051\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{(x_1,x_2): x_2 \\ge \\frac{1}{x_1}\\}<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.273ex\" height=\"2.461ex\" viewBox=\"0 -755.9 1409.1 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E82-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E82-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E82-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E82-MJMAIN-2212\" x=\"1073\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">R_-<\/script><\/p><p><strong><span>Theorem 8 (Set-Set Separation):<\/span><\/strong><span>\t<\/span><span>If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.477ex\" height=\"2.344ex\" viewBox=\"0 -806.1 1066.6 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E83-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E83-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E83-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E83-MJMAIN-31\" x=\"866\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S_1<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.555ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3683.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E84-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E84-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E84-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E84-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E84-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E84-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E84-MJMAIN-32\" x=\"866\" y=\"-213\"><\/use><use xlink:href=\"#E84-MJMAIN-2286\" x=\"1344\" y=\"0\"><\/use><g transform=\"translate(2400,0)\"><use xlink:href=\"#E84-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E84-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">S_2 \\subseteq R^n<\/script><span> is non-empty. closed and convex with <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.794ex\" height=\"2.461ex\" viewBox=\"0 -856.4 5078.1 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E85-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E85-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E85-MJMAIN-2227\" d=\"M318 591Q325 598 333 598Q344 598 348 591Q349 590 414 445T545 151T611 -4Q609 -22 591 -22Q588 -22 586 -21T581 -20T577 -17T575 -13T572 -9T570 -4L333 528L96 -4Q87 -20 80 -21Q78 -22 75 -22Q57 -22 55 -4Q55 2 120 150T251 444T318 591Z\"><\/path><path stroke-width=\"0\" id=\"E85-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E85-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E85-MJMAIN-2205\" d=\"M331 696Q335 708 339 722T345 744T350 759T357 769T367 772Q374 772 381 767T388 754Q388 746 377 712L366 673L378 661Q460 575 460 344Q460 281 456 234T432 126T373 27Q319 -22 250 -22Q214 -22 180 -7Q168 -3 168 -4L159 -33Q148 -71 142 -75Q138 -78 132 -78Q124 -78 118 -72T111 -60Q111 -52 122 -18L133 21L125 29Q39 111 39 344Q39 596 137 675Q187 716 251 716Q265 716 278 714T296 710T315 703T331 696ZM276 676Q264 679 246 679Q196 679 159 631Q134 597 128 536T121 356Q121 234 127 174T151 80L234 366Q253 430 275 506T308 618L318 654Q318 656 294 669L276 676ZM181 42Q207 16 250 16Q291 16 324 47Q354 78 366 136T378 356Q378 470 372 528T349 616L348 613Q348 611 264 326L181 42Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E85-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E85-MJMAIN-31\" x=\"866\" y=\"-213\"><\/use><use xlink:href=\"#E85-MJMAIN-2227\" x=\"1288\" y=\"0\"><\/use><g transform=\"translate(2177,0)\"><use xlink:href=\"#E85-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E85-MJMAIN-32\" x=\"866\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E85-MJMAIN-3D\" x=\"3522\" y=\"0\"><\/use><use xlink:href=\"#E85-MJMAIN-2205\" x=\"4578\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S_1 \\and S_2 = \\empty<\/script><span>. <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.477ex\" height=\"2.344ex\" viewBox=\"0 -806.1 1066.6 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E86-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E86-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E86-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E86-MJMAIN-32\" x=\"866\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S_2<\/script><span> is bounded. Then there exists a <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.974ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3002.8 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E87-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E87-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E87-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E87-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E87-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E87-MJMAIN-2208\" x=\"774\" y=\"0\"><\/use><g transform=\"translate(1719,0)\"><use xlink:href=\"#E87-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E87-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y\\in R^n<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"27.305ex\" height=\"2.928ex\" viewBox=\"0 -906.7 11756.5 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E88-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-7A\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E88-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E88-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E88-MJMATHI-61\" x=\"878\" y=\"0\"><\/use><g transform=\"translate(1407,0)\"><use xlink:href=\"#E88-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(572,-155)\"><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMATHI-7A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMAIN-2208\" x=\"467\" y=\"0\"><\/use><g transform=\"translate(802,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E88-MJMAIN-31\" x=\"866\" y=\"-213\"><\/use><\/g><\/g><\/g><g transform=\"translate(3635,0)\"><use xlink:href=\"#E88-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMATHI-54\" x=\"705\" y=\"513\"><\/use><\/g><use xlink:href=\"#E88-MJMATHI-7A\" x=\"4732\" y=\"0\"><\/use><use xlink:href=\"#E88-MJMAIN-3C\" x=\"5478\" y=\"0\"><\/use><use xlink:href=\"#E88-MJMATHI-6D\" x=\"6534\" y=\"0\"><\/use><use xlink:href=\"#E88-MJMATHI-69\" x=\"7412\" y=\"0\"><\/use><g transform=\"translate(7757,0)\"><use xlink:href=\"#E88-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(600,-155)\"><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(876,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E88-MJMAIN-32\" x=\"866\" y=\"-213\"><\/use><\/g><\/g><\/g><g transform=\"translate(10087,0)\"><use xlink:href=\"#E88-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E88-MJMATHI-54\" x=\"705\" y=\"513\"><\/use><\/g><use xlink:href=\"#E88-MJMATHI-75\" x=\"11184\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">max _{z\\in S_1}y^Tz \\lt min _{u \\in S_2} y^T u<\/script><\/p><h3><a name=\"basic-definitions-and-propeerties-of-convex-functions\" class=\"md-header-anchor\"><\/a><span>Basic Definitions and Propeerties of Convex Functions<\/span><\/h3><p><strong><span>Convex Functions:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"45.118ex\" height=\"2.577ex\" viewBox=\"0 -806.1 19425.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E89-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E89-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E89-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMATHI-3B1\" x=\"939\" y=\"0\"><\/use><g transform=\"translate(1579,0)\"><use xlink:href=\"#E89-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E89-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E89-MJMAIN-2B\" x=\"2826\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-28\" x=\"3826\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-31\" x=\"4215\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-2212\" x=\"4938\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMATHI-3B1\" x=\"5938\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-29\" x=\"6578\" y=\"0\"><\/use><g transform=\"translate(6967,0)\"><use xlink:href=\"#E89-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E89-MJMAIN-32\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E89-MJMAIN-29\" x=\"7992\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-2264\" x=\"8659\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMATHI-3B1\" x=\"9715\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMATHI-66\" x=\"10355\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-28\" x=\"10905\" y=\"0\"><\/use><g transform=\"translate(11294,0)\"><use xlink:href=\"#E89-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E89-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E89-MJMAIN-29\" x=\"12320\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-2B\" x=\"12931\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-28\" x=\"13931\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-31\" x=\"14320\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-2212\" x=\"15042\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMATHI-3B1\" x=\"16042\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-29\" x=\"16682\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMATHI-66\" x=\"17071\" y=\"0\"><\/use><use xlink:href=\"#E89-MJMAIN-28\" x=\"17621\" y=\"0\"><\/use><g transform=\"translate(18010,0)\"><use xlink:href=\"#E89-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E89-MJMAIN-32\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E89-MJMAIN-29\" x=\"19036\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(\\alpha x_1 + (1-\\alpha)x_2) \\le \\alpha f(x_1) + (1-\\alpha)f(x_2)<\/script><\/p><p><strong><span>Concave Functions:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.084ex\" height=\"2.577ex\" viewBox=\"0 -806.1 1328 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E90-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E90-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E90-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E90-MJMATHI-66\" x=\"778\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-f<\/script><span> is convex<\/span><\/p><p><strong><span>Epigraph:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"37.27ex\" height=\"2.577ex\" viewBox=\"0 -806.1 16046.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E91-MJMATHI-65\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E91-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E91-MJMATHI-65\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-70\" x=\"466\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-69\" x=\"969\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-28\" x=\"1314\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-66\" x=\"1703\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-29\" x=\"2253\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-3D\" x=\"2919\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-7B\" x=\"3975\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-28\" x=\"4475\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-78\" x=\"4864\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-2C\" x=\"5436\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-74\" x=\"5881\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-29\" x=\"6242\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-2208\" x=\"6909\" y=\"0\"><\/use><g transform=\"translate(7853,0)\"><use xlink:href=\"#E91-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E91-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E91-MJMAIN-D7\" x=\"9359\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-52\" x=\"10359\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-3A\" x=\"11396\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-66\" x=\"11952\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-28\" x=\"12502\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-78\" x=\"12891\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-29\" x=\"13463\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-2264\" x=\"14129\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMATHI-74\" x=\"15185\" y=\"0\"><\/use><use xlink:href=\"#E91-MJMAIN-7D\" x=\"15546\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">epi(f) = \\{(x,t) \\in R^n\\times R: f(x) \\le t\\}<\/script><\/p><p><strong><span>Effective Domain:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"33.609ex\" height=\"2.577ex\" viewBox=\"0 -806.1 14470.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E92-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path stroke-width=\"0\" id=\"E92-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E92-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMATHI-6F\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMATHI-6D\" x=\"1008\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-28\" x=\"1886\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMATHI-66\" x=\"2275\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-29\" x=\"2825\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-3D\" x=\"3491\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-7B\" x=\"4547\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMATHI-78\" x=\"5047\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-2208\" x=\"5897\" y=\"0\"><\/use><g transform=\"translate(6842,0)\"><use xlink:href=\"#E92-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E92-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E92-MJMAIN-3A\" x=\"8403\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMATHI-66\" x=\"8958\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-28\" x=\"9508\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMATHI-78\" x=\"9897\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-29\" x=\"10469\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-3C\" x=\"11136\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-2B\" x=\"12192\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-221E\" x=\"12970\" y=\"0\"><\/use><use xlink:href=\"#E92-MJMAIN-7D\" x=\"13970\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dom(f) = \\{x \\in R^n:f(x) \\lt +\\infty \\}<\/script><\/p><p><strong><span>Proposition 9:<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.344ex\" height=\"2.577ex\" viewBox=\"0 -806.1 8759 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E97-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJSZ1-22C3\" d=\"M96 750Q103 750 109 748T120 744T127 737T133 730T137 723T139 718V395L140 73L142 60Q159 -43 237 -104T416 -166Q521 -166 597 -103T690 60L692 73L694 718Q708 749 735 749Q765 749 775 720Q777 714 777 398Q777 78 776 71Q766 -51 680 -140Q571 -249 416 -249H411Q261 -249 152 -140Q66 -51 56 71Q55 78 55 398Q55 714 57 720Q60 734 70 740Q80 750 96 750Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E97-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-3A\" x=\"827\" y=\"0\"><\/use><g transform=\"translate(1383,0)\"><use xlink:href=\"#E97-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E97-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E97-MJMAIN-2192\" x=\"2944\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMATHI-52\" x=\"4222\" y=\"0\"><\/use><use xlink:href=\"#E97-MJSZ1-22C3\" x=\"5148\" y=\"-1\"><\/use><use xlink:href=\"#E97-MJMAIN-7B\" x=\"5981\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-2B\" x=\"6481\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-221E\" x=\"7259\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-7D\" x=\"8259\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f:R^n \\rightarrow R \\bigcup \\{+\\infty\\}<\/script><span>, it is convex iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.136ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2642 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E94-MJMATHI-65\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path stroke-width=\"0\" id=\"E94-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E94-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E94-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E94-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E94-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E94-MJMATHI-65\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E94-MJMATHI-70\" x=\"466\" y=\"0\"><\/use><use xlink:href=\"#E94-MJMATHI-69\" x=\"969\" y=\"0\"><\/use><use xlink:href=\"#E94-MJMAIN-28\" x=\"1314\" y=\"0\"><\/use><use xlink:href=\"#E94-MJMATHI-66\" x=\"1703\" y=\"0\"><\/use><use xlink:href=\"#E94-MJMAIN-29\" x=\"2253\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">epi(f)<\/script><span> is convex<\/span><\/p><p><span>Note that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.465ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3214 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E95-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E95-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E95-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E95-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E95-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E95-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E95-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E95-MJMATHI-6F\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E95-MJMATHI-6D\" x=\"1008\" y=\"0\"><\/use><use xlink:href=\"#E95-MJMAIN-28\" x=\"1886\" y=\"0\"><\/use><use xlink:href=\"#E95-MJMATHI-66\" x=\"2275\" y=\"0\"><\/use><use xlink:href=\"#E95-MJMAIN-29\" x=\"2825\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dom(f)<\/script><span> is also convex if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is convex<\/span><\/p><p><strong><span>Corollary 2: (Jensen's Inequality)<\/span><\/strong><span>\t\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.344ex\" height=\"2.577ex\" viewBox=\"0 -806.1 8759 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E97-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJSZ1-22C3\" d=\"M96 750Q103 750 109 748T120 744T127 737T133 730T137 723T139 718V395L140 73L142 60Q159 -43 237 -104T416 -166Q521 -166 597 -103T690 60L692 73L694 718Q708 749 735 749Q765 749 775 720Q777 714 777 398Q777 78 776 71Q766 -51 680 -140Q571 -249 416 -249H411Q261 -249 152 -140Q66 -51 56 71Q55 78 55 398Q55 714 57 720Q60 734 70 740Q80 750 96 750Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path stroke-width=\"0\" id=\"E97-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E97-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-3A\" x=\"827\" y=\"0\"><\/use><g transform=\"translate(1383,0)\"><use xlink:href=\"#E97-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E97-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E97-MJMAIN-2192\" x=\"2944\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMATHI-52\" x=\"4222\" y=\"0\"><\/use><use xlink:href=\"#E97-MJSZ1-22C3\" x=\"5148\" y=\"-1\"><\/use><use xlink:href=\"#E97-MJMAIN-7B\" x=\"5981\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-2B\" x=\"6481\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-221E\" x=\"7259\" y=\"0\"><\/use><use xlink:href=\"#E97-MJMAIN-7D\" x=\"8259\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f:R^n \\rightarrow R \\bigcup \\{+\\infty\\}<\/script><span>, it is convex iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"29.566ex\" height=\"3.278ex\" viewBox=\"0 -1057.4 12730 1411.3\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E98-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E98-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E98-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E98-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><g transform=\"translate(939,0)\"><use xlink:href=\"#E98-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-6B\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(3409,0)\"><use xlink:href=\"#E98-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-69\" x=\"905\" y=\"-213\"><\/use><\/g><g transform=\"translate(4393,0)\"><use xlink:href=\"#E98-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E98-MJMAIN-29\" x=\"5309\" y=\"0\"><\/use><use xlink:href=\"#E98-MJMAIN-2264\" x=\"5975\" y=\"0\"><\/use><g transform=\"translate(7031,0)\"><use xlink:href=\"#E98-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-6B\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(9502,0)\"><use xlink:href=\"#E98-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-69\" x=\"905\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E98-MJMATHI-66\" x=\"10486\" y=\"0\"><\/use><use xlink:href=\"#E98-MJMAIN-28\" x=\"11036\" y=\"0\"><\/use><g transform=\"translate(11425,0)\"><use xlink:href=\"#E98-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E98-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E98-MJMAIN-29\" x=\"12340\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(\\sum ^k _{i=1} \\alpha _i x_i) \\le \\sum ^k _{i=1} \\alpha_i f(x_i)<\/script><span> for any <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.782ex\" height=\"2.344ex\" viewBox=\"0 -755.9 6795.1 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E99-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E99-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E99-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E99-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><use xlink:href=\"#E99-MJMAIN-2C\" x=\"1025\" y=\"0\"><\/use><use xlink:href=\"#E99-MJMAIN-2E\" x=\"1470\" y=\"0\"><\/use><use xlink:href=\"#E99-MJMAIN-2E\" x=\"1914\" y=\"0\"><\/use><use xlink:href=\"#E99-MJMAIN-2E\" x=\"2359\" y=\"0\"><\/use><use xlink:href=\"#E99-MJMAIN-2C\" x=\"2804\" y=\"0\"><\/use><g transform=\"translate(3248,0)\"><use xlink:href=\"#E99-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E99-MJMATHI-6B\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E99-MJMAIN-2208\" x=\"4567\" y=\"0\"><\/use><g transform=\"translate(5511,0)\"><use xlink:href=\"#E99-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E99-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x_1,...,x_k \\in R^n<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.764ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7648.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E100-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E100-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E100-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E100-MJMAIN-31\" x=\"905\" y=\"-213\"><\/use><use xlink:href=\"#E100-MJMAIN-2C\" x=\"1093\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-2E\" x=\"1538\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-2E\" x=\"1982\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-2E\" x=\"2427\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-2C\" x=\"2872\" y=\"0\"><\/use><g transform=\"translate(3316,0)\"><use xlink:href=\"#E100-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E100-MJMATHI-6B\" x=\"905\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E100-MJMAIN-2208\" x=\"4703\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-5B\" x=\"5647\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-30\" x=\"5925\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-2C\" x=\"6425\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-31\" x=\"6870\" y=\"0\"><\/use><use xlink:href=\"#E100-MJMAIN-5D\" x=\"7370\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha_1,...,\\alpha_k \\in [0,1]<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.281ex\" height=\"3.278ex\" viewBox=\"0 -1057.4 5287.8 1411.3\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E101-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E101-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E101-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E101-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E101-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E101-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E101-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E101-MJMATHI-6B\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E101-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E101-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E101-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><g transform=\"translate(2470,0)\"><use xlink:href=\"#E101-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E101-MJMATHI-69\" x=\"905\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E101-MJMAIN-3D\" x=\"3732\" y=\"0\"><\/use><use xlink:href=\"#E101-MJMAIN-31\" x=\"4787\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\sum ^k _{i=1} \\alpha _i = 1<\/script><\/p><h3><a name=\"convexity-preserving-transformations\" class=\"md-header-anchor\"><\/a><span>Convexity-Preserving Transformations<\/span><\/h3><p><strong><span>Theorem 11:<\/span><\/strong><span>\t<\/span><\/p><p><strong><span>Non-Negative Combinations:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.605ex\" height=\"2.811ex\" viewBox=\"0 -856.4 8871.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E102-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E102-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E102-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E102-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E102-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E102-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E102-MJMAIN-3D\" x=\"2177\" y=\"0\"><\/use><g transform=\"translate(3233,0)\"><use xlink:href=\"#E102-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E102-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E102-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E102-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E102-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(5703,0)\"><use xlink:href=\"#E102-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E102-MJMATHI-69\" x=\"905\" y=\"-213\"><\/use><\/g><g transform=\"translate(6687,0)\"><use xlink:href=\"#E102-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E102-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E102-MJMAIN-28\" x=\"7521\" y=\"0\"><\/use><use xlink:href=\"#E102-MJMATHI-78\" x=\"7910\" y=\"0\"><\/use><use xlink:href=\"#E102-MJMAIN-29\" x=\"8482\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) = \\sum ^m _{i=1} \\alpha _i f_i(x)<\/script><span> is onvex if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.937ex\" height=\"2.577ex\" viewBox=\"0 -806.1 834 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E103-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E103-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E103-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E103-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f_i<\/script><span> is convex and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.544ex\" height=\"2.344ex\" viewBox=\"0 -755.9 2817.5 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E104-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E104-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E104-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E104-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E104-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E104-MJMATHI-69\" x=\"905\" y=\"-213\"><\/use><use xlink:href=\"#E104-MJMAIN-2265\" x=\"1261\" y=\"0\"><\/use><use xlink:href=\"#E104-MJMAIN-30\" x=\"2317\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha_i \\ge 0<\/script><\/p><p><strong><span>Pointwise Supremum:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.891ex\" height=\"2.577ex\" viewBox=\"0 -806.1 8133.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E105-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E105-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E105-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMAIN-3D\" x=\"2177\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMATHI-73\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMATHI-75\" x=\"3702\" y=\"0\"><\/use><g transform=\"translate(4274,0)\"><use xlink:href=\"#E105-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(503,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E105-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E105-MJMAIN-2208\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E105-MJMATHI-49\" x=\"1012\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(5949,0)\"><use xlink:href=\"#E105-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E105-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E105-MJMAIN-28\" x=\"6783\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMATHI-78\" x=\"7172\" y=\"0\"><\/use><use xlink:href=\"#E105-MJMAIN-29\" x=\"7744\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) = sup _{i\\in I}f_i(x)<\/script><\/p><p><strong><span>Affine Composition:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.309ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6591.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E106-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E106-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E106-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E106-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E106-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E106-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E106-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E106-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-3D\" x=\"2177\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMATHI-67\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-28\" x=\"3713\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMATHI-41\" x=\"4102\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-28\" x=\"4852\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMATHI-78\" x=\"5241\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-29\" x=\"5813\" y=\"0\"><\/use><use xlink:href=\"#E106-MJMAIN-29\" x=\"6202\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) = g(A(x))<\/script><\/p><p><strong><span>Composition with an Increasing Convex Function:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.338ex\" height=\"1.994ex\" viewBox=\"0 -755.9 576 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E113-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E113-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">h<\/script><span> is increasing on <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.525ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3240 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E108-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E108-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E108-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E108-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E108-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E108-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E108-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E108-MJMATHI-6F\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E108-MJMATHI-6D\" x=\"1008\" y=\"0\"><\/use><use xlink:href=\"#E108-MJMAIN-28\" x=\"1886\" y=\"0\"><\/use><use xlink:href=\"#E108-MJMATHI-68\" x=\"2275\" y=\"0\"><\/use><use xlink:href=\"#E108-MJMAIN-29\" x=\"2851\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dom(h)<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.905ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6417.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E109-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E109-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E109-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E109-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E109-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E109-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E109-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E109-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-3D\" x=\"2177\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMATHI-67\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-28\" x=\"3713\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMATHI-68\" x=\"4102\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-28\" x=\"4678\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMATHI-78\" x=\"5067\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-29\" x=\"5639\" y=\"0\"><\/use><use xlink:href=\"#E109-MJMAIN-29\" x=\"6028\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) = g(h(x))<\/script><\/p><p><strong><span>Restriction on Lines:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is convex iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.482ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4513 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E111-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E111-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E111-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E111-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><g transform=\"translate(939,0)\"><use xlink:href=\"#E111-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E111-MJMAIN-30\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E111-MJMAIN-2B\" x=\"2186\" y=\"0\"><\/use><use xlink:href=\"#E111-MJMATHI-74\" x=\"3186\" y=\"0\"><\/use><use xlink:href=\"#E111-MJMATHI-68\" x=\"3547\" y=\"0\"><\/use><use xlink:href=\"#E111-MJMAIN-29\" x=\"4123\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x_0 + th)<\/script><span> is convex for any <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.382ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1025.6 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E112-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E112-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E112-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E112-MJMAIN-30\" x=\"808\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x_0<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.338ex\" height=\"1.994ex\" viewBox=\"0 -755.9 576 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E113-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E113-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">h<\/script><\/p><h3><a name=\"differentiable-convex-functions\" class=\"md-header-anchor\"><\/a><span>Differentiable Convex Functions<\/span><\/h3><p><strong><span>Theorem 12:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is differentiable on the open set, then it is convex iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.609ex\" height=\"2.811ex\" viewBox=\"0 -906.7 13609.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E115-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E115-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E115-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-2265\" x=\"2177\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMATHI-66\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-28\" x=\"3783\" y=\"0\"><\/use><g transform=\"translate(4172,0)\"><use xlink:href=\"#E115-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E115-MJMAIN-29\" x=\"4744\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-2B\" x=\"5355\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-28\" x=\"6356\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-2207\" x=\"6745\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMATHI-66\" x=\"7578\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-28\" x=\"8128\" y=\"0\"><\/use><g transform=\"translate(8517,0)\"><use xlink:href=\"#E115-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E115-MJMAIN-29\" x=\"9089\" y=\"0\"><\/use><g transform=\"translate(9478,0)\"><use xlink:href=\"#E115-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E115-MJMATHI-54\" x=\"550\" y=\"513\"><\/use><\/g><use xlink:href=\"#E115-MJMAIN-28\" x=\"10464\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMATHI-78\" x=\"10853\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-2212\" x=\"11648\" y=\"0\"><\/use><g transform=\"translate(12648,0)\"><use xlink:href=\"#E115-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E115-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E115-MJMAIN-29\" x=\"13220\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) \\ge f(\\bar{x}) + (\\nabla f(\\bar{x}))^T(x-\\bar{x})<\/script><span> for all <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.027ex\" height=\"2.461ex\" viewBox=\"0 -806.1 3456.2 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E116-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E116-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E116-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E116-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E116-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E116-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E116-MJMAIN-2C\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(1016,0)\"><use xlink:href=\"#E116-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E116-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E116-MJMAIN-2208\" x=\"1866\" y=\"0\"><\/use><use xlink:href=\"#E116-MJMATHI-53\" x=\"2811\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x,\\bar{x} \\in S<\/script><\/p><p><strong><span>Theorem 13:<\/span><\/strong><span>\t<\/span><span>When <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is twice continuously differentiable on convex set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.576ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3261.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E118-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E118-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E118-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E118-MJMAIN-2286\" x=\"922\" y=\"0\"><\/use><g transform=\"translate(1978,0)\"><use xlink:href=\"#E118-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E118-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">S \\subseteq R^n<\/script><span>. Then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is convex on <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.66ex\" height=\"2.811ex\" viewBox=\"0 -906.7 5020.1 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E121-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E121-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E121-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E121-MJMAIN-32\" x=\"1178\" y=\"513\"><\/use><use xlink:href=\"#E121-MJMATHI-66\" x=\"1286\" y=\"0\"><\/use><use xlink:href=\"#E121-MJMAIN-28\" x=\"1836\" y=\"0\"><\/use><use xlink:href=\"#E121-MJMATHI-78\" x=\"2225\" y=\"0\"><\/use><use xlink:href=\"#E121-MJMAIN-29\" x=\"2797\" y=\"0\"><\/use><use xlink:href=\"#E121-MJMAIN-2265\" x=\"3464\" y=\"0\"><\/use><use xlink:href=\"#E121-MJMAIN-30\" x=\"4520\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\nabla ^2 f(x) \\ge 0<\/script><span> for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.666ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2439.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E122-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E122-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E122-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E122-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E122-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E122-MJMATHI-53\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in S<\/script><\/p><h3><a name=\"non-differentiable-convex-functions\" class=\"md-header-anchor\"><\/a><span>Non-Differentiable Convex Functions<\/span><\/h3><p><strong><span>Subgradient:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.089ex\" height=\"1.41ex\" viewBox=\"0 -504.6 469 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E127-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E127-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">s<\/script><span> is subgradient of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"24.543ex\" height=\"2.811ex\" viewBox=\"0 -906.7 10567.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E126-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E126-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E126-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-2265\" x=\"2177\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMATHI-66\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-28\" x=\"3783\" y=\"0\"><\/use><g transform=\"translate(4172,0)\"><use xlink:href=\"#E126-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E126-MJMAIN-29\" x=\"4744\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-2B\" x=\"5355\" y=\"0\"><\/use><g transform=\"translate(6356,0)\"><use xlink:href=\"#E126-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E126-MJMATHI-54\" x=\"663\" y=\"513\"><\/use><\/g><use xlink:href=\"#E126-MJMAIN-28\" x=\"7422\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMATHI-78\" x=\"7811\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-2212\" x=\"8606\" y=\"0\"><\/use><g transform=\"translate(9606,0)\"><use xlink:href=\"#E126-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E126-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E126-MJMAIN-29\" x=\"10178\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) \\ge f(\\bar{x}) + s^T(x-\\bar{x})<\/script><span>, the set of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.089ex\" height=\"1.41ex\" viewBox=\"0 -504.6 469 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E127-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E127-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">s<\/script><span> is called subdifferetial of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> and is denoted by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.73ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2467 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E130-MJMAIN-2202\" d=\"M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z\"><\/path><path stroke-width=\"0\" id=\"E130-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E130-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E130-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E130-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E130-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E130-MJMAIN-2202\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E130-MJMATHI-66\" x=\"567\" y=\"0\"><\/use><use xlink:href=\"#E130-MJMAIN-28\" x=\"1117\" y=\"0\"><\/use><g transform=\"translate(1506,0)\"><use xlink:href=\"#E130-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E130-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E130-MJMAIN-29\" x=\"2078\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\partial f(\\bar{x}) <\/script><\/p><p><strong><span>Theorem 14:<\/span><\/strong><span>\t<\/span><\/p><p><span>1) The convex function <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> is differentiable at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E132-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E132-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E132-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E132-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E132-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E132-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E132-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E132-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x\\in R^n<\/script><span> iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.497ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7533.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E133-MJMAIN-2202\" d=\"M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E133-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E133-MJMAIN-2202\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMATHI-66\" x=\"567\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-28\" x=\"1117\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMATHI-78\" x=\"1506\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-29\" x=\"2078\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-3D\" x=\"2744\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-7B\" x=\"3800\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-2207\" x=\"4300\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMATHI-66\" x=\"5133\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-28\" x=\"5683\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMATHI-78\" x=\"6072\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-29\" x=\"6644\" y=\"0\"><\/use><use xlink:href=\"#E133-MJMAIN-7D\" x=\"7033\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\partial f(x) = \\{\\nabla f(x)\\}<\/script><\/p><p><span>2) Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> be convex and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"28.47ex\" height=\"3.861ex\" viewBox=\"0 -1208.2 12257.7 1662.6\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.055ex;\"><defs><path stroke-width=\"0\" id=\"E136-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-69\" d=\"M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E136-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E136-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-2032\" x=\"803\" y=\"513\"><\/use><use xlink:href=\"#E136-MJMAIN-28\" x=\"862\" y=\"0\"><\/use><use xlink:href=\"#E136-MJMATHI-78\" x=\"1251\" y=\"0\"><\/use><use xlink:href=\"#E136-MJMAIN-2C\" x=\"1823\" y=\"0\"><\/use><use xlink:href=\"#E136-MJMATHI-64\" x=\"2268\" y=\"0\"><\/use><use xlink:href=\"#E136-MJMAIN-29\" x=\"2791\" y=\"0\"><\/use><use xlink:href=\"#E136-MJMAIN-3D\" x=\"3457\" y=\"0\"><\/use><g transform=\"translate(4513,0)\"><use xlink:href=\"#E136-MJMAIN-6C\"><\/use><use xlink:href=\"#E136-MJMAIN-69\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E136-MJMAIN-6D\" x=\"556\" y=\"0\"><\/use><g transform=\"translate(1389,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-74\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-2192\" x=\"361\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-30\" x=\"1361\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(7318,0)\"><g transform=\"translate(286,0)\"><rect stroke=\"none\" width=\"4532\" height=\"60\" x=\"0\" y=\"220\"><\/rect><g transform=\"translate(60,581)\"><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-2B\" x=\"1511\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-74\" x=\"2289\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-64\" x=\"2649\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-29\" x=\"3173\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-2212\" x=\"3562\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-66\" x=\"4340\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-28\" x=\"4890\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-78\" x=\"5279\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMAIN-29\" x=\"5851\" y=\"0\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E136-MJMATHI-74\" x=\"3024\" y=\"-491\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">f'(x,d) = \\lim _{t \\rightarrow 0} \\frac{f(x+td) - f(x)}{t}<\/script><span> be the directional derivative of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E168-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E168-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> in the direction <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.68ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5028.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E171-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-2216\" d=\"M56 731Q56 740 62 745T75 750Q85 750 92 740Q96 733 270 255T444 -231Q444 -239 438 -244T424 -250Q414 -250 407 -240Q404 -236 230 242T56 731Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E171-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-2208\" x=\"800\" y=\"0\"><\/use><g transform=\"translate(1745,0)\"><use xlink:href=\"#E171-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E171-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E171-MJMAIN-2216\" x=\"3028\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-7B\" x=\"3528\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-30\" x=\"4028\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-7D\" x=\"4528\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">d \\in R^n \\backslash \\{0\\}<\/script><span>. Then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.73ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2467 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E140-MJMAIN-2202\" d=\"M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z\"><\/path><path stroke-width=\"0\" id=\"E140-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E140-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E140-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E140-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E140-MJMAIN-2202\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E140-MJMATHI-66\" x=\"567\" y=\"0\"><\/use><use xlink:href=\"#E140-MJMAIN-28\" x=\"1117\" y=\"0\"><\/use><use xlink:href=\"#E140-MJMATHI-78\" x=\"1506\" y=\"0\"><\/use><use xlink:href=\"#E140-MJMAIN-29\" x=\"2078\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\partial f(x)<\/script><span> is a non-empty compact convex set and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"24.922ex\" height=\"3.161ex\" viewBox=\"0 -906.7 10730.2 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.055ex;\"><defs><path stroke-width=\"0\" id=\"E141-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMAIN-2202\" d=\"M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z\"><\/path><path stroke-width=\"0\" id=\"E141-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E141-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMAIN-2032\" x=\"803\" y=\"513\"><\/use><use xlink:href=\"#E141-MJMAIN-28\" x=\"862\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMATHI-78\" x=\"1251\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMAIN-2C\" x=\"1823\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMATHI-64\" x=\"2268\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMAIN-29\" x=\"2791\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMAIN-3D\" x=\"3457\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMATHI-6D\" x=\"4513\" y=\"0\"><\/use><use xlink:href=\"#E141-MJMATHI-61\" x=\"5391\" y=\"0\"><\/use><g transform=\"translate(5920,0)\"><use xlink:href=\"#E141-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(572,-186)\"><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMAIN-2208\" x=\"469\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMAIN-2202\" x=\"1136\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMATHI-66\" x=\"1703\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMAIN-28\" x=\"2253\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMATHI-78\" x=\"2642\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMAIN-29\" x=\"3213\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(9140,0)\"><use xlink:href=\"#E141-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E141-MJMATHI-54\" x=\"663\" y=\"513\"><\/use><\/g><use xlink:href=\"#E141-MJMATHI-64\" x=\"10207\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f'(x,d) = max _{s \\in \\partial f(x)}s^Td<\/script><span> for any <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.215ex\" height=\"1.994ex\" viewBox=\"0 -755.9 523 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E310-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E310-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">d<\/script><\/p><h3><a name=\"calculus-and-linear-algebra-preparations\" class=\"md-header-anchor\"><\/a><span>Calculus and Linear Algebra Preparations<\/span><\/h3><p><strong><span>Cachy-Schwarz Inequality:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"35.443ex\" height=\"3.044ex\" viewBox=\"0 -906.7 15260.2 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E143-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E143-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E143-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E143-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-6E\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(2859,0)\"><use xlink:href=\"#E143-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><g transform=\"translate(3775,0)\"><use xlink:href=\"#E143-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(4609,0)\"><use xlink:href=\"#E143-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-32\" x=\"550\" y=\"513\"><\/use><\/g><use xlink:href=\"#E143-MJMAIN-2264\" x=\"5729\" y=\"0\"><\/use><use xlink:href=\"#E143-MJMAIN-28\" x=\"6785\" y=\"0\"><\/use><g transform=\"translate(7174,0)\"><use xlink:href=\"#E143-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-6E\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(9644,0)\"><use xlink:href=\"#E143-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-32\" x=\"808\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"808\" y=\"-429\"><\/use><\/g><use xlink:href=\"#E143-MJMAIN-29\" x=\"10670\" y=\"0\"><\/use><use xlink:href=\"#E143-MJMAIN-28\" x=\"11059\" y=\"0\"><\/use><g transform=\"translate(11448,0)\"><use xlink:href=\"#E143-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-6E\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(13918,0)\"><use xlink:href=\"#E143-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMAIN-32\" x=\"705\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E143-MJMATHI-69\" x=\"692\" y=\"-429\"><\/use><\/g><use xlink:href=\"#E143-MJMAIN-29\" x=\"14871\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(\\sum ^n _{i=1} x_iy_i)^2 \\le (\\sum^n _{i=1} x_i ^2)(\\sum ^n _{i=1} y_i^2)<\/script><\/p><p><strong><span>Vector Norm:<\/span><\/strong><\/p><p><span>1-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.218ex\" height=\"2.811ex\" viewBox=\"0 -856.4 7413.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E144-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E144-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E144-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E144-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E144-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E144-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E144-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E144-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E144-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E144-MJMATHI-78\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E144-MJMAIN-7C\" x=\"1128\" y=\"0\"><\/use><g transform=\"translate(1406,0)\"><use xlink:href=\"#E144-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E144-MJMAIN-31\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E144-MJMAIN-3D\" x=\"2415\" y=\"0\"><\/use><g transform=\"translate(3471,0)\"><use xlink:href=\"#E144-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E144-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E144-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E144-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E144-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E144-MJMAIN-7C\" x=\"5941\" y=\"0\"><\/use><g transform=\"translate(6219,0)\"><use xlink:href=\"#E144-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E144-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E144-MJMAIN-7C\" x=\"7135\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||x||_1 = \\sum ^m _{i=1} |x_i|<\/script><\/p><p><span>2-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.504ex\" height=\"4.679ex\" viewBox=\"0 -1308.7 7967 2014.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.639ex;\"><defs><path stroke-width=\"0\" id=\"E145-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E145-MJSZ2-221A\" d=\"M1001 1150Q1017 1150 1020 1132Q1020 1127 741 244L460 -643Q453 -650 436 -650H424Q423 -647 423 -645T421 -640T419 -631T415 -617T408 -594T399 -560T385 -512T367 -448T343 -364T312 -259L203 119L138 41L111 67L212 188L264 248L472 -474L983 1140Q988 1150 1001 1150Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E145-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E145-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E145-MJMATHI-78\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E145-MJMAIN-7C\" x=\"1128\" y=\"0\"><\/use><g transform=\"translate(1406,0)\"><use xlink:href=\"#E145-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMAIN-32\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E145-MJMAIN-3D\" x=\"2415\" y=\"0\"><\/use><g transform=\"translate(3471,0)\"><use xlink:href=\"#E145-MJSZ2-221A\" x=\"0\" y=\"24\"><\/use><rect stroke=\"none\" width=\"3495\" height=\"60\" x=\"1000\" y=\"1114\"><\/rect><g transform=\"translate(1000,0)\"><use xlink:href=\"#E145-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><g transform=\"translate(2470,0)\"><use xlink:href=\"#E145-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMAIN-32\" x=\"808\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E145-MJMATHI-69\" x=\"808\" y=\"-429\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">||x||_2 = \\sqrt{\\sum ^m _{i=1} x_i^2}<\/script><\/p><p><span>p-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"21.741ex\" height=\"3.978ex\" viewBox=\"0 -1208.2 9360.7 1712.8\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E146-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E146-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E146-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E146-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E146-MJMATHI-78\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E146-MJMAIN-7C\" x=\"1128\" y=\"0\"><\/use><g transform=\"translate(1406,0)\"><use xlink:href=\"#E146-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMATHI-70\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E146-MJMAIN-3D\" x=\"2417\" y=\"0\"><\/use><use xlink:href=\"#E146-MJMAIN-28\" x=\"3473\" y=\"0\"><\/use><g transform=\"translate(3862,0)\"><use xlink:href=\"#E146-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E146-MJMAIN-7C\" x=\"6332\" y=\"0\"><\/use><g transform=\"translate(6610,0)\"><use xlink:href=\"#E146-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><g transform=\"translate(7526,0)\"><use xlink:href=\"#E146-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E146-MJMATHI-70\" x=\"393\" y=\"674\"><\/use><\/g><g transform=\"translate(8260,0)\"><use xlink:href=\"#E146-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,448)\"><g transform=\"translate(120,0)\"><rect stroke=\"none\" width=\"371\" height=\"60\" x=\"0\" y=\"146\"><\/rect><use transform=\"scale(0.5)\" xlink:href=\"#E146-MJMAIN-31\" x=\"121\" y=\"662\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E146-MJMATHI-70\" x=\"119\" y=\"-488\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">||x||_p = (\\sum ^m _{i=1} |x_i|^p)^{\\frac{1}{p}}<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.323ex\" height=\"1.41ex\" viewBox=\"0 -504.6 1000 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E155-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E155-MJMAIN-221E\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\infty<\/script><span>-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.697ex\" height=\"2.694ex\" viewBox=\"0 -806.1 7619.6 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E148-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E148-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E148-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path stroke-width=\"0\" id=\"E148-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E148-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E148-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E148-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E148-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E148-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E148-MJMATHI-78\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E148-MJMAIN-7C\" x=\"1128\" y=\"0\"><\/use><g transform=\"translate(1406,0)\"><use xlink:href=\"#E148-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E148-MJMAIN-221E\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E148-MJMAIN-3D\" x=\"2768\" y=\"0\"><\/use><use xlink:href=\"#E148-MJMATHI-6D\" x=\"3824\" y=\"0\"><\/use><use xlink:href=\"#E148-MJMATHI-61\" x=\"4702\" y=\"0\"><\/use><g transform=\"translate(5231,0)\"><use xlink:href=\"#E148-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E148-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E148-MJMAIN-7C\" x=\"6147\" y=\"0\"><\/use><g transform=\"translate(6425,0)\"><use xlink:href=\"#E148-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E148-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E148-MJMAIN-7C\" x=\"7341\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||x||_{\\infty} = max _i |x_i|<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.13ex\" height=\"1.877ex\" viewBox=\"0 -655.4 1778 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E206-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E206-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E206-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E206-MJMAIN-221E\" x=\"778\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-\\infty<\/script><span>-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.613ex\" height=\"2.811ex\" viewBox=\"0 -806.1 8013.7 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E150-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E150-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E150-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E150-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E150-MJMATHI-78\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E150-MJMAIN-7C\" x=\"1128\" y=\"0\"><\/use><g transform=\"translate(1406,0)\"><use xlink:href=\"#E150-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(278,-285)\"><use transform=\"scale(0.707)\" xlink:href=\"#E150-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E150-MJMAIN-221E\" x=\"778\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E150-MJMAIN-3D\" x=\"3319\" y=\"0\"><\/use><use xlink:href=\"#E150-MJMATHI-6D\" x=\"4374\" y=\"0\"><\/use><use xlink:href=\"#E150-MJMATHI-69\" x=\"5252\" y=\"0\"><\/use><g transform=\"translate(5597,0)\"><use xlink:href=\"#E150-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E150-MJMATHI-69\" x=\"848\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E150-MJMAIN-7C\" x=\"6541\" y=\"0\"><\/use><g transform=\"translate(6819,0)\"><use xlink:href=\"#E150-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E150-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E150-MJMAIN-7C\" x=\"7735\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||x||_{-\\infty} = min _i |x_i|<\/script><\/p><p><span>0-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.417ex\" height=\"2.694ex\" viewBox=\"0 -806.1 3193.3 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E151-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E151-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E151-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E151-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E151-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E151-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E151-MJMATHI-78\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E151-MJMAIN-7C\" x=\"1128\" y=\"0\"><\/use><g transform=\"translate(1406,0)\"><use xlink:href=\"#E151-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E151-MJMAIN-30\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E151-MJMAIN-3D\" x=\"2415\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||x||_0 =<\/script><span>Nums of non-zero element of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E168-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E168-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><\/p><p><strong><span>Matrix Norm:<\/span><\/strong><\/p><p><span>1-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"24.557ex\" height=\"2.811ex\" viewBox=\"0 -856.4 10573.3 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E153-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E153-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E153-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E153-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E153-MJMATHI-41\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E153-MJMAIN-7C\" x=\"1306\" y=\"0\"><\/use><g transform=\"translate(1584,0)\"><use xlink:href=\"#E153-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMAIN-31\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E153-MJMAIN-3D\" x=\"2593\" y=\"0\"><\/use><use xlink:href=\"#E153-MJMATHI-6D\" x=\"3649\" y=\"0\"><\/use><use xlink:href=\"#E153-MJMATHI-61\" x=\"4527\" y=\"0\"><\/use><g transform=\"translate(5056,0)\"><use xlink:href=\"#E153-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><g transform=\"translate(6186,0)\"><use xlink:href=\"#E153-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E153-MJMAIN-7C\" x=\"8656\" y=\"0\"><\/use><g transform=\"translate(8934,0)\"><use xlink:href=\"#E153-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(529,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMAIN-2C\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E153-MJMATHI-6A\" x=\"623\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E153-MJMAIN-7C\" x=\"10295\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||A||_1 = max _j \\sum ^m _{i=1} |a _{i,j}|<\/script><\/p><p><span>2-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.818ex\" height=\"2.928ex\" viewBox=\"0 -906.7 5518.7 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E154-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E154-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E154-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E154-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E154-MJMATHI-3BB\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path stroke-width=\"0\" id=\"E154-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E154-MJMAIN-221A\" d=\"M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E154-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E154-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E154-MJMATHI-41\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E154-MJMAIN-7C\" x=\"1306\" y=\"0\"><\/use><g transform=\"translate(1584,0)\"><use xlink:href=\"#E154-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E154-MJMAIN-32\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E154-MJMAIN-3D\" x=\"2593\" y=\"0\"><\/use><g transform=\"translate(3649,0)\"><use xlink:href=\"#E154-MJMAIN-221A\" x=\"0\" y=\"-6\"><\/use><rect stroke=\"none\" width=\"1036\" height=\"60\" x=\"833\" y=\"734\"><\/rect><g transform=\"translate(833,0)\"><use xlink:href=\"#E154-MJMATHI-3BB\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E154-MJMAIN-31\" x=\"824\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">||A||_2 = \\sqrt{\\lambda _1}<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.323ex\" height=\"1.41ex\" viewBox=\"0 -504.6 1000 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E155-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E155-MJMAIN-221E\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\infty<\/script><span>-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"25.378ex\" height=\"3.161ex\" viewBox=\"0 -856.4 10926.8 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E156-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E156-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E156-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E156-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E156-MJMATHI-41\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E156-MJMAIN-7C\" x=\"1306\" y=\"0\"><\/use><g transform=\"translate(1584,0)\"><use xlink:href=\"#E156-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMAIN-221E\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E156-MJMAIN-3D\" x=\"2946\" y=\"0\"><\/use><use xlink:href=\"#E156-MJMATHI-6D\" x=\"4002\" y=\"0\"><\/use><use xlink:href=\"#E156-MJMATHI-61\" x=\"4880\" y=\"0\"><\/use><g transform=\"translate(5409,0)\"><use xlink:href=\"#E156-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><g transform=\"translate(6492,0)\"><use xlink:href=\"#E156-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMAIN-3D\" x=\"412\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMAIN-31\" x=\"1189\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E156-MJMAIN-7C\" x=\"9009\" y=\"0\"><\/use><g transform=\"translate(9287,0)\"><use xlink:href=\"#E156-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(529,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMAIN-2C\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E156-MJMATHI-6A\" x=\"623\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E156-MJMAIN-7C\" x=\"10648\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||A||_{\\infty} = max_i \\sum ^m _{j=1} |a_{i,j}|<\/script><\/p><p><span>F-Norm:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"27.086ex\" height=\"3.861ex\" viewBox=\"0 -1158 11662 1662.6\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E157-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-46\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E157-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E157-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E157-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E157-MJMATHI-41\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E157-MJMAIN-7C\" x=\"1306\" y=\"0\"><\/use><g transform=\"translate(1584,0)\"><use xlink:href=\"#E157-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-46\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E157-MJMAIN-3D\" x=\"2769\" y=\"0\"><\/use><use xlink:href=\"#E157-MJMAIN-28\" x=\"3825\" y=\"0\"><\/use><g transform=\"translate(4214,0)\"><use xlink:href=\"#E157-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(6684,0)\"><use xlink:href=\"#E157-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-6E\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMAIN-3D\" x=\"412\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMAIN-31\" x=\"1189\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(9202,0)\"><use xlink:href=\"#E157-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMAIN-32\" x=\"748\" y=\"487\"><\/use><g transform=\"translate(529,-303)\"><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMAIN-2C\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E157-MJMATHI-6A\" x=\"623\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(10563,0)\"><use xlink:href=\"#E157-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,418)\"><g transform=\"translate(120,0)\"><rect stroke=\"none\" width=\"370\" height=\"60\" x=\"0\" y=\"146\"><\/rect><use transform=\"scale(0.5)\" xlink:href=\"#E157-MJMAIN-31\" x=\"119\" y=\"662\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E157-MJMAIN-32\" x=\"119\" y=\"-621\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">||A||_F = (\\sum ^m _{i=1} \\sum ^n _{j=1} a_{i,j}^2)^{\\frac{1}{2}}<\/script><\/p><p><strong><span>Taylor's Formula:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"20.778ex\" height=\"2.577ex\" viewBox=\"0 -806.1 8946.3 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E158-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E158-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E158-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-3D\" x=\"2177\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMATHI-66\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-28\" x=\"3783\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMATHI-61\" x=\"4172\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-29\" x=\"4701\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-2B\" x=\"5312\" y=\"0\"><\/use><g transform=\"translate(6313,0)\"><use xlink:href=\"#E158-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E158-MJMATHI-6E\" x=\"1073\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E158-MJMAIN-28\" x=\"7596\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMATHI-78\" x=\"7985\" y=\"0\"><\/use><use xlink:href=\"#E158-MJMAIN-29\" x=\"8557\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x) = f(a) + R_n(x)<\/script><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"28.246ex\" height=\"4.795ex\" viewBox=\"0 -1359 12161.3 2064.6\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.639ex;\"><defs><path stroke-width=\"0\" id=\"E159-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-21\" d=\"M78 661Q78 682 96 699T138 716T180 700T199 661Q199 654 179 432T158 206Q156 198 139 198Q121 198 119 206Q118 209 98 431T78 661ZM79 61Q79 89 97 105T141 121Q164 119 181 104T198 61Q198 31 181 16T139 1Q114 1 97 16T79 61Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E159-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E159-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMATHI-6E\" x=\"1073\" y=\"-213\"><\/use><use xlink:href=\"#E159-MJMAIN-28\" x=\"1283\" y=\"0\"><\/use><use xlink:href=\"#E159-MJMATHI-78\" x=\"1672\" y=\"0\"><\/use><use xlink:href=\"#E159-MJMAIN-29\" x=\"2244\" y=\"0\"><\/use><use xlink:href=\"#E159-MJMAIN-3D\" x=\"2911\" y=\"0\"><\/use><g transform=\"translate(3689,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"2874\" height=\"60\" x=\"0\" y=\"220\"><\/rect><g transform=\"translate(60,581)\"><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(401,305)\"><use transform=\"scale(0.5)\" xlink:href=\"#E159-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E159-MJMATHI-6E\" x=\"389\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E159-MJMAIN-2B\" x=\"989\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E159-MJMAIN-31\" x=\"1767\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E159-MJMAIN-29\" x=\"2267\" y=\"0\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-28\" x=\"2546\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMATHI-78\" x=\"2935\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-29\" x=\"3507\" y=\"0\"><\/use><\/g><g transform=\"translate(400,-435)\"><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMATHI-6E\" x=\"389\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-2B\" x=\"989\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-31\" x=\"1766\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-29\" x=\"2267\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-21\" x=\"2656\" y=\"0\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E159-MJMAIN-28\" x=\"7081\" y=\"0\"><\/use><use xlink:href=\"#E159-MJMATHI-78\" x=\"7470\" y=\"0\"><\/use><use xlink:href=\"#E159-MJMAIN-2212\" x=\"8265\" y=\"0\"><\/use><use xlink:href=\"#E159-MJMATHI-61\" x=\"9265\" y=\"0\"><\/use><g transform=\"translate(9794,0)\"><use xlink:href=\"#E159-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMATHI-6E\" x=\"389\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-2B\" x=\"989\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-31\" x=\"1766\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E159-MJMAIN-29\" x=\"2267\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">R_n(x) = \\frac{f^{(n+1)}(x)}{(n+1)!}(x-a) ^{(n+1)}<\/script><\/p><p><strong><span>Semidefinite:<\/span><\/strong><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.691ex\" height=\"6.079ex\" viewBox=\"0 -1560 6325.4 2617.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -2.456ex;\"><defs><path stroke-width=\"0\" id=\"E160-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMATHI-42\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJSZ3-5B\" d=\"M247 -949V1450H516V1388H309V-887H516V-949H247Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJSZ3-5D\" d=\"M11 1388V1450H280V-949H11V-887H218V1388H11Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E160-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E160-MJSZ3-5B\"><\/use><g transform=\"translate(695,0)\"><g transform=\"translate(-15,0)\"><use xlink:href=\"#E160-MJMATHI-41\" x=\"303\" y=\"670\"><\/use><g transform=\"translate(0,-771)\"><use xlink:href=\"#E160-MJMATHI-42\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E160-MJMATHI-54\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><g transform=\"translate(2341,0)\"><use xlink:href=\"#E160-MJMATHI-42\" x=\"0\" y=\"670\"><\/use><use xlink:href=\"#E160-MJMATHI-43\" x=\"0\" y=\"-771\"><\/use><\/g><\/g><use xlink:href=\"#E160-MJSZ3-5D\" x=\"3963\" y=\"-1\"><\/use><use xlink:href=\"#E160-MJMAIN-2265\" x=\"4769\" y=\"0\"><\/use><use xlink:href=\"#E160-MJMAIN-30\" x=\"5825\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\left [ \\begin{matrix} A & B \\\\ B^T & C \\end{matrix} \\right ] \\ge 0<\/script><span>\t\t\t<\/span><span><=><\/span><span>\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.85ex\" height=\"2.577ex\" viewBox=\"0 -906.7 7685.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E161-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMATHI-42\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E161-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E161-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E161-MJMAIN-2212\" x=\"982\" y=\"0\"><\/use><g transform=\"translate(1982,0)\"><use xlink:href=\"#E161-MJMATHI-42\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E161-MJMATHI-54\" x=\"1073\" y=\"513\"><\/use><\/g><g transform=\"translate(3339,0)\"><use xlink:href=\"#E161-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(750,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E161-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E161-MJMAIN-31\" x=\"778\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E161-MJMATHI-42\" x=\"5092\" y=\"0\"><\/use><use xlink:href=\"#E161-MJMAIN-2265\" x=\"6129\" y=\"0\"><\/use><use xlink:href=\"#E161-MJMAIN-30\" x=\"7185\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C - B^TA^{-1}B \\ge 0<\/script><\/p><h2><a name=\"handout-3\telements-of-linear-programming\" class=\"md-header-anchor\"><\/a><span>Handout 3<\/span><span>\t<\/span><span>Elements of Linear Programming<\/span><\/h2><h3><a name=\"basic-definitions-and-properties\" class=\"md-header-anchor\"><\/a><span>Basic Definitions and Properties<\/span><\/h3><p><strong><span>Polyhedron:<\/span><\/strong><span>\t<\/span><span>The intersection of a finite set of halfspaces<\/span><\/p><p><strong><span>Polytope:<\/span><\/strong><span>\t<\/span><span>A bounded polyhedron<\/span><\/p><p><span>Note that a closed convex set is the intersection of all the halfspaces containing it but this does not mean that any closed convex set is a polyhedron<\/span><\/p><h3><a name=\"external-elements-of-a-polyheedron\" class=\"md-header-anchor\"><\/a><span>External Elements of a Polyheedron<\/span><\/h3><p><strong><span>Active Set:<\/span><\/strong><span>\t<\/span><span>The index set of all constraints such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.838ex\" height=\"3.044ex\" viewBox=\"0 -906.7 3805.3 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E162-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E162-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E162-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E162-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E162-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E162-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E162-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E162-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E162-MJMATHI-54\" x=\"748\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E162-MJMATHI-69\" x=\"748\" y=\"-429\"><\/use><g transform=\"translate(1126,0)\"><use xlink:href=\"#E162-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E162-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E162-MJMAIN-3D\" x=\"1976\" y=\"0\"><\/use><g transform=\"translate(3032,0)\"><use xlink:href=\"#E162-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E162-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">a_i^T\\bar{x} = b_i<\/script><\/p><p><strong><span>Theorem 1:<\/span><\/strong><span>\t<\/span><span>The following are equivalent:<\/span><\/p><p><span>1)<\/span><span>\t<\/span><span>There exists n vectors in the set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.7ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6759.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E163-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path stroke-width=\"0\" id=\"E163-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E163-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(500,0)\"><use xlink:href=\"#E163-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E163-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E163-MJMAIN-2208\" x=\"1650\" y=\"0\"><\/use><use xlink:href=\"#E163-MJMATHI-52\" x=\"2595\" y=\"0\"><\/use><use xlink:href=\"#E163-MJMAIN-3A\" x=\"3632\" y=\"0\"><\/use><use xlink:href=\"#E163-MJMATHI-69\" x=\"4188\" y=\"0\"><\/use><use xlink:href=\"#E163-MJMAIN-2208\" x=\"4810\" y=\"0\"><\/use><use xlink:href=\"#E163-MJMATHI-49\" x=\"5755\" y=\"0\"><\/use><use xlink:href=\"#E163-MJMAIN-7D\" x=\"6259\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{a_i \\in R: i \\in I\\}<\/script><span> that are linearly independent<\/span><\/p><p><span>2)<\/span><span>\t<\/span><span>The point <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E397-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E397-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E397-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E397-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E397-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E397-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in R^n<\/script><span> is the unique solution to the following system of linear equations: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.838ex\" height=\"3.044ex\" viewBox=\"0 -906.7 3805.3 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E165-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E165-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E165-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E165-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E165-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E165-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E165-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E165-MJMATHI-54\" x=\"748\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E165-MJMATHI-69\" x=\"748\" y=\"-429\"><\/use><use xlink:href=\"#E165-MJMATHI-78\" x=\"1126\" y=\"0\"><\/use><use xlink:href=\"#E165-MJMAIN-3D\" x=\"1976\" y=\"0\"><\/use><g transform=\"translate(3032,0)\"><use xlink:href=\"#E165-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E165-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">a_i^Tx = b_i<\/script><\/p><p><strong><span>Basic Solution:<\/span><\/strong><span>\t<\/span><span>The vector <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E168-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E168-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> is called basic solution if there are n linearly independent active constraints to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E168-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E168-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span>, if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E168-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E168-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> is in the polyhedron, then it is a basic feasible solution<\/span><\/p><p><span>Note that not every polyhedron has basic feasible solution<\/span><\/p><p><strong><span>Line:<\/span><\/strong><span>\t<\/span><span>A polyhedron <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.744ex\" height=\"1.877ex\" viewBox=\"0 -755.9 751 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E182-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E182-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P<\/script><span> contains a line if there exists <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.912ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2545.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E170-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E170-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E170-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E170-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E170-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E170-MJMATHI-50\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in P<\/script><span> and a vector <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.68ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5028.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E171-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-2216\" d=\"M56 731Q56 740 62 745T75 750Q85 750 92 740Q96 733 270 255T444 -231Q444 -239 438 -244T424 -250Q414 -250 407 -240Q404 -236 230 242T56 731Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E171-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E171-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-2208\" x=\"800\" y=\"0\"><\/use><g transform=\"translate(1745,0)\"><use xlink:href=\"#E171-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E171-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E171-MJMAIN-2216\" x=\"3028\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-7B\" x=\"3528\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-30\" x=\"4028\" y=\"0\"><\/use><use xlink:href=\"#E171-MJMAIN-7D\" x=\"4528\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">d \\in R^n \\backslash \\{0\\}<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.453ex\" height=\"2.11ex\" viewBox=\"0 -755.9 4931 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E172-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E172-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E172-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E172-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E172-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E172-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E172-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E172-MJMAIN-2B\" x=\"794\" y=\"0\"><\/use><use xlink:href=\"#E172-MJMATHI-3B1\" x=\"1794\" y=\"0\"><\/use><use xlink:href=\"#E172-MJMATHI-64\" x=\"2434\" y=\"0\"><\/use><use xlink:href=\"#E172-MJMAIN-2208\" x=\"3235\" y=\"0\"><\/use><use xlink:href=\"#E172-MJMATHI-50\" x=\"4180\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x+ \\alpha d \\in P<\/script><span> for all <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.089ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2621.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E173-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E173-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E173-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E173-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E173-MJMAIN-2208\" x=\"917\" y=\"0\"><\/use><use xlink:href=\"#E173-MJMATHI-52\" x=\"1862\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha \\in R<\/script><\/p><p><strong><span>Theorem 3:<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.822ex\" height=\"2.227ex\" viewBox=\"0 -755.9 3367.8 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E174-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E174-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E174-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E174-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E174-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E174-MJMAIN-2286\" x=\"1028\" y=\"0\"><\/use><g transform=\"translate(2084,0)\"><use xlink:href=\"#E174-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E174-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">P \\subseteq R^n<\/script><span> is a non-empty polyhedron, then the following are equivalent:<\/span><\/p><p><span>1)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.744ex\" height=\"1.877ex\" viewBox=\"0 -755.9 751 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E182-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E182-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P<\/script><span> has at least one vertex<\/span><\/p><p><span>2)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.744ex\" height=\"1.877ex\" viewBox=\"0 -755.9 751 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E182-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E182-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P<\/script><span> does not contain a line<\/span><\/p><p><span>3)<\/span><span>\t<\/span><span>There exists n linearly independent vectors in <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.248ex\" height=\"2.811ex\" viewBox=\"0 -806.1 3120.6 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E177-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E177-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E177-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E177-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E177-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E177-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E177-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E177-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(500,0)\"><use xlink:href=\"#E177-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E177-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><g transform=\"translate(1372,0)\"><use xlink:href=\"#E177-MJMAIN-7D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E177-MJMATHI-6D\" x=\"707\" y=\"498\"><\/use><g transform=\"translate(500,-307)\"><use transform=\"scale(0.707)\" xlink:href=\"#E177-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E177-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E177-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\{a_i\\}^m _{i=1}<\/script><\/p><h3><a name=\"existence-of-optimal-solutions-to-linear-programs\" class=\"md-header-anchor\"><\/a><span>Existence of Optimal Solutions to Linear Programs<\/span><\/h3><p><strong><span>Theorem 4:<\/span><\/strong><span>\t<\/span><span> Consider the LP <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.789ex\" height=\"2.694ex\" viewBox=\"0 -906.7 5075.9 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E181-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E181-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E181-MJMATHI-69\" x=\"878\" y=\"0\"><\/use><g transform=\"translate(1223,0)\"><use xlink:href=\"#E181-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(600,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMATHI-50\" x=\"1239\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(3330,0)\"><use xlink:href=\"#E181-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMATHI-54\" x=\"814\" y=\"513\"><\/use><\/g><use xlink:href=\"#E181-MJMATHI-78\" x=\"4503\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">min _{x\\in P} h^Tx<\/script><span>. Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.744ex\" height=\"1.877ex\" viewBox=\"0 -755.9 751 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E182-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E182-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P<\/script><span> has at least one vertex, then either the optimal value is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.13ex\" height=\"1.877ex\" viewBox=\"0 -655.4 1778 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E206-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E206-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E206-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E206-MJMAIN-221E\" x=\"778\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-\\infty<\/script><span> or there exists a vertex that is optimal.<\/span><\/p><p><span>Note that there could be non-vertex optimal solutions but at least one vertex optimal solution exists<\/span><\/p><p><strong><span>Corollary 1:<\/span><\/strong><span>\t<\/span><span>Consider the LP <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.789ex\" height=\"2.694ex\" viewBox=\"0 -906.7 5075.9 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E181-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E181-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E181-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E181-MJMATHI-69\" x=\"878\" y=\"0\"><\/use><g transform=\"translate(1223,0)\"><use xlink:href=\"#E181-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(600,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMATHI-50\" x=\"1239\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(3330,0)\"><use xlink:href=\"#E181-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E181-MJMATHI-54\" x=\"814\" y=\"513\"><\/use><\/g><use xlink:href=\"#E181-MJMATHI-78\" x=\"4503\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">min _{x\\in P} h^Tx<\/script><span>. Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.744ex\" height=\"1.877ex\" viewBox=\"0 -755.9 751 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E182-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E182-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P<\/script><span> is non-empty. Then either the optimal value is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.13ex\" height=\"1.877ex\" viewBox=\"0 -655.4 1778 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E206-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E206-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E206-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E206-MJMAIN-221E\" x=\"778\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-\\infty<\/script><span> or there exists an optimal solution.<\/span><\/p><p><strong><span>Standard LP:<\/span><\/strong><span>\t<\/span><\/p><p><span>minimize<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.548ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1527.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E184-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E184-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E184-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E184-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E184-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><use xlink:href=\"#E184-MJMATHI-79\" x=\"1030\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">c^Ty<\/script><span>\t<\/span><\/p><p><span>subject to <\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.99ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3009.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E185-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E185-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E185-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E185-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E185-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E185-MJMATHI-79\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E185-MJMAIN-3D\" x=\"1524\" y=\"0\"><\/use><use xlink:href=\"#E185-MJMATHI-62\" x=\"2580\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Ay = b<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.413ex\" height=\"2.461ex\" viewBox=\"0 -755.9 2330.6 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E196-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E196-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E196-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E196-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E196-MJMAIN-2265\" x=\"774\" y=\"0\"><\/use><use xlink:href=\"#E196-MJMAIN-30\" x=\"1830\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">y \\ge 0<\/script><\/p><h3><a name=\"theorems-of-alternatives\" class=\"md-header-anchor\"><\/a><span>Theorems of Alternatives<\/span><\/h3><p><strong><span>Theorem 5: (Farkas' Lemma)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.282ex\" height=\"2.11ex\" viewBox=\"0 -806.1 4426.8 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E187-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E187-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E187-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E187-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E187-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E187-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E187-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E187-MJMAIN-2208\" x=\"1027\" y=\"0\"><\/use><g transform=\"translate(1972,0)\"><use xlink:href=\"#E187-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E187-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E187-MJMAIN-D7\" x=\"878\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E187-MJMATHI-6E\" x=\"1655\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">A \\in R ^{m \\times n}<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.273ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3131.4 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E188-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E188-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E188-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E188-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E188-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E188-MJMAIN-2208\" x=\"706\" y=\"0\"><\/use><g transform=\"translate(1651,0)\"><use xlink:href=\"#E188-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E188-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">b \\in R^m<\/script><span> be given. Then exactly one of the following systems has a solution:<\/span><\/p><p><span>1)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.164ex\" height=\"2.11ex\" viewBox=\"0 -806.1 3084.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E199-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E199-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E199-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E199-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E199-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E199-MJMATHI-78\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E199-MJMAIN-3D\" x=\"1599\" y=\"0\"><\/use><use xlink:href=\"#E199-MJMATHI-62\" x=\"2655\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Ax = b<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.587ex\" height=\"2.227ex\" viewBox=\"0 -755.9 2405.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E200-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E200-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E200-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E200-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E200-MJMAIN-2265\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E200-MJMAIN-30\" x=\"1905\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\ge 0<\/script><\/p><p><span>2)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.543ex\" height=\"2.811ex\" viewBox=\"0 -906.7 3678.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E191-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E191-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E191-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E191-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E191-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E191-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E191-MJMATHI-54\" x=\"1060\" y=\"513\"><\/use><use xlink:href=\"#E191-MJMATHI-79\" x=\"1347\" y=\"0\"><\/use><use xlink:href=\"#E191-MJMAIN-2264\" x=\"2122\" y=\"0\"><\/use><use xlink:href=\"#E191-MJMAIN-30\" x=\"3178\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A^Ty \\le 0<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.798ex\" height=\"2.811ex\" viewBox=\"0 -906.7 3357.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E192-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E192-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E192-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E192-MJMAIN-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path stroke-width=\"0\" id=\"E192-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E192-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E192-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E192-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><use xlink:href=\"#E192-MJMAIN-3E\" x=\"1801\" y=\"0\"><\/use><use xlink:href=\"#E192-MJMAIN-30\" x=\"2857\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty \\gt 0<\/script><\/p><p><span>Note that 2) is not a polyhedron since polyhedrons can not have strict inequalites<\/span><\/p><p><strong><span>Corollary 2: (Gordan's Theorem)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.282ex\" height=\"2.11ex\" viewBox=\"0 -806.1 4426.8 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E193-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E193-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E193-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E193-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E193-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E193-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E193-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E193-MJMAIN-2208\" x=\"1027\" y=\"0\"><\/use><g transform=\"translate(1972,0)\"><use xlink:href=\"#E193-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E193-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E193-MJMAIN-D7\" x=\"878\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E193-MJMATHI-6E\" x=\"1655\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">A \\in R^{m \\times n}<\/script><span> be given. Then exactly one of the following systems has a solution:<\/span><\/p><p><span>1)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.329ex\" height=\"2.11ex\" viewBox=\"0 -806.1 3155.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E194-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E194-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E194-MJMAIN-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path stroke-width=\"0\" id=\"E194-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E194-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E194-MJMATHI-78\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E194-MJMAIN-3E\" x=\"1599\" y=\"0\"><\/use><use xlink:href=\"#E194-MJMAIN-30\" x=\"2655\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Ax \\gt 0<\/script><\/p><p><span>2)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.543ex\" height=\"2.811ex\" viewBox=\"0 -906.7 3678.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E195-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E195-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E195-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E195-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E195-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E195-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E195-MJMATHI-54\" x=\"1060\" y=\"513\"><\/use><use xlink:href=\"#E195-MJMATHI-79\" x=\"1347\" y=\"0\"><\/use><use xlink:href=\"#E195-MJMAIN-3D\" x=\"2122\" y=\"0\"><\/use><use xlink:href=\"#E195-MJMAIN-30\" x=\"3178\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A^Ty = 0<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.413ex\" height=\"2.461ex\" viewBox=\"0 -755.9 2330.6 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E196-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E196-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E196-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E196-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E196-MJMAIN-2265\" x=\"774\" y=\"0\"><\/use><use xlink:href=\"#E196-MJMAIN-30\" x=\"1830\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">y \\ge 0<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.413ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2330.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E197-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E197-MJMAIN-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path stroke-width=\"0\" id=\"E197-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E197-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E197-MJMAIN-2260\" x=\"774\" y=\"0\"><\/use><use xlink:href=\"#E197-MJMAIN-30\" x=\"1830\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">y \\neq 0<\/script><\/p><h3><a name=\"lp-duality-theory\" class=\"md-header-anchor\"><\/a><span>LP Duality Theory<\/span><\/h3><p><strong><span>Primal Problem and Dual Problem:<\/span><\/strong><\/p><p><span>(P)<\/span><span>\t<\/span><span>minimize<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.723ex\" height=\"2.344ex\" viewBox=\"0 -906.7 1602.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E261-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E261-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E261-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><use xlink:href=\"#E261-MJMATHI-78\" x=\"1030\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">c^Tx<\/script><\/p><p><span>\t\t<\/span><span>subject to<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.164ex\" height=\"2.11ex\" viewBox=\"0 -806.1 3084.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E199-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E199-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E199-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E199-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E199-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E199-MJMATHI-78\" x=\"750\" y=\"0\"><\/use><use xlink:href=\"#E199-MJMAIN-3D\" x=\"1599\" y=\"0\"><\/use><use xlink:href=\"#E199-MJMATHI-62\" x=\"2655\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Ax = b<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.587ex\" height=\"2.227ex\" viewBox=\"0 -755.9 2405.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E200-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E200-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E200-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E200-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E200-MJMAIN-2265\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E200-MJMAIN-30\" x=\"1905\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\ge 0<\/script><\/p><p><span>(D)<\/span><span>\t<\/span><span>maximize<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.539ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1523.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E271-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E271-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E271-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E271-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty<\/script><\/p><p><span>\t\t<\/span><span>subject to<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.388ex\" height=\"2.811ex\" viewBox=\"0 -906.7 3611.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E202-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E202-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E202-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E202-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E202-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E202-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E202-MJMATHI-54\" x=\"1060\" y=\"513\"><\/use><use xlink:href=\"#E202-MJMATHI-79\" x=\"1347\" y=\"0\"><\/use><use xlink:href=\"#E202-MJMAIN-2264\" x=\"2122\" y=\"0\"><\/use><use xlink:href=\"#E202-MJMATHI-63\" x=\"3178\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">A^Ty \\le c<\/script><\/p><p><strong><span>Theorem 6: (LP Weak Duality)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E397-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E397-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E397-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E397-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E397-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E397-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in R^n<\/script><span> be feasible for (P) and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.578ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3262.6 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E278-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E278-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E278-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><use xlink:href=\"#E278-MJMAIN-2208\" x=\"837\" y=\"0\"><\/use><g transform=\"translate(1782,0)\"><use xlink:href=\"#E278-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E278-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{y} \\in R^m<\/script><span> be feasible for (D). Then we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.506ex\" height=\"2.811ex\" viewBox=\"0 -906.7 4523.3 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E205-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E205-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E205-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E205-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E205-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E205-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E205-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E205-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E205-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><g transform=\"translate(1026,0)\"><use xlink:href=\"#E205-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E205-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E205-MJMAIN-2264\" x=\"1864\" y=\"0\"><\/use><g transform=\"translate(2920,0)\"><use xlink:href=\"#E205-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E205-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><\/g><g transform=\"translate(3951,0)\"><use xlink:href=\"#E205-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E205-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">b^T\\bar{y} \\le c^T \\bar{x}<\/script><span>. <\/span><\/p><p><strong><span>Corollary 3:<\/span><\/strong><span>\t<\/span><\/p><p><span>1)<\/span><span>\t<\/span><span>If the optimal value of (P) is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.13ex\" height=\"1.877ex\" viewBox=\"0 -655.4 1778 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E206-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E206-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E206-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E206-MJMAIN-221E\" x=\"778\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-\\infty<\/script><span>, then (D) must be infeasible<\/span><\/p><p><span>2)<\/span><span>\t<\/span><span>If the optimal value of (D) is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.13ex\" height=\"1.877ex\" viewBox=\"0 -655.4 1778 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E207-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E207-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E207-MJMAIN-2B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E207-MJMAIN-221E\" x=\"778\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">+\\infty<\/script><span>, then (P) must be infeasible<\/span><\/p><p><span>3)<\/span><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> and<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.301ex\" height=\"2.227ex\" viewBox=\"0 -655.4 560.2 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E217-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E217-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E217-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E217-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{y}<\/script><span> be feasible for (P) and (D). Suppose that the duality gap <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"25.008ex\" height=\"2.811ex\" viewBox=\"0 -906.7 10767.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E210-MJMAIN-394\" d=\"M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E210-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E210-MJMAIN-394\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-28\" x=\"833\" y=\"0\"><\/use><g transform=\"translate(1222,0)\"><use xlink:href=\"#E210-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E210-MJMAIN-2C\" x=\"1794\" y=\"0\"><\/use><g transform=\"translate(2238,0)\"><use xlink:href=\"#E210-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E210-MJMAIN-29\" x=\"2798\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-3D\" x=\"3465\" y=\"0\"><\/use><g transform=\"translate(4521,0)\"><use xlink:href=\"#E210-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E210-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><\/g><g transform=\"translate(5552,0)\"><use xlink:href=\"#E210-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E210-MJMAIN-2212\" x=\"6346\" y=\"0\"><\/use><g transform=\"translate(7346,0)\"><use xlink:href=\"#E210-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E210-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><\/g><g transform=\"translate(8373,0)\"><use xlink:href=\"#E210-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E210-MJMAIN-3D\" x=\"9211\" y=\"0\"><\/use><use xlink:href=\"#E210-MJMAIN-30\" x=\"10267\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\Delta(\\bar{x},\\bar{y})=c^T\\bar{x}-b^T\\bar{y} = 0<\/script><span>. Then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.301ex\" height=\"2.227ex\" viewBox=\"0 -655.4 560.2 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E217-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E217-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E217-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E217-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{y}<\/script><span> are optimal solutions to (P) and (D)<\/span><\/p><p><span>Note that if (P) is infeasible, it is possible that (D) is also infeasible<\/span><\/p><p><strong><span>Theorem 7: (LP Strong Duality)<\/span><\/strong><span>\t<\/span><span>Suppose that (P) has an optimal solution <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.202ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3531.4 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E213-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E213-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E213-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E213-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E213-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E213-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E213-MJMAIN-2217\" x=\"808\" y=\"513\"><\/use><use xlink:href=\"#E213-MJMAIN-2208\" x=\"1303\" y=\"0\"><\/use><g transform=\"translate(2248,0)\"><use xlink:href=\"#E213-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E213-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x^* \\in R^n<\/script><span>. Then (D) also has an optimal solution <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.489ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3655.1 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E214-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E214-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E214-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E214-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E214-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E214-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E214-MJMAIN-2217\" x=\"705\" y=\"513\"><\/use><use xlink:href=\"#E214-MJMAIN-2208\" x=\"1230\" y=\"0\"><\/use><g transform=\"translate(2175,0)\"><use xlink:href=\"#E214-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E214-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y^* \\in R^m<\/script><span>, and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.471ex\" height=\"2.811ex\" viewBox=\"0 -906.7 5369.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E215-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E215-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E215-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E215-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E215-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E215-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E215-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E215-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E215-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><g transform=\"translate(1030,0)\"><use xlink:href=\"#E215-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E215-MJMAIN-2217\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E215-MJMAIN-3D\" x=\"2334\" y=\"0\"><\/use><g transform=\"translate(3389,0)\"><use xlink:href=\"#E215-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E215-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><\/g><g transform=\"translate(4416,0)\"><use xlink:href=\"#E215-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E215-MJMAIN-2217\" x=\"705\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">c^Tx^* = b^Ty^*<\/script><\/p><p><strong><span>Corollary 4:<\/span><\/strong><span>\t<\/span><span>Suppose both (P) and (D) are feasible. Then both (P) and (D) have optimal solutions and their respective optimal values are equal<\/span><\/p><p><strong><span>Theorem 8: (Complementory Slackness)<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.301ex\" height=\"2.227ex\" viewBox=\"0 -655.4 560.2 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E217-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E217-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E217-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E217-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{y}<\/script><span> are optimal for their respective problems iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.268ex\" height=\"2.811ex\" viewBox=\"0 -906.7 7434.9 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E218-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E218-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E218-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E218-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E218-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><use xlink:href=\"#E218-MJMAIN-28\" x=\"915\" y=\"0\"><\/use><use xlink:href=\"#E218-MJMATHI-63\" x=\"1304\" y=\"0\"><\/use><use xlink:href=\"#E218-MJMAIN-2212\" x=\"1960\" y=\"0\"><\/use><g transform=\"translate(2960,0)\"><use xlink:href=\"#E218-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E218-MJMATHI-54\" x=\"1060\" y=\"513\"><\/use><\/g><g transform=\"translate(4308,0)\"><use xlink:href=\"#E218-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E218-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><g transform=\"translate(4868,0)\"><use xlink:href=\"#E218-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E218-MJMATHI-69\" x=\"550\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E218-MJMAIN-3D\" x=\"5879\" y=\"0\"><\/use><use xlink:href=\"#E218-MJMAIN-30\" x=\"6934\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}_i(c-A^T\\bar{y})_i = 0<\/script><span> for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.617ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5001.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E232-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E232-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMATHI-6E\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,n<\/script><\/p><h2><a name=\"handout-5\telements-of-conic-linear-programming\" class=\"md-header-anchor\"><\/a><span>Handout 5<\/span><span>\t<\/span><span>Elements of Conic Linear Programming<\/span><\/h2><h3><a name=\"introduction\" class=\"md-header-anchor\"><\/a><span>Introduction<\/span><\/h3><p><strong><span>Pointed Cone:<\/span><\/strong><span>\t<\/span><\/p><p><span>1) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.065ex\" height=\"1.877ex\" viewBox=\"0 -755.9 889 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E251-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E251-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K<\/script><span> is non-empty and closed under addition<\/span><\/p><p><span>2) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.065ex\" height=\"1.877ex\" viewBox=\"0 -755.9 889 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E251-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E251-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K<\/script><span> is a cone<\/span><\/p><p><span>3) <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.065ex\" height=\"1.877ex\" viewBox=\"0 -755.9 889 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E251-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E251-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K<\/script><span> is pointed, if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.233ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2683.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E223-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E223-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E223-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E223-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E223-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E223-MJMATHI-4B\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">u \\in K<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.04ex\" height=\"2.11ex\" viewBox=\"0 -755.9 3461.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.355ex;\"><defs><path stroke-width=\"0\" id=\"E224-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E224-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E224-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E224-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E224-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E224-MJMATHI-75\" x=\"778\" y=\"0\"><\/use><use xlink:href=\"#E224-MJMAIN-2208\" x=\"1627\" y=\"0\"><\/use><use xlink:href=\"#E224-MJMATHI-4B\" x=\"2572\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-u \\in K<\/script><span>, then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.587ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2405.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E225-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E225-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E225-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E225-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E225-MJMAIN-3D\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E225-MJMAIN-30\" x=\"1905\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">u = 0<\/script><\/p><p><span>A pointed cone is automatically convex<\/span><\/p><p><strong><span>Examples of Pointed Cone:<\/span><\/strong><\/p><p><span>1) Non-Negative Orthant: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.273ex\" height=\"2.694ex\" viewBox=\"0 -755.9 1409.1 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E226-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E226-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E226-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E226-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E226-MJMATHI-6E\" x=\"1073\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E226-MJMAIN-2B\" x=\"1073\" y=\"-351\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">R^n _+<\/script><\/p><p><span>2) Lorentz Cone\/Second-Order Cone\/Ice Cream Cone: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"36.839ex\" height=\"2.928ex\" viewBox=\"0 -906.7 15861.1 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E227-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E227-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E227-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(791,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E227-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E227-MJMAIN-2B\" x=\"600\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E227-MJMAIN-31\" x=\"1378\" y=\"0\"><\/use><\/g><use xlink:href=\"#E227-MJMAIN-3D\" x=\"2496\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-7B\" x=\"3552\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-28\" x=\"4052\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMATHI-74\" x=\"4441\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-2C\" x=\"4802\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMATHI-78\" x=\"5247\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-29\" x=\"5819\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-2208\" x=\"6485\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMATHI-52\" x=\"7430\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-D7\" x=\"8411\" y=\"0\"><\/use><g transform=\"translate(9412,0)\"><use xlink:href=\"#E227-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E227-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E227-MJMAIN-3A\" x=\"10973\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMATHI-74\" x=\"11528\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-2265\" x=\"12167\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-7C\" x=\"13223\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-7C\" x=\"13501\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMATHI-78\" x=\"13779\" y=\"0\"><\/use><use xlink:href=\"#E227-MJMAIN-7C\" x=\"14351\" y=\"0\"><\/use><g transform=\"translate(14629,0)\"><use xlink:href=\"#E227-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E227-MJMAIN-32\" x=\"393\" y=\"-403\"><\/use><\/g><use xlink:href=\"#E227-MJMAIN-7D\" x=\"15361\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">Q^{n+1}= \\{(t,x) \\in R \\times R^n:t \\ge ||x||_2\\}<\/script><\/p><p><span>3) Positive Semidefinte Cone: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"48.31ex\" height=\"3.044ex\" viewBox=\"0 -906.7 20799.9 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E228-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-72\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-6C\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E228-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E228-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E228-MJMATHI-6E\" x=\"925\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E228-MJMAIN-2B\" x=\"866\" y=\"-351\"><\/use><use xlink:href=\"#E228-MJMAIN-3D\" x=\"1540\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMAIN-7B\" x=\"2596\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-58\" x=\"3096\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMAIN-2208\" x=\"4226\" y=\"0\"><\/use><g transform=\"translate(5171,0)\"><use xlink:href=\"#E228-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E228-MJMATHI-6E\" x=\"925\" y=\"513\"><\/use><\/g><use xlink:href=\"#E228-MJMAIN-3A\" x=\"6627\" y=\"0\"><\/use><g transform=\"translate(7183,0)\"><use xlink:href=\"#E228-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E228-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E228-MJMATHI-58\" x=\"8353\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-75\" x=\"9205\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMAIN-2265\" x=\"10055\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMAIN-30\" x=\"11111\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-66\" x=\"12611\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-6F\" x=\"13161\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-72\" x=\"13646\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-61\" x=\"15097\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-6C\" x=\"15626\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-6C\" x=\"15924\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMATHI-75\" x=\"17222\" y=\"0\"><\/use><use xlink:href=\"#E228-MJMAIN-2208\" x=\"18071\" y=\"0\"><\/use><g transform=\"translate(19016,0)\"><use xlink:href=\"#E228-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E228-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E228-MJMAIN-7D\" x=\"20299\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S^n _+ = \\{X \\in S^n: u^TXu \\ge 0 \\quad for \\quad all \\quad u \\in R^n\\}<\/script><\/p><p><span>Note that these three cone are all self-dual<\/span><\/p><p><span>Frobenius inner product: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.97ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6876 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E229-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMATHI-59\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMATHI-72\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E229-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E229-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMAIN-22C5\" x=\"1074\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMATHI-59\" x=\"1574\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMAIN-3D\" x=\"2615\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMATHI-74\" x=\"3671\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMATHI-72\" x=\"4032\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMAIN-28\" x=\"4483\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMATHI-58\" x=\"4872\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMATHI-59\" x=\"5724\" y=\"0\"><\/use><use xlink:href=\"#E229-MJMAIN-29\" x=\"6487\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X \\cdot Y = tr(XY)<\/script><\/p><p><strong><span>Proposition 1:<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.863ex\" height=\"2.344ex\" viewBox=\"0 -755.9 4677.2 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E230-MJMATHI-45\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path stroke-width=\"0\" id=\"E230-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E230-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E230-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E230-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E230-MJMATHI-45\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E230-MJMAIN-31\" x=\"1043\" y=\"-213\"><\/use><use xlink:href=\"#E230-MJMAIN-2C\" x=\"1191\" y=\"0\"><\/use><use xlink:href=\"#E230-MJMAIN-2E\" x=\"1636\" y=\"0\"><\/use><use xlink:href=\"#E230-MJMAIN-2E\" x=\"2080\" y=\"0\"><\/use><use xlink:href=\"#E230-MJMAIN-2E\" x=\"2525\" y=\"0\"><\/use><use xlink:href=\"#E230-MJMAIN-2C\" x=\"2970\" y=\"0\"><\/use><g transform=\"translate(3414,0)\"><use xlink:href=\"#E230-MJMATHI-45\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E230-MJMATHI-6E\" x=\"1043\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">E_1,...,E_n<\/script><span> be finite-dimensional Euclidean spaces and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.381ex\" height=\"2.344ex\" viewBox=\"0 -755.9 3608.5 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E231-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E231-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E231-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E231-MJMATHI-45\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E231-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E231-MJMATHI-69\" x=\"1200\" y=\"-213\"><\/use><use xlink:href=\"#E231-MJMAIN-2286\" x=\"1470\" y=\"0\"><\/use><g transform=\"translate(2526,0)\"><use xlink:href=\"#E231-MJMATHI-45\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E231-MJMATHI-69\" x=\"1043\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">K_i \\sube E_i<\/script><span> be closed pointed cone with non-empty interiors, where <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.617ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5001.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E232-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E232-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E232-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E232-MJMATHI-6E\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,n<\/script><span>. Then the set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"60.333ex\" height=\"2.577ex\" viewBox=\"0 -806.1 25976.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E233-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMATHI-45\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E233-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E233-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-3D\" x=\"1166\" y=\"0\"><\/use><g transform=\"translate(2222,0)\"><use xlink:href=\"#E233-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMAIN-31\" x=\"1200\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-D7\" x=\"3525\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"4303\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"4747\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"5192\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-D7\" x=\"5637\" y=\"0\"><\/use><g transform=\"translate(6415,0)\"><use xlink:href=\"#E233-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMATHI-6E\" x=\"1200\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-3D\" x=\"8066\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-7B\" x=\"9121\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-28\" x=\"9621\" y=\"0\"><\/use><g transform=\"translate(10010,0)\"><use xlink:href=\"#E233-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-2C\" x=\"11036\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"11481\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"11925\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"12370\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2C\" x=\"12815\" y=\"0\"><\/use><g transform=\"translate(13259,0)\"><use xlink:href=\"#E233-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMATHI-6E\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-29\" x=\"14356\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2208\" x=\"15022\" y=\"0\"><\/use><g transform=\"translate(15967,0)\"><use xlink:href=\"#E233-MJMATHI-45\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMAIN-31\" x=\"1043\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-D7\" x=\"17159\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"17937\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"18381\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-2E\" x=\"18826\" y=\"0\"><\/use><use xlink:href=\"#E233-MJMAIN-D7\" x=\"19271\" y=\"0\"><\/use><g transform=\"translate(20049,0)\"><use xlink:href=\"#E233-MJMATHI-45\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMATHI-6E\" x=\"1043\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-3A\" x=\"21589\" y=\"0\"><\/use><g transform=\"translate(22145,0)\"><use xlink:href=\"#E233-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-2208\" x=\"23338\" y=\"0\"><\/use><g transform=\"translate(24283,0)\"><use xlink:href=\"#E233-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E233-MJMATHI-69\" x=\"1200\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E233-MJMAIN-7D\" x=\"25476\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K = K_1 \\times ... \\times K_n =\\{(x_1,...,x_n) \\in E_1 \\times ... \\times E_n: x_i \\in K_i \\}<\/script><span> is a closed pointed cone with non-empty interior.<\/span><\/p><h3><a name=\"conic-linear-programming\" class=\"md-header-anchor\"><\/a><span>Conic Linear Programming<\/span><\/h3><p><strong><span>Standard Form\uff1a<\/span><\/strong><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.185ex\" height=\"2.461ex\" viewBox=\"0 -705.6 940.7 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E385-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E385-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E385-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E385-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E385-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E385-MJMATHI-70\" x=\"685\" y=\"-211\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v^* _p<\/script><span>\t\t<\/span><span>=<\/span><span>\t\t\t<\/span><span>inf<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.012ex\" height=\"1.41ex\" viewBox=\"0 -504.6 1727.4 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E235-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E235-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E235-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E235-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E235-MJMAIN-22C5\" x=\"655\" y=\"0\"><\/use><use xlink:href=\"#E235-MJMATHI-78\" x=\"1155\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">c \\cdot x<\/script><\/p><p><span>\t\t\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.927ex\" height=\"2.344ex\" viewBox=\"0 -755.9 4273.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E280-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E280-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E280-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><use xlink:href=\"#E280-MJMAIN-22C5\" x=\"1095\" y=\"0\"><\/use><use xlink:href=\"#E280-MJMATHI-78\" x=\"1595\" y=\"0\"><\/use><use xlink:href=\"#E280-MJMAIN-3D\" x=\"2445\" y=\"0\"><\/use><g transform=\"translate(3500,0)\"><use xlink:href=\"#E280-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E280-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">a_i \\cdot x = b_i<\/script><span>\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E296-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E296-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i=1,...,m<\/script><\/p><p><span>\t\t\t\t\t\t\t\t\t<\/span><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.279ex\" height=\"2.227ex\" viewBox=\"0 -755.9 3134.2 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E238-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E238-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E238-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E238-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E238-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(849,0)\"><use xlink:href=\"#E238-MJMAIN-2265\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E238-MJMATHI-4B\" x=\"1100\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E238-MJMAIN-30\" x=\"2634\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\ge _K 0<\/script><\/p><p><strong><span>Dual:<\/span><\/strong><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.218ex\" height=\"2.577ex\" viewBox=\"0 -705.6 954.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E239-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E239-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E239-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E239-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E239-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E239-MJMATHI-64\" x=\"685\" y=\"-462\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v^* _d<\/script><span>\t\t<\/span><span>=<\/span><span>\t\t\t<\/span><span>sup<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.539ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1523.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E271-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E271-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E271-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E271-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty<\/script><\/p><p><span>\t\t\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.733ex\" height=\"2.811ex\" viewBox=\"0 -856.4 7635.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E241-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E241-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E241-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E241-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E241-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E241-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E241-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><g transform=\"translate(2470,0)\"><use xlink:href=\"#E241-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E241-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(3304,0)\"><use xlink:href=\"#E241-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E241-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E241-MJMAIN-2B\" x=\"4399\" y=\"0\"><\/use><use xlink:href=\"#E241-MJMATHI-73\" x=\"5399\" y=\"0\"><\/use><use xlink:href=\"#E241-MJMAIN-3D\" x=\"6146\" y=\"0\"><\/use><use xlink:href=\"#E241-MJMATHI-63\" x=\"7202\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\sum ^m _{i=1} y_ia_i + s = c<\/script><\/p><p><span>\t\t\t\t\t\t\t\t\t\t<\/span><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.431ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3199.4 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E273-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E273-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E273-MJMAIN-2208\" x=\"774\" y=\"0\"><\/use><g transform=\"translate(1719,0)\"><use xlink:href=\"#E273-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E273-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y \\in R^m<\/script><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.805ex\" height=\"2.344ex\" viewBox=\"0 -755.9 3360.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E243-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E243-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E243-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E243-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E243-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E243-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(746,0)\"><use xlink:href=\"#E243-MJMAIN-2265\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(778,-178)\"><use transform=\"scale(0.707)\" xlink:href=\"#E243-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E243-MJMAIN-2217\" x=\"1274\" y=\"579\"><\/use><\/g><\/g><use xlink:href=\"#E243-MJMAIN-30\" x=\"2860\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">s \\ge _{K^*} 0<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"44.972ex\" height=\"2.577ex\" viewBox=\"0 -806.1 19362.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E244-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-45\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-72\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMATHI-6C\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path stroke-width=\"0\" id=\"E244-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E244-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E244-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><use xlink:href=\"#E244-MJMAIN-3D\" x=\"1632\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-7B\" x=\"2688\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-77\" x=\"3188\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-2208\" x=\"4181\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-45\" x=\"5126\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-3A\" x=\"6168\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-78\" x=\"6724\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-22C5\" x=\"7518\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-77\" x=\"8018\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-2265\" x=\"9012\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-30\" x=\"10068\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-66\" x=\"11568\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-6F\" x=\"12118\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-72\" x=\"12603\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-61\" x=\"14054\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-6C\" x=\"14583\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-6C\" x=\"14881\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-78\" x=\"16179\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-2208\" x=\"17029\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMATHI-4B\" x=\"17973\" y=\"0\"><\/use><use xlink:href=\"#E244-MJMAIN-7D\" x=\"18862\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K^* = \\{w \\in E: x \\cdot w \\ge 0 \\quad for \\quad all \\quad x \\in K\\}<\/script><\/p><p><strong><span>Proposition 2:<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.937ex\" height=\"2.227ex\" viewBox=\"0 -755.9 2986.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E245-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E245-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E245-MJMATHI-45\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E245-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E245-MJMAIN-2286\" x=\"1166\" y=\"0\"><\/use><use xlink:href=\"#E245-MJMATHI-45\" x=\"2222\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K \\sube E<\/script><span> be a non-empty set. Then the following hold:<\/span><\/p><p><span>1) The set <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.146ex\" height=\"1.877ex\" viewBox=\"0 -755.9 1354.6 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E252-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E252-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E252-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E252-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K^*<\/script><span> is a closed convex cone<\/span><\/p><p><span>2) If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.065ex\" height=\"1.877ex\" viewBox=\"0 -755.9 889 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E251-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E251-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K<\/script><span> is a closed convex cone, then so is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.146ex\" height=\"1.877ex\" viewBox=\"0 -755.9 1354.6 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E252-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E252-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E252-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E252-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K^*<\/script><\/p><p><span>3) If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.065ex\" height=\"1.877ex\" viewBox=\"0 -755.9 889 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E251-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E251-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K<\/script><span> has a non-empty interior, then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.146ex\" height=\"1.877ex\" viewBox=\"0 -755.9 1354.6 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E252-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E252-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E252-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E252-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K^*<\/script><span> is pointed<\/span><\/p><p><span>4) If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.065ex\" height=\"1.877ex\" viewBox=\"0 -755.9 889 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E251-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E251-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K<\/script><span> is a closed pointed cone, then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.146ex\" height=\"1.877ex\" viewBox=\"0 -755.9 1354.6 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E252-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E252-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E252-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E252-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K^*<\/script><span> has a mon-empty interior<\/span><\/p><p><strong><span>Examples:<\/span><\/strong><\/p><p><strong><span>Linear Programming:<\/span><\/strong><\/p><p><span>(P)<\/span><span>\t\t\t<\/span><span>inf<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.723ex\" height=\"2.344ex\" viewBox=\"0 -906.7 1602.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E261-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E261-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E261-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><use xlink:href=\"#E261-MJMATHI-78\" x=\"1030\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">c^Tx<\/script><\/p><p><span>\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.838ex\" height=\"3.044ex\" viewBox=\"0 -906.7 3805.3 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E254-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E254-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E254-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E254-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E254-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E254-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E254-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E254-MJMATHI-54\" x=\"748\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E254-MJMATHI-69\" x=\"748\" y=\"-429\"><\/use><use xlink:href=\"#E254-MJMATHI-78\" x=\"1126\" y=\"0\"><\/use><use xlink:href=\"#E254-MJMAIN-3D\" x=\"1976\" y=\"0\"><\/use><g transform=\"translate(3032,0)\"><use xlink:href=\"#E254-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E254-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">a^T_i x = b_i<\/script><span>\t\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E296-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E296-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i=1,...,m<\/script><\/p><p><span>\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.441ex\" height=\"2.694ex\" viewBox=\"0 -755.9 3203.7 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E256-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E256-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E256-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E256-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E256-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E256-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E256-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E256-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E256-MJMATHI-6E\" x=\"1073\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E256-MJMAIN-2B\" x=\"1073\" y=\"-351\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in R^n _+<\/script><\/p><p><span>(D)<\/span><span>\t\t\t<\/span><span>sup<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.539ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1523.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E271-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E271-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E271-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E271-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty<\/script><\/p><p><span>\t\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.733ex\" height=\"2.811ex\" viewBox=\"0 -856.4 7635.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E258-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E258-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E258-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E258-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E258-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E258-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E258-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><g transform=\"translate(2470,0)\"><use xlink:href=\"#E258-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E258-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(3304,0)\"><use xlink:href=\"#E258-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E258-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E258-MJMAIN-2B\" x=\"4399\" y=\"0\"><\/use><use xlink:href=\"#E258-MJMATHI-73\" x=\"5399\" y=\"0\"><\/use><use xlink:href=\"#E258-MJMAIN-3D\" x=\"6146\" y=\"0\"><\/use><use xlink:href=\"#E258-MJMATHI-63\" x=\"7202\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\sum ^m _{i=1} y_ia_i +s =c<\/script><\/p><p><span>\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.431ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3199.4 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E273-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E273-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E273-MJMAIN-2208\" x=\"774\" y=\"0\"><\/use><g transform=\"translate(1719,0)\"><use xlink:href=\"#E273-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E273-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y \\in R^m<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.202ex\" height=\"2.694ex\" viewBox=\"0 -755.9 3100.7 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E260-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E260-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E260-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E260-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E260-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E260-MJMATHI-73\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E260-MJMAIN-2208\" x=\"746\" y=\"0\"><\/use><g transform=\"translate(1691,0)\"><use xlink:href=\"#E260-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E260-MJMATHI-6E\" x=\"1073\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E260-MJMAIN-2B\" x=\"1073\" y=\"-351\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">s \\in R^n _+<\/script><\/p><p><strong><span>Second-Order Cone Programming (SOCP):<\/span><\/strong><\/p><p><span>(P)<\/span><span>\t\t\t<\/span><span>inf<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.723ex\" height=\"2.344ex\" viewBox=\"0 -906.7 1602.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E261-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E261-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E261-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E261-MJMATHI-54\" x=\"612\" y=\"513\"><\/use><use xlink:href=\"#E261-MJMATHI-78\" x=\"1030\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">c^Tx<\/script><\/p><p><span>\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.838ex\" height=\"3.044ex\" viewBox=\"0 -906.7 3805.3 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E262-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E262-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E262-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E262-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E262-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E262-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E262-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E262-MJMATHI-54\" x=\"748\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E262-MJMATHI-69\" x=\"748\" y=\"-429\"><\/use><use xlink:href=\"#E262-MJMATHI-78\" x=\"1126\" y=\"0\"><\/use><use xlink:href=\"#E262-MJMAIN-3D\" x=\"1976\" y=\"0\"><\/use><g transform=\"translate(3032,0)\"><use xlink:href=\"#E262-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E262-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">a^T_ix = b_i<\/script><span>\t\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E334-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E334-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m<\/script><\/p><p><span>\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.322ex\" height=\"2.694ex\" viewBox=\"0 -906.7 4013.5 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E264-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E264-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E264-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E264-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E264-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E264-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E264-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E264-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E264-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(791,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E264-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E264-MJMAIN-2B\" x=\"600\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E264-MJMAIN-31\" x=\"1378\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in Q^{n+1}<\/script><\/p><p><span>(D)<\/span><span>\t\t\t<\/span><span>sup<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.539ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1523.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E271-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E271-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E271-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E271-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty<\/script><\/p><p><span>\t\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"27.009ex\" height=\"2.811ex\" viewBox=\"0 -906.7 11628.7 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E266-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-51\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E266-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E266-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMATHI-76\" x=\"389\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMAIN-2212\" x=\"1096\" y=\"0\"><\/use><g transform=\"translate(2096,0)\"><use xlink:href=\"#E266-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E266-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E266-MJMATHI-79\" x=\"3266\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMAIN-2C\" x=\"3763\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMATHI-64\" x=\"4207\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMAIN-2212\" x=\"4953\" y=\"0\"><\/use><g transform=\"translate(5953,0)\"><use xlink:href=\"#E266-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E266-MJMATHI-54\" x=\"1060\" y=\"513\"><\/use><\/g><use xlink:href=\"#E266-MJMATHI-79\" x=\"7301\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMAIN-29\" x=\"7798\" y=\"0\"><\/use><use xlink:href=\"#E266-MJMAIN-2208\" x=\"8464\" y=\"0\"><\/use><g transform=\"translate(9409,0)\"><use xlink:href=\"#E266-MJMATHI-51\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(791,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E266-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E266-MJMAIN-2B\" x=\"600\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E266-MJMAIN-31\" x=\"1378\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">(v - u^Ty, d - A^Ty) \\in Q^{n+1}<\/script><\/p><p><strong><span>Semidefinite Programming (SDP):<\/span><\/strong><\/p><p><span>(P)<\/span><span>\t\t\t<\/span><span>inf<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.422ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2334.4 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E267-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E267-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E267-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E267-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E267-MJMAIN-22C5\" x=\"982\" y=\"0\"><\/use><use xlink:href=\"#E267-MJMATHI-58\" x=\"1482\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C \\cdot X<\/script><\/p><p><span>\t\t\t<\/span><span>subject to<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.09ex\" height=\"2.461ex\" viewBox=\"0 -806.1 4774.9 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E268-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E268-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E268-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E268-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E268-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E268-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E268-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E268-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><use xlink:href=\"#E268-MJMAIN-22C5\" x=\"1316\" y=\"0\"><\/use><use xlink:href=\"#E268-MJMATHI-58\" x=\"1816\" y=\"0\"><\/use><use xlink:href=\"#E268-MJMAIN-3D\" x=\"2946\" y=\"0\"><\/use><g transform=\"translate(4001,0)\"><use xlink:href=\"#E268-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E268-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">A_i \\cdot X = b_i<\/script><span>\t\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E334-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E334-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m<\/script><\/p><p><span>\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.752ex\" height=\"2.811ex\" viewBox=\"0 -806.1 3337.7 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E270-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E270-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E270-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E270-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E270-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E270-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E270-MJMAIN-2208\" x=\"1129\" y=\"0\"><\/use><g transform=\"translate(2074,0)\"><use xlink:href=\"#E270-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E270-MJMATHI-6E\" x=\"925\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E270-MJMAIN-2B\" x=\"866\" y=\"-351\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">X \\in S^n _+<\/script><\/p><p><span>(D)<\/span><span>\t\t\t<\/span><span>sup<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.539ex\" height=\"2.811ex\" viewBox=\"0 -906.7 1523.8 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E271-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E271-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E271-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E271-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><use xlink:href=\"#E271-MJMATHI-79\" x=\"1026\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b^Ty<\/script><\/p><p><span>\t\t\t<\/span><span>subject to<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"19.415ex\" height=\"2.811ex\" viewBox=\"0 -856.4 8359.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E272-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E272-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E272-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E272-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E272-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E272-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E272-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><g transform=\"translate(2470,0)\"><use xlink:href=\"#E272-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E272-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(3304,0)\"><use xlink:href=\"#E272-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E272-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E272-MJMAIN-2B\" x=\"4620\" y=\"0\"><\/use><use xlink:href=\"#E272-MJMATHI-53\" x=\"5620\" y=\"0\"><\/use><use xlink:href=\"#E272-MJMAIN-3D\" x=\"6543\" y=\"0\"><\/use><use xlink:href=\"#E272-MJMATHI-43\" x=\"7599\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\sum ^m _{i=1} y_iA_i + S = C<\/script><\/p><p><span>\t\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.431ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3199.4 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E273-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E273-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E273-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E273-MJMAIN-2208\" x=\"774\" y=\"0\"><\/use><g transform=\"translate(1719,0)\"><use xlink:href=\"#E273-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E273-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">y \\in R^m<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.271ex\" height=\"2.811ex\" viewBox=\"0 -806.1 3130.7 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E274-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E274-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E274-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E274-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E274-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E274-MJMAIN-2208\" x=\"922\" y=\"0\"><\/use><g transform=\"translate(1867,0)\"><use xlink:href=\"#E274-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E274-MJMATHI-6E\" x=\"925\" y=\"498\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E274-MJMAIN-2B\" x=\"866\" y=\"-351\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">S \\in S^n _+<\/script><\/p><p><strong><span>Theorem 1: (CLP Weak Duality)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.233ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2683.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E275-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E275-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E275-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E275-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E275-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E275-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E275-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E275-MJMATHI-4B\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in K<\/script><span> be feasible for (P) and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.693ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7617.9 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E276-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E276-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E276-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E276-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E276-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E276-MJMAIN-2C\" x=\"949\" y=\"0\"><\/use><g transform=\"translate(1393,0)\"><use xlink:href=\"#E276-MJMATHI-73\" x=\"15\" y=\"0\"><\/use><use xlink:href=\"#E276-MJMAIN-AF\" x=\"55\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E276-MJMAIN-29\" x=\"1949\" y=\"0\"><\/use><use xlink:href=\"#E276-MJMAIN-2208\" x=\"2616\" y=\"0\"><\/use><g transform=\"translate(3560,0)\"><use xlink:href=\"#E276-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E276-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E276-MJMAIN-D7\" x=\"5263\" y=\"0\"><\/use><g transform=\"translate(6263,0)\"><use xlink:href=\"#E276-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E276-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{y}, \\bar{s}) \\in R^m \\times K^*<\/script><span> be feasible for (D). Then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.795ex\" height=\"2.811ex\" viewBox=\"0 -906.7 4648 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E277-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E277-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E277-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E277-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><g transform=\"translate(1026,0)\"><use xlink:href=\"#E277-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E277-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E277-MJMAIN-2264\" x=\"1864\" y=\"0\"><\/use><use xlink:href=\"#E277-MJMATHI-63\" x=\"2920\" y=\"0\"><\/use><use xlink:href=\"#E277-MJMAIN-22C5\" x=\"3575\" y=\"0\"><\/use><g transform=\"translate(4075,0)\"><use xlink:href=\"#E277-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E277-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">b^T\\bar{y} \\le c \\cdot \\bar{x}<\/script><\/p><p><strong><span>Theorem 2: (CLP Farkas Lemma)<\/span><\/strong><span> Suppose that there exists <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.578ex\" height=\"2.461ex\" viewBox=\"0 -755.9 3262.6 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E278-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E278-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E278-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E278-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><use xlink:href=\"#E278-MJMAIN-2208\" x=\"837\" y=\"0\"><\/use><g transform=\"translate(1782,0)\"><use xlink:href=\"#E278-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E278-MJMATHI-6D\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{y} \\in R^m<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"22.885ex\" height=\"2.811ex\" viewBox=\"0 -856.4 9853.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E279-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E279-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E279-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(944,0)\"><use xlink:href=\"#E279-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(3414,0)\"><use xlink:href=\"#E279-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E279-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMATHI-69\" x=\"792\" y=\"-340\"><\/use><\/g><g transform=\"translate(4319,0)\"><use xlink:href=\"#E279-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E279-MJMAIN-2208\" x=\"5469\" y=\"0\"><\/use><use xlink:href=\"#E279-MJMATHI-69\" x=\"6414\" y=\"0\"><\/use><use xlink:href=\"#E279-MJMATHI-6E\" x=\"6759\" y=\"0\"><\/use><use xlink:href=\"#E279-MJMATHI-74\" x=\"7359\" y=\"0\"><\/use><use xlink:href=\"#E279-MJMAIN-28\" x=\"7720\" y=\"0\"><\/use><g transform=\"translate(8109,0)\"><use xlink:href=\"#E279-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E279-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><use xlink:href=\"#E279-MJMAIN-29\" x=\"9464\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-\\sum ^m _{i=1} \\bar{y}_i a_i \\in int(K^*)<\/script><span>, then exactly one of the following holds:<\/span><\/p><p><span>1)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.927ex\" height=\"2.344ex\" viewBox=\"0 -755.9 4273.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E280-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E280-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E280-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E280-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><use xlink:href=\"#E280-MJMAIN-22C5\" x=\"1095\" y=\"0\"><\/use><use xlink:href=\"#E280-MJMATHI-78\" x=\"1595\" y=\"0\"><\/use><use xlink:href=\"#E280-MJMAIN-3D\" x=\"2445\" y=\"0\"><\/use><g transform=\"translate(3500,0)\"><use xlink:href=\"#E280-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E280-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">a_i \\cdot x = b_i<\/script><span>\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E334-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E334-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.883ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2963.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E282-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E282-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E282-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E282-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E282-MJMAIN-2208\" x=\"1129\" y=\"0\"><\/use><use xlink:href=\"#E282-MJMATHI-4B\" x=\"2074\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X \\in K<\/script><\/p><p><span>2)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"26.875ex\" height=\"2.928ex\" viewBox=\"0 -906.7 11571.2 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E283-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path stroke-width=\"0\" id=\"E283-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E283-MJMAIN-2212\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(944,0)\"><use xlink:href=\"#E283-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMATHI-6D\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(3414,0)\"><use xlink:href=\"#E283-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E283-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMATHI-69\" x=\"792\" y=\"-340\"><\/use><\/g><g transform=\"translate(4319,0)\"><use xlink:href=\"#E283-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E283-MJMAIN-2208\" x=\"5469\" y=\"0\"><\/use><g transform=\"translate(6414,0)\"><use xlink:href=\"#E283-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMAIN-2217\" x=\"1274\" y=\"513\"><\/use><\/g><use xlink:href=\"#E283-MJMAIN-2C\" x=\"7769\" y=\"0\"><\/use><g transform=\"translate(8213,0)\"><use xlink:href=\"#E283-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E283-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><\/g><use xlink:href=\"#E283-MJMATHI-79\" x=\"9240\" y=\"0\"><\/use><use xlink:href=\"#E283-MJMAIN-3E\" x=\"10015\" y=\"0\"><\/use><use xlink:href=\"#E283-MJMAIN-30\" x=\"11071\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">-\\sum ^m _{i=1} \\bar{y}_i a_i \\in K^*, b^Ty > 0<\/script><\/p><p><strong><span>Theorem 3: (CLP Strong Duality)<\/span><\/strong><span>\t<\/span><\/p><p><span>1) Suppose (P) is bounded below and strictly feasible. Then we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.5ex\" height=\"2.577ex\" viewBox=\"0 -705.6 3229 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E287-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E287-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMATHI-70\" x=\"685\" y=\"-211\"><\/use><use xlink:href=\"#E287-MJMAIN-3D\" x=\"1218\" y=\"0\"><\/use><g transform=\"translate(2274,0)\"><use xlink:href=\"#E287-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMATHI-64\" x=\"685\" y=\"-462\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">v_p ^* = v^* _d<\/script><span> and there exists a feasible solution <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.431ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2338.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E291-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E291-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E291-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E291-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E291-MJMAIN-2C\" x=\"949\" y=\"0\"><\/use><g transform=\"translate(1393,0)\"><use xlink:href=\"#E291-MJMATHI-73\" x=\"15\" y=\"0\"><\/use><use xlink:href=\"#E291-MJMAIN-AF\" x=\"55\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E291-MJMAIN-29\" x=\"1949\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{y}, \\bar{s})<\/script><span> to (D) such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.283ex\" height=\"3.044ex\" viewBox=\"0 -906.7 6149.6 1310.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E286-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E286-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E286-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E286-MJMATHI-54\" x=\"606\" y=\"513\"><\/use><g transform=\"translate(1026,0)\"><use xlink:href=\"#E286-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E286-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E286-MJMAIN-3D\" x=\"1864\" y=\"0\"><\/use><g transform=\"translate(2920,0)\"><use xlink:href=\"#E286-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E286-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E286-MJMATHI-70\" x=\"685\" y=\"-211\"><\/use><\/g><use xlink:href=\"#E286-MJMAIN-3D\" x=\"4138\" y=\"0\"><\/use><g transform=\"translate(5194,0)\"><use xlink:href=\"#E286-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E286-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E286-MJMATHI-64\" x=\"685\" y=\"-462\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">b^T\\bar{y} = v_p ^* = v^* _d<\/script><\/p><p><span>2) Suppose (D) is bonded above and stictly feasible. Then we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.5ex\" height=\"2.577ex\" viewBox=\"0 -705.6 3229 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E287-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E287-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E287-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMATHI-70\" x=\"685\" y=\"-211\"><\/use><use xlink:href=\"#E287-MJMAIN-3D\" x=\"1218\" y=\"0\"><\/use><g transform=\"translate(2274,0)\"><use xlink:href=\"#E287-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E287-MJMATHI-64\" x=\"685\" y=\"-462\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">v_p ^* = v^* _d<\/script><span> and there exists a feasible solution <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.431ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2338.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E291-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E291-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E291-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E291-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E291-MJMAIN-2C\" x=\"949\" y=\"0\"><\/use><g transform=\"translate(1393,0)\"><use xlink:href=\"#E291-MJMATHI-73\" x=\"15\" y=\"0\"><\/use><use xlink:href=\"#E291-MJMAIN-AF\" x=\"55\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E291-MJMAIN-29\" x=\"1949\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{y}, \\bar{s})<\/script><span> to (D) such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.609ex\" height=\"2.577ex\" viewBox=\"0 -705.6 6290 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E289-MJMATHI-63\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E289-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E289-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E289-MJMAIN-22C5\" x=\"655\" y=\"0\"><\/use><g transform=\"translate(1155,0)\"><use xlink:href=\"#E289-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E289-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E289-MJMAIN-3D\" x=\"2005\" y=\"0\"><\/use><g transform=\"translate(3061,0)\"><use xlink:href=\"#E289-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E289-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E289-MJMATHI-70\" x=\"685\" y=\"-211\"><\/use><\/g><use xlink:href=\"#E289-MJMAIN-3D\" x=\"4279\" y=\"0\"><\/use><g transform=\"translate(5335,0)\"><use xlink:href=\"#E289-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E289-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E289-MJMATHI-64\" x=\"685\" y=\"-462\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">c \\cdot \\bar{x} = v_p ^* = v^* _d<\/script><\/p><p><span>3) Suppose (P) and (D) are bounded and strictly feasible. Then given a feasible solution <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> to (p) and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.431ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2338.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E291-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E291-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E291-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E291-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use xlink:href=\"#E291-MJMAIN-AF\" x=\"60\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E291-MJMAIN-2C\" x=\"949\" y=\"0\"><\/use><g transform=\"translate(1393,0)\"><use xlink:href=\"#E291-MJMATHI-73\" x=\"15\" y=\"0\"><\/use><use xlink:href=\"#E291-MJMAIN-AF\" x=\"55\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E291-MJMAIN-29\" x=\"1949\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{y}, \\bar{s})<\/script><span> to (D), the following are equivalent:<\/span><\/p><ul><li><span>They are optimal solutions<\/span><\/li><li><span>The duality gap is zero<\/span><\/li><li><span>We have complementary slackness: <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.555ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3683.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.051ex; margin-bottom: -0.187ex;\"><defs><path stroke-width=\"0\" id=\"E292-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E292-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E292-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E292-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E292-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E292-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E292-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E292-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E292-MJMAIN-22C5\" x=\"794\" y=\"0\"><\/use><g transform=\"translate(1294,0)\"><use xlink:href=\"#E292-MJMATHI-73\" x=\"15\" y=\"0\"><\/use><use xlink:href=\"#E292-MJMAIN-AF\" x=\"55\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E292-MJMAIN-3D\" x=\"2127\" y=\"0\"><\/use><use xlink:href=\"#E292-MJMAIN-30\" x=\"3183\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\cdot \\bar{s} = 0<\/script><\/li><\/ul><h2><a name=\"handout-6----some-applications-of-conic-linear-programming\" class=\"md-header-anchor\"><\/a><span>Handout 6 Some Applications of Conic Linear Programming<\/span><\/h2><h3><a name=\"quadratically-constrained-quadratic-optimization\" class=\"md-header-anchor\"><\/a><span>Quadratically Constrained Quadratic Optimization<\/span><\/h3><p><strong><span>Original Problem:<\/span><\/strong><\/p><p><span>minimize<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.811ex\" height=\"2.344ex\" viewBox=\"0 -906.7 2501.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E293-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E293-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E293-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E293-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E293-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><use xlink:href=\"#E293-MJMATHI-43\" x=\"1169\" y=\"0\"><\/use><use xlink:href=\"#E293-MJMATHI-78\" x=\"1929\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x^TCx<\/script><span>\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span><span>`<\/span><span>\t\t\t\t\t\t\t\t\t<\/span><span>minimize<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.422ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2334.4 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E300-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E300-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E300-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E300-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E300-MJMAIN-22C5\" x=\"982\" y=\"0\"><\/use><use xlink:href=\"#E300-MJMATHI-58\" x=\"1482\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C\\cdot X<\/script><\/p><p><span>subject to<\/span><span>\t<\/span><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.479ex\" height=\"2.694ex\" viewBox=\"0 -906.7 4942.3 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E295-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E295-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E295-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E295-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E295-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E295-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E295-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E295-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><g transform=\"translate(1169,0)\"><use xlink:href=\"#E295-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E295-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E295-MJMATHI-78\" x=\"2263\" y=\"0\"><\/use><use xlink:href=\"#E295-MJMAIN-2265\" x=\"3113\" y=\"0\"><\/use><g transform=\"translate(4169,0)\"><use xlink:href=\"#E295-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E295-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">x^TA_ix \\ge b_i<\/script><span>\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E296-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E296-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E296-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E296-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i=1,...,m<\/script><span>\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span><span>subject to <\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.09ex\" height=\"2.461ex\" viewBox=\"0 -806.1 4774.9 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E301-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E301-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E301-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><use xlink:href=\"#E301-MJMAIN-22C5\" x=\"1316\" y=\"0\"><\/use><use xlink:href=\"#E301-MJMATHI-58\" x=\"1816\" y=\"0\"><\/use><use xlink:href=\"#E301-MJMAIN-2265\" x=\"2946\" y=\"0\"><\/use><g transform=\"translate(4001,0)\"><use xlink:href=\"#E301-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E301-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">A_i \\cdot X \\ge b_i<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.237ex\" height=\"2.227ex\" viewBox=\"0 -755.9 2685.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E302-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E302-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E302-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E302-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E302-MJMAIN-2265\" x=\"1129\" y=\"0\"><\/use><use xlink:href=\"#E302-MJMAIN-30\" x=\"2185\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X \\ge 0<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.924ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5564.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E299-MJMATHI-72\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E299-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E299-MJMATHI-72\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMATHI-61\" x=\"451\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMATHI-6E\" x=\"980\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMATHI-6B\" x=\"1580\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMAIN-28\" x=\"2101\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMATHI-58\" x=\"2490\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMAIN-29\" x=\"3342\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMAIN-2264\" x=\"4008\" y=\"0\"><\/use><use xlink:href=\"#E299-MJMAIN-31\" x=\"5064\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">rank(X)\\le 1<\/script><\/p><p><strong><span>Semidefinite Relaxation:<\/span><\/strong><\/p><p><span>minimize<\/span><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.422ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2334.4 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E300-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E300-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E300-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E300-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E300-MJMAIN-22C5\" x=\"982\" y=\"0\"><\/use><use xlink:href=\"#E300-MJMATHI-58\" x=\"1482\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">C\\cdot X<\/script><\/p><p><span>subject to <\/span><span>\t<\/span><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.09ex\" height=\"2.461ex\" viewBox=\"0 -806.1 4774.9 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E301-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMAIN-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E301-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E301-MJMATHI-41\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E301-MJMATHI-69\" x=\"1060\" y=\"-213\"><\/use><use xlink:href=\"#E301-MJMAIN-22C5\" x=\"1316\" y=\"0\"><\/use><use xlink:href=\"#E301-MJMATHI-58\" x=\"1816\" y=\"0\"><\/use><use xlink:href=\"#E301-MJMAIN-2265\" x=\"2946\" y=\"0\"><\/use><g transform=\"translate(4001,0)\"><use xlink:href=\"#E301-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E301-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">A_i \\cdot X \\ge b_i<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.237ex\" height=\"2.227ex\" viewBox=\"0 -755.9 2685.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E302-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E302-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E302-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E302-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E302-MJMAIN-2265\" x=\"1129\" y=\"0\"><\/use><use xlink:href=\"#E302-MJMAIN-30\" x=\"2185\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X \\ge 0<\/script><\/p><h2><a name=\"handout-7-\toptimality-conditions-and-lagrangian-duality\" class=\"md-header-anchor\"><\/a><span>handout 7 <\/span><span>\t<\/span><span>Optimality Conditions and Lagrangian Duality<\/span><\/h2><h3><a name=\"unconstrained-optimizationm-problems\" class=\"md-header-anchor\"><\/a><span>Unconstrained Optimizationm Problems<\/span><\/h3><p><strong><span>Proposition 1:<\/span><\/strong><span>\t<\/span><span>Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.57ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4981.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E325-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E325-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMAIN-3A\" x=\"827\" y=\"0\"><\/use><g transform=\"translate(1383,0)\"><use xlink:href=\"#E325-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E325-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E325-MJMAIN-2192\" x=\"2944\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMATHI-52\" x=\"4222\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f: R^n \\rightarrow R<\/script><span> is continuously differentiable at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E397-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E397-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E397-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E397-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E397-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E397-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in R^n<\/script><span>. If there exists a <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.035ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3028.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E305-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E305-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E305-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E305-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E305-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E305-MJMAIN-2208\" x=\"800\" y=\"0\"><\/use><g transform=\"translate(1745,0)\"><use xlink:href=\"#E305-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E305-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">d \\in R^n<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.209ex\" height=\"2.811ex\" viewBox=\"0 -906.7 5687.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E306-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E306-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E306-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E306-MJMATHI-66\" x=\"833\" y=\"0\"><\/use><use xlink:href=\"#E306-MJMAIN-28\" x=\"1383\" y=\"0\"><\/use><g transform=\"translate(1772,0)\"><use xlink:href=\"#E306-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E306-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><g transform=\"translate(2344,0)\"><use xlink:href=\"#E306-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E306-MJMATHI-54\" x=\"550\" y=\"513\"><\/use><\/g><use xlink:href=\"#E306-MJMATHI-64\" x=\"3330\" y=\"0\"><\/use><use xlink:href=\"#E306-MJMAIN-3C\" x=\"4131\" y=\"0\"><\/use><use xlink:href=\"#E306-MJMAIN-30\" x=\"5187\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\nabla f (\\bar{x})^Td < 0<\/script><span>, then there exists an <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.798ex\" height=\"2.344ex\" viewBox=\"0 -755.9 2927.1 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E307-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E307-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E307-MJMAIN-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E307-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E307-MJMAIN-30\" x=\"905\" y=\"-213\"><\/use><use xlink:href=\"#E307-MJMAIN-3E\" x=\"1371\" y=\"0\"><\/use><use xlink:href=\"#E307-MJMAIN-30\" x=\"2427\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha _0 >0<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.464ex\" height=\"2.577ex\" viewBox=\"0 -806.1 7519 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E308-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E308-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E308-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><g transform=\"translate(939,0)\"><use xlink:href=\"#E308-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E308-MJMAIN-2B\" x=\"1733\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMATHI-3B1\" x=\"2733\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMATHI-64\" x=\"3373\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMAIN-29\" x=\"3896\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMAIN-3C\" x=\"4563\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMATHI-66\" x=\"5618\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMAIN-28\" x=\"6168\" y=\"0\"><\/use><g transform=\"translate(6557,0)\"><use xlink:href=\"#E308-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E308-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E308-MJMAIN-29\" x=\"7129\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(\\bar{x} + \\alpha d) < f(\\bar{x})<\/script><span> for all <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.867ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4678.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E309-MJMATHI-3B1\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path stroke-width=\"0\" id=\"E309-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E309-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E309-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E309-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E309-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E309-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E309-MJMAIN-2208\" x=\"917\" y=\"0\"><\/use><use xlink:href=\"#E309-MJMAIN-28\" x=\"1862\" y=\"0\"><\/use><use xlink:href=\"#E309-MJMAIN-30\" x=\"2251\" y=\"0\"><\/use><use xlink:href=\"#E309-MJMAIN-2C\" x=\"2751\" y=\"0\"><\/use><g transform=\"translate(3196,0)\"><use xlink:href=\"#E309-MJMATHI-3B1\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E309-MJMAIN-30\" x=\"905\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E309-MJMAIN-29\" x=\"4289\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\alpha \\in (0,\\alpha_0)<\/script><span>. In other words, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.215ex\" height=\"1.994ex\" viewBox=\"0 -755.9 523 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E310-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E310-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">d<\/script><span> is a descent direction of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.277ex\" height=\"2.577ex\" viewBox=\"0 -806.1 550 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E311-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E311-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f<\/script><span> at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span>.<\/span><\/p><p><strong><span>Corollary 1: (First Order Necessary Condition for Unconstrained Optimization)<\/span><\/strong><span>\t<\/span><span>Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.57ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4981.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E325-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E325-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMAIN-3A\" x=\"827\" y=\"0\"><\/use><g transform=\"translate(1383,0)\"><use xlink:href=\"#E325-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E325-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E325-MJMAIN-2192\" x=\"2944\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMATHI-52\" x=\"4222\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f: R^n \\rightarrow R<\/script><span> is continuously differentiable at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E397-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E397-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E397-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E397-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E397-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E397-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in R^n<\/script><span>. If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> is a local minimum, then we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.606ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4566.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E327-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E327-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMATHI-66\" x=\"833\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-28\" x=\"1383\" y=\"0\"><\/use><g transform=\"translate(1772,0)\"><use xlink:href=\"#E327-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E327-MJMAIN-29\" x=\"2344\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-3D\" x=\"3010\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-30\" x=\"4066\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\nabla f(\\bar{x}) = 0<\/script><span>. In particular, we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"28.761ex\" height=\"2.811ex\" viewBox=\"0 -906.7 12383.3 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E317-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path stroke-width=\"0\" id=\"E317-MJMAIN-2205\" d=\"M331 696Q335 708 339 722T345 744T350 759T357 769T367 772Q374 772 381 767T388 754Q388 746 377 712L366 673L378 661Q460 575 460 344Q460 281 456 234T432 126T373 27Q319 -22 250 -22Q214 -22 180 -7Q168 -3 168 -4L159 -33Q148 -71 142 -75Q138 -78 132 -78Q124 -78 118 -72T111 -60Q111 -52 122 -18L133 21L125 29Q39 111 39 344Q39 596 137 675Q187 716 251 716Q265 716 278 714T296 710T315 703T331 696ZM276 676Q264 679 246 679Q196 679 159 631Q134 597 128 536T121 356Q121 234 127 174T151 80L234 366Q253 430 275 506T308 618L318 654Q318 656 294 669L276 676ZM181 42Q207 16 250 16Q291 16 324 47Q354 78 366 136T378 356Q378 470 372 528T349 616L348 613Q348 611 264 326L181 42Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E317-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMATHI-64\" x=\"500\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-2208\" x=\"1300\" y=\"0\"><\/use><g transform=\"translate(2245,0)\"><use xlink:href=\"#E317-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E317-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E317-MJMAIN-3A\" x=\"3806\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-2207\" x=\"4362\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMATHI-66\" x=\"5195\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-28\" x=\"5745\" y=\"0\"><\/use><g transform=\"translate(6134,0)\"><use xlink:href=\"#E317-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><g transform=\"translate(6706,0)\"><use xlink:href=\"#E317-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E317-MJMATHI-54\" x=\"550\" y=\"513\"><\/use><\/g><use xlink:href=\"#E317-MJMATHI-64\" x=\"7693\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-3C\" x=\"8493\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-30\" x=\"9549\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-7D\" x=\"10049\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-2260\" x=\"10827\" y=\"0\"><\/use><use xlink:href=\"#E317-MJMAIN-2205\" x=\"11883\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\{d \\in R^n: \\nabla f(\\bar{x}) ^Td < 0 \\} \\neq \\empty<\/script><span>.<\/span><\/p><p><strong><span>Proposition 2:<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.576ex\" height=\"2.344ex\" viewBox=\"0 -806.1 3261.8 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E318-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E318-MJMAIN-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path stroke-width=\"0\" id=\"E318-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E318-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E318-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E318-MJMAIN-2286\" x=\"922\" y=\"0\"><\/use><g transform=\"translate(1978,0)\"><use xlink:href=\"#E318-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E318-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">S \\sube R^n<\/script><span> be an open convex set. Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.57ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4981.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E325-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E325-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMAIN-3A\" x=\"827\" y=\"0\"><\/use><g transform=\"translate(1383,0)\"><use xlink:href=\"#E325-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E325-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E325-MJMAIN-2192\" x=\"2944\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMATHI-52\" x=\"4222\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f: R^n \\rightarrow R<\/script><span> is convex on <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> and continuously differentiable at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.666ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2439.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E368-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E368-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E368-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMATHI-53\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in S<\/script><span>. Then, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> is a global minimum in <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.498ex\" height=\"2.11ex\" viewBox=\"0 -806.1 645 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E323-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E323-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S<\/script><span> iff <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.606ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4566.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E327-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E327-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMATHI-66\" x=\"833\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-28\" x=\"1383\" y=\"0\"><\/use><g transform=\"translate(1772,0)\"><use xlink:href=\"#E327-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E327-MJMAIN-29\" x=\"2344\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-3D\" x=\"3010\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-30\" x=\"4066\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\nabla f(\\bar{x}) = 0<\/script><span>.<\/span><\/p><p><strong><span>Proposition 4: (Second Order Sufficient Condition for Unconstrained Optimization)<\/span><\/strong><span>\t<\/span><span>Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.57ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4981.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E325-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E325-MJMAIN-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E325-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMAIN-3A\" x=\"827\" y=\"0\"><\/use><g transform=\"translate(1383,0)\"><use xlink:href=\"#E325-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E325-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><use xlink:href=\"#E325-MJMAIN-2192\" x=\"2944\" y=\"0\"><\/use><use xlink:href=\"#E325-MJMATHI-52\" x=\"4222\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f: R^n \\rightarrow R<\/script><span> is twice continuously differentiable at <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E397-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E397-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E397-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E397-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E397-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E397-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in R^n<\/script><span>. If <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.606ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4566.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E327-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E327-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E327-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMATHI-66\" x=\"833\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-28\" x=\"1383\" y=\"0\"><\/use><g transform=\"translate(1772,0)\"><use xlink:href=\"#E327-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E327-MJMAIN-29\" x=\"2344\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-3D\" x=\"3010\" y=\"0\"><\/use><use xlink:href=\"#E327-MJMAIN-30\" x=\"4066\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\nabla f(\\bar{x}) = 0<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.401ex\" height=\"2.811ex\" viewBox=\"0 -906.7 3186.6 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E328-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E328-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E328-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E328-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E328-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E328-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E328-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E328-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E328-MJMAIN-32\" x=\"1178\" y=\"513\"><\/use><use xlink:href=\"#E328-MJMATHI-66\" x=\"1286\" y=\"0\"><\/use><use xlink:href=\"#E328-MJMAIN-28\" x=\"1836\" y=\"0\"><\/use><g transform=\"translate(2225,0)\"><use xlink:href=\"#E328-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E328-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E328-MJMAIN-29\" x=\"2797\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\nabla ^2 f(\\bar{x})<\/script><span> is positive definite, then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> is a local minimum.<\/span><\/p><h3><a name=\"constrained-optimization-problems\" class=\"md-header-anchor\"><\/a><span>Constrained Optimization Problems<\/span><\/h3><p><strong><span>Problem:<\/span><\/strong><\/p><p><span>inf<\/span><span>\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.413ex\" height=\"2.577ex\" viewBox=\"0 -806.1 1900 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E386-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E386-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x)<\/script><\/p><p><span>subject to<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.301ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4004.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E387-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E387-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E387-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><use xlink:href=\"#E387-MJMAIN-28\" x=\"820\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMATHI-78\" x=\"1209\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMAIN-29\" x=\"1781\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMAIN-2264\" x=\"2448\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMAIN-30\" x=\"3504\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g_i(x) \\le 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E334-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E334-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m<\/script><\/p><p><span>\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.641ex\" height=\"2.694ex\" viewBox=\"0 -806.1 4150.9 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E389-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E389-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E389-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><use xlink:href=\"#E389-MJMAIN-28\" x=\"967\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMATHI-78\" x=\"1356\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMAIN-29\" x=\"1928\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMAIN-3D\" x=\"2595\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMAIN-30\" x=\"3650\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">h_j(x) = 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.263ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5279.9 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E334-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E334-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E334-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><use xlink:href=\"#E334-MJMATHI-6D\" x=\"4401\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m<\/script><\/p><p><span>\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.147ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2646.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E406-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E406-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E406-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E406-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E406-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E406-MJMATHI-58\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in X<\/script><\/p><p><strong><span>Theorem 2: (The Fritz John Necessary Conditions)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.666ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2439.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E368-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E368-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E368-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMATHI-53\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in S<\/script><span> be a local minimum of the above problem. Then there exists <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.931ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2553.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E337-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E337-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E337-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E337-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E337-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E337-MJMATHI-52\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">u \\in R<\/script><span>, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.492ex\" height=\"2.577ex\" viewBox=\"0 -755.9 6670 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E338-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E338-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E338-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E338-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E338-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E338-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E338-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E338-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E338-MJMAIN-31\" x=\"685\" y=\"-213\"><\/use><use xlink:href=\"#E338-MJMAIN-2C\" x=\"938\" y=\"0\"><\/use><use xlink:href=\"#E338-MJMAIN-2E\" x=\"1383\" y=\"0\"><\/use><use xlink:href=\"#E338-MJMAIN-2E\" x=\"1827\" y=\"0\"><\/use><use xlink:href=\"#E338-MJMAIN-2E\" x=\"2272\" y=\"0\"><\/use><use xlink:href=\"#E338-MJMAIN-2C\" x=\"2717\" y=\"0\"><\/use><g transform=\"translate(3161,0)\"><use xlink:href=\"#E338-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(485,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E338-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E338-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E338-MJMAIN-2208\" x=\"4966\" y=\"0\"><\/use><use xlink:href=\"#E338-MJMATHI-52\" x=\"5910\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v_1,...,v_{m_1} \\in R<\/script><span>, and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"16.565ex\" height=\"2.577ex\" viewBox=\"0 -755.9 7132 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E352-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E352-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E352-MJMAIN-31\" x=\"1012\" y=\"-213\"><\/use><use xlink:href=\"#E352-MJMAIN-2C\" x=\"1169\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2E\" x=\"1614\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2E\" x=\"2058\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2E\" x=\"2503\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2C\" x=\"2948\" y=\"0\"><\/use><g transform=\"translate(3392,0)\"><use xlink:href=\"#E352-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(716,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E352-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E352-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E352-MJMAIN-2208\" x=\"5428\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMATHI-52\" x=\"6372\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">w_1,...,w_{m_2} \\in R<\/script><span> such that<\/span><\/p><p><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"47.989ex\" height=\"3.161ex\" viewBox=\"0 -856.4 20662 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E340-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E340-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E340-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-2207\" x=\"572\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMATHI-66\" x=\"1405\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-28\" x=\"1955\" y=\"0\"><\/use><g transform=\"translate(2344,0)\"><use xlink:href=\"#E340-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-29\" x=\"2916\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-2B\" x=\"3527\" y=\"0\"><\/use><g transform=\"translate(4527,0)\"><use xlink:href=\"#E340-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,476)\"><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E340-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(6997,0)\"><use xlink:href=\"#E340-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-69\" x=\"685\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-2207\" x=\"7826\" y=\"0\"><\/use><g transform=\"translate(8659,0)\"><use xlink:href=\"#E340-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-28\" x=\"9480\" y=\"0\"><\/use><g transform=\"translate(9869,0)\"><use xlink:href=\"#E340-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-29\" x=\"10441\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-2B\" x=\"11052\" y=\"0\"><\/use><g transform=\"translate(12053,0)\"><use xlink:href=\"#E340-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,476)\"><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E340-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMAIN-3D\" x=\"412\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMAIN-31\" x=\"1189\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(14570,0)\"><use xlink:href=\"#E340-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-2207\" x=\"15678\" y=\"0\"><\/use><g transform=\"translate(16511,0)\"><use xlink:href=\"#E340-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E340-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-28\" x=\"17478\" y=\"0\"><\/use><g transform=\"translate(17867,0)\"><use xlink:href=\"#E340-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E340-MJMAIN-29\" x=\"18439\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-3D\" x=\"19106\" y=\"0\"><\/use><use xlink:href=\"#E340-MJMAIN-30\" x=\"20161\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">u \\nabla f(\\bar{x}) + \\sum ^{m_1} _{i=1} v_i \\nabla g_i(\\bar{x}) + \\sum ^{m_2} _{j=1} w_j\\nabla h_j (\\bar{x}) = 0 <\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.545ex\" height=\"2.344ex\" viewBox=\"0 -755.9 3679.2 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E341-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E341-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E341-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E341-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E341-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E341-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E341-MJMATHI-75\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E341-MJMAIN-2C\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(1016,0)\"><use xlink:href=\"#E341-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E341-MJMATHI-69\" x=\"685\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E341-MJMAIN-2265\" x=\"2123\" y=\"0\"><\/use><use xlink:href=\"#E341-MJMAIN-30\" x=\"3179\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">u, v_i \\ge 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"32.311ex\" height=\"2.694ex\" viewBox=\"0 -806.1 13911.8 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E343-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMATHI-75\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path stroke-width=\"0\" id=\"E343-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E343-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMATHI-75\" x=\"389\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2C\" x=\"961\" y=\"0\"><\/use><g transform=\"translate(1405,0)\"><use xlink:href=\"#E343-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E343-MJMAIN-31\" x=\"685\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E343-MJMAIN-2C\" x=\"2344\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2E\" x=\"2788\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2E\" x=\"3233\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2E\" x=\"3678\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2C\" x=\"4122\" y=\"0\"><\/use><g transform=\"translate(4567,0)\"><use xlink:href=\"#E343-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(485,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E343-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E343-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E343-MJMAIN-2C\" x=\"6094\" y=\"0\"><\/use><g transform=\"translate(6538,0)\"><use xlink:href=\"#E343-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E343-MJMAIN-31\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E343-MJMAIN-2C\" x=\"7708\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2E\" x=\"8152\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2E\" x=\"8597\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2E\" x=\"9042\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2C\" x=\"9486\" y=\"0\"><\/use><g transform=\"translate(9931,0)\"><use xlink:href=\"#E343-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(716,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E343-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E343-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E343-MJMAIN-29\" x=\"11689\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-2260\" x=\"12355\" y=\"0\"><\/use><use xlink:href=\"#E343-MJMAIN-30\" x=\"13411\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(u, v_1,...,v_{m_1},w_1,...,w_{m_2}) \\neq 0<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.226ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4833.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E356-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E356-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E356-MJMATHI-69\" x=\"685\" y=\"-213\"><\/use><g transform=\"translate(828,0)\"><use xlink:href=\"#E356-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E356-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E356-MJMAIN-28\" x=\"1649\" y=\"0\"><\/use><g transform=\"translate(2038,0)\"><use xlink:href=\"#E356-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E356-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E356-MJMAIN-29\" x=\"2610\" y=\"0\"><\/use><use xlink:href=\"#E356-MJMAIN-3D\" x=\"3277\" y=\"0\"><\/use><use xlink:href=\"#E356-MJMAIN-30\" x=\"4333\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v_ig_i(\\bar{x}) = 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><strong><span>Theorem 3: (The Karush-Kuhn-Tucker Necessary Conditions)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.666ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2439.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E368-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E368-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E368-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMATHI-53\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in S<\/script><span> \u200b be a local minimum of the above problem. Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"33.209ex\" height=\"2.577ex\" viewBox=\"0 -806.1 14298.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E361-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E361-MJMATHI-49\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-3D\" x=\"781\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-7B\" x=\"1837\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMATHI-69\" x=\"2337\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2208\" x=\"2960\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-7B\" x=\"3905\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-31\" x=\"4405\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2C\" x=\"4905\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2E\" x=\"5349\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2E\" x=\"5794\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2E\" x=\"6239\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2C\" x=\"6683\" y=\"0\"><\/use><g transform=\"translate(7128,0)\"><use xlink:href=\"#E361-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E361-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E361-MJMAIN-7D\" x=\"8459\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-3A\" x=\"9237\" y=\"0\"><\/use><g transform=\"translate(9793,0)\"><use xlink:href=\"#E361-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E361-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E361-MJMAIN-28\" x=\"10614\" y=\"0\"><\/use><g transform=\"translate(11003,0)\"><use xlink:href=\"#E361-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E361-MJMAIN-29\" x=\"11575\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-3D\" x=\"12242\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-30\" x=\"13298\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-7D\" x=\"13798\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">I = \\{i \\in \\{1,...,m_1\\}:g_i(\\bar{x}) = 0\\}<\/script><span> be the index set for active constraints. Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> is regular which means <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.022ex\" height=\"2.577ex\" viewBox=\"0 -806.1 5175.9 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E349-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E349-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E349-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E349-MJMAIN-2207\" x=\"500\" y=\"0\"><\/use><g transform=\"translate(1333,0)\"><use xlink:href=\"#E349-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E349-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E349-MJMAIN-28\" x=\"2153\" y=\"0\"><\/use><g transform=\"translate(2542,0)\"><use xlink:href=\"#E349-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E349-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E349-MJMAIN-29\" x=\"3114\" y=\"0\"><\/use><g transform=\"translate(3503,0)\"><use xlink:href=\"#E349-MJMAIN-7D\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(500,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E349-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E349-MJMAIN-2208\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E349-MJMATHI-49\" x=\"1012\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\{\\nabla g_i(\\bar{x}) \\} _{i \\in I}<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.647ex\" height=\"3.161ex\" viewBox=\"0 -856.4 5445.3 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E350-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E350-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E350-MJMAIN-7B\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E350-MJMAIN-2207\" x=\"500\" y=\"0\"><\/use><g transform=\"translate(1333,0)\"><use xlink:href=\"#E350-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E350-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E350-MJMAIN-28\" x=\"2300\" y=\"0\"><\/use><g transform=\"translate(2689,0)\"><use xlink:href=\"#E350-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E350-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E350-MJMAIN-29\" x=\"3261\" y=\"0\"><\/use><g transform=\"translate(3650,0)\"><use xlink:href=\"#E350-MJMAIN-7D\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(500,450)\"><use transform=\"scale(0.707)\" xlink:href=\"#E350-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E350-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(500,-307)\"><use transform=\"scale(0.707)\" xlink:href=\"#E350-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E350-MJMAIN-3D\" x=\"412\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E350-MJMAIN-31\" x=\"1189\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\{\\nabla h_j(\\bar{x}) \\}_{j=1} ^{m_2}<\/script><span> of vectors is linearly independent. Then there exist <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"16.524ex\" height=\"2.577ex\" viewBox=\"0 -755.9 7114.7 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E351-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E351-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E351-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E351-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E351-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E351-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E351-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E351-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E351-MJMAIN-31\" x=\"685\" y=\"-213\"><\/use><use xlink:href=\"#E351-MJMAIN-2C\" x=\"938\" y=\"0\"><\/use><use xlink:href=\"#E351-MJMAIN-2E\" x=\"1383\" y=\"0\"><\/use><use xlink:href=\"#E351-MJMAIN-2E\" x=\"1827\" y=\"0\"><\/use><use xlink:href=\"#E351-MJMAIN-2E\" x=\"2272\" y=\"0\"><\/use><use xlink:href=\"#E351-MJMAIN-2E\" x=\"2717\" y=\"0\"><\/use><use xlink:href=\"#E351-MJMAIN-2C\" x=\"3161\" y=\"0\"><\/use><g transform=\"translate(3606,0)\"><use xlink:href=\"#E351-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(485,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E351-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E351-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E351-MJMAIN-2208\" x=\"5410\" y=\"0\"><\/use><use xlink:href=\"#E351-MJMATHI-52\" x=\"6355\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v_1,....,v_{m_1} \\in R<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"16.565ex\" height=\"2.577ex\" viewBox=\"0 -755.9 7132 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E352-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E352-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E352-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E352-MJMAIN-31\" x=\"1012\" y=\"-213\"><\/use><use xlink:href=\"#E352-MJMAIN-2C\" x=\"1169\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2E\" x=\"1614\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2E\" x=\"2058\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2E\" x=\"2503\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMAIN-2C\" x=\"2948\" y=\"0\"><\/use><g transform=\"translate(3392,0)\"><use xlink:href=\"#E352-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(716,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E352-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E352-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E352-MJMAIN-2208\" x=\"5428\" y=\"0\"><\/use><use xlink:href=\"#E352-MJMATHI-52\" x=\"6372\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">w_1,...,w_{m_2} \\in R<\/script><span> such that<\/span><\/p><p><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"46.661ex\" height=\"3.161ex\" viewBox=\"0 -856.4 20090 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E353-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E353-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E353-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMATHI-66\" x=\"833\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-28\" x=\"1383\" y=\"0\"><\/use><g transform=\"translate(1772,0)\"><use xlink:href=\"#E353-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-29\" x=\"2344\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-2B\" x=\"2955\" y=\"0\"><\/use><g transform=\"translate(3955,0)\"><use xlink:href=\"#E353-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,476)\"><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E353-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(6425,0)\"><use xlink:href=\"#E353-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-69\" x=\"685\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-2207\" x=\"7254\" y=\"0\"><\/use><g transform=\"translate(8087,0)\"><use xlink:href=\"#E353-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-28\" x=\"8908\" y=\"0\"><\/use><g transform=\"translate(9297,0)\"><use xlink:href=\"#E353-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-29\" x=\"9869\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-2B\" x=\"10480\" y=\"0\"><\/use><g transform=\"translate(11481,0)\"><use xlink:href=\"#E353-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,476)\"><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E353-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMAIN-3D\" x=\"412\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMAIN-31\" x=\"1189\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(13998,0)\"><use xlink:href=\"#E353-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-2207\" x=\"15106\" y=\"0\"><\/use><g transform=\"translate(15939,0)\"><use xlink:href=\"#E353-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E353-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-28\" x=\"16906\" y=\"0\"><\/use><g transform=\"translate(17295,0)\"><use xlink:href=\"#E353-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E353-MJMAIN-29\" x=\"17867\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-3D\" x=\"18534\" y=\"0\"><\/use><use xlink:href=\"#E353-MJMAIN-30\" x=\"19589\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\"> \\nabla f(\\bar{x}) + \\sum ^{m_1} _{i=1} v_i \\nabla g_i(\\bar{x}) + \\sum ^{m_2} _{j=1} w_j\\nabla h_j (\\bar{x}) = 0 <\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.184ex\" height=\"2.344ex\" viewBox=\"0 -755.9 2662.5 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E354-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E354-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E354-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E354-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E354-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E354-MJMATHI-69\" x=\"685\" y=\"-213\"><\/use><use xlink:href=\"#E354-MJMAIN-2265\" x=\"1106\" y=\"0\"><\/use><use xlink:href=\"#E354-MJMAIN-30\" x=\"2162\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v_i \\ge 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.226ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4833.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E356-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E356-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E356-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E356-MJMATHI-69\" x=\"685\" y=\"-213\"><\/use><g transform=\"translate(828,0)\"><use xlink:href=\"#E356-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E356-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E356-MJMAIN-28\" x=\"1649\" y=\"0\"><\/use><g transform=\"translate(2038,0)\"><use xlink:href=\"#E356-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E356-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E356-MJMAIN-29\" x=\"2610\" y=\"0\"><\/use><use xlink:href=\"#E356-MJMAIN-3D\" x=\"3277\" y=\"0\"><\/use><use xlink:href=\"#E356-MJMAIN-30\" x=\"4333\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v_ig_i(\\bar{x}) = 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><strong><span>Theorem 4:<\/span><\/strong><span>\t<\/span><span>Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.852ex\" height=\"1.994ex\" viewBox=\"0 -504.6 4672.4 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E366-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E366-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E366-MJMAIN-31\" x=\"674\" y=\"-213\"><\/use><use xlink:href=\"#E366-MJMAIN-2C\" x=\"930\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2E\" x=\"1375\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2E\" x=\"1819\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2E\" x=\"2264\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2C\" x=\"2709\" y=\"0\"><\/use><g transform=\"translate(3153,0)\"><use xlink:href=\"#E366-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(477,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E366-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E366-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">g_1,...,g_{m_1}<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.312ex\" height=\"2.577ex\" viewBox=\"0 -755.9 4870.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E373-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E373-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E373-MJMAIN-31\" x=\"814\" y=\"-213\"><\/use><use xlink:href=\"#E373-MJMAIN-2C\" x=\"1029\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"1474\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"1918\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"2363\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2C\" x=\"2808\" y=\"0\"><\/use><g transform=\"translate(3252,0)\"><use xlink:href=\"#E373-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(576,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E373-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E373-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">h_1,...,h_{m_2}<\/script><span> are affine. Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.666ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2439.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E368-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E368-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E368-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMATHI-53\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in S<\/script><span> be a local minimum and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"33.209ex\" height=\"2.577ex\" viewBox=\"0 -806.1 14298.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E361-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E361-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E361-MJMATHI-49\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-3D\" x=\"781\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-7B\" x=\"1837\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMATHI-69\" x=\"2337\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2208\" x=\"2960\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-7B\" x=\"3905\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-31\" x=\"4405\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2C\" x=\"4905\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2E\" x=\"5349\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2E\" x=\"5794\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2E\" x=\"6239\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-2C\" x=\"6683\" y=\"0\"><\/use><g transform=\"translate(7128,0)\"><use xlink:href=\"#E361-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E361-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E361-MJMAIN-7D\" x=\"8459\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-3A\" x=\"9237\" y=\"0\"><\/use><g transform=\"translate(9793,0)\"><use xlink:href=\"#E361-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E361-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E361-MJMAIN-28\" x=\"10614\" y=\"0\"><\/use><g transform=\"translate(11003,0)\"><use xlink:href=\"#E361-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E361-MJMAIN-29\" x=\"11575\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-3D\" x=\"12242\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-30\" x=\"13298\" y=\"0\"><\/use><use xlink:href=\"#E361-MJMAIN-7D\" x=\"13798\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">I = \\{i \\in \\{1,...,m_1\\}:g_i(\\bar{x}) = 0\\}<\/script><span> . Suppose that the Slater condition is satisfied which means there exists an <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.35ex\" height=\"2.227ex\" viewBox=\"0 -856.4 2734 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E362-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E362-MJMAIN-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path stroke-width=\"0\" id=\"E362-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E362-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E362-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E362-MJMAIN-2032\" x=\"808\" y=\"513\"><\/use><use xlink:href=\"#E362-MJMAIN-2208\" x=\"1144\" y=\"0\"><\/use><use xlink:href=\"#E362-MJMATHI-53\" x=\"2089\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x' \\in S<\/script><span> such that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.985ex\" height=\"2.694ex\" viewBox=\"0 -856.4 4299 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E363-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMAIN-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E363-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E363-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E363-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><use xlink:href=\"#E363-MJMAIN-28\" x=\"820\" y=\"0\"><\/use><g transform=\"translate(1209,0)\"><use xlink:href=\"#E363-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E363-MJMAIN-2032\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E363-MJMAIN-29\" x=\"2076\" y=\"0\"><\/use><use xlink:href=\"#E363-MJMAIN-3C\" x=\"2743\" y=\"0\"><\/use><use xlink:href=\"#E363-MJMAIN-30\" x=\"3798\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g_i(x') <0<\/script><span> for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.811ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2071.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E364-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E364-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E364-MJMATHI-49\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E364-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E364-MJMAIN-2208\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E364-MJMATHI-49\" x=\"1567\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i \\in I<\/script><span>. Then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> satisfies the KKT conditions.<\/span><\/p><p><strong><span>Theorem 5:<\/span><\/strong><span>\t<\/span><span>Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.852ex\" height=\"1.994ex\" viewBox=\"0 -504.6 4672.4 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E366-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E366-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E366-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E366-MJMAIN-31\" x=\"674\" y=\"-213\"><\/use><use xlink:href=\"#E366-MJMAIN-2C\" x=\"930\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2E\" x=\"1375\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2E\" x=\"1819\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2E\" x=\"2264\" y=\"0\"><\/use><use xlink:href=\"#E366-MJMAIN-2C\" x=\"2709\" y=\"0\"><\/use><g transform=\"translate(3153,0)\"><use xlink:href=\"#E366-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(477,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E366-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E366-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">g_1,...,g_{m_1}<\/script><span> are concave and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.312ex\" height=\"2.577ex\" viewBox=\"0 -755.9 4870.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E373-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E373-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E373-MJMAIN-31\" x=\"814\" y=\"-213\"><\/use><use xlink:href=\"#E373-MJMAIN-2C\" x=\"1029\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"1474\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"1918\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"2363\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2C\" x=\"2808\" y=\"0\"><\/use><g transform=\"translate(3252,0)\"><use xlink:href=\"#E373-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(576,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E373-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E373-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">h_1,...,h_{m_2}<\/script><span> are affine. Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.666ex\" height=\"2.11ex\" viewBox=\"0 -806.1 2439.6 908.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E368-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E368-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E368-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E368-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E368-MJMATHI-53\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in S<\/script><span> be a local minimum. Then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> satisfies the KKT conditions.<\/span><\/p><p><strong><span>Theorem 6:<\/span><\/strong><span>\t<\/span><span>Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.979ex\" height=\"1.877ex\" viewBox=\"0 -755.9 852 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E372-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E372-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X<\/script><span> is open and convex, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.162ex\" height=\"2.694ex\" viewBox=\"0 -806.1 5667.1 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E371-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E371-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E371-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E371-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E371-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E371-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E371-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E371-MJMAIN-2C\" x=\"550\" y=\"0\"><\/use><g transform=\"translate(994,0)\"><use xlink:href=\"#E371-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E371-MJMAIN-31\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E371-MJMAIN-2C\" x=\"1925\" y=\"0\"><\/use><use xlink:href=\"#E371-MJMAIN-2E\" x=\"2369\" y=\"0\"><\/use><use xlink:href=\"#E371-MJMAIN-2E\" x=\"2814\" y=\"0\"><\/use><use xlink:href=\"#E371-MJMAIN-2E\" x=\"3259\" y=\"0\"><\/use><use xlink:href=\"#E371-MJMAIN-2C\" x=\"3703\" y=\"0\"><\/use><g transform=\"translate(4148,0)\"><use xlink:href=\"#E371-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(477,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E371-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E371-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">f,g_1,...,g_{m_1}<\/script><span> are convex on <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.979ex\" height=\"1.877ex\" viewBox=\"0 -755.9 852 808.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.121ex;\"><defs><path stroke-width=\"0\" id=\"E372-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E372-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">X<\/script><span>, and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.312ex\" height=\"2.577ex\" viewBox=\"0 -755.9 4870.4 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E373-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E373-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E373-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E373-MJMAIN-31\" x=\"814\" y=\"-213\"><\/use><use xlink:href=\"#E373-MJMAIN-2C\" x=\"1029\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"1474\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"1918\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2E\" x=\"2363\" y=\"0\"><\/use><use xlink:href=\"#E373-MJMAIN-2C\" x=\"2808\" y=\"0\"><\/use><g transform=\"translate(3252,0)\"><use xlink:href=\"#E373-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(576,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E373-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E373-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">h_1,...,h_{m_2}<\/script><span> are affine. Suppose that <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"26.951ex\" height=\"2.577ex\" viewBox=\"0 -806.1 11603.7 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E374-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E374-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E374-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E374-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E374-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E374-MJMAIN-2C\" x=\"961\" y=\"0\"><\/use><g transform=\"translate(1405,0)\"><use xlink:href=\"#E374-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E374-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E374-MJMAIN-2C\" x=\"1933\" y=\"0\"><\/use><g transform=\"translate(2378,0)\"><use xlink:href=\"#E374-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E374-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E374-MJMAIN-29\" x=\"3094\" y=\"0\"><\/use><use xlink:href=\"#E374-MJMAIN-2208\" x=\"3760\" y=\"0\"><\/use><use xlink:href=\"#E374-MJMATHI-58\" x=\"4705\" y=\"0\"><\/use><use xlink:href=\"#E374-MJMAIN-D7\" x=\"5779\" y=\"0\"><\/use><g transform=\"translate(6780,0)\"><use xlink:href=\"#E374-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E374-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E374-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E374-MJMAIN-D7\" x=\"8802\" y=\"0\"><\/use><g transform=\"translate(9803,0)\"><use xlink:href=\"#E374-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E374-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E374-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{x},\\bar{v},\\bar{w}) \\in X \\times R^{m_1} \\times R^{m_2}<\/script><span> is a solution to the KKT conditions (includes primal conditions), then <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> is an optimal solution.<\/span><\/p><p><span> <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"46.76ex\" height=\"3.161ex\" viewBox=\"0 -856.4 20132.8 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E376-MJMAIN-2207\" d=\"M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E376-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E376-MJMAIN-2207\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMATHI-66\" x=\"833\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-28\" x=\"1383\" y=\"0\"><\/use><g transform=\"translate(1772,0)\"><use xlink:href=\"#E376-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-29\" x=\"2344\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-2B\" x=\"2955\" y=\"0\"><\/use><g transform=\"translate(3955,0)\"><use xlink:href=\"#E376-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,476)\"><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E376-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(6425,0)\"><use xlink:href=\"#E376-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-69\" x=\"746\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-2207\" x=\"7297\" y=\"0\"><\/use><g transform=\"translate(8130,0)\"><use xlink:href=\"#E376-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-28\" x=\"8951\" y=\"0\"><\/use><g transform=\"translate(9340,0)\"><use xlink:href=\"#E376-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-29\" x=\"9912\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-2B\" x=\"10523\" y=\"0\"><\/use><g transform=\"translate(11523,0)\"><use xlink:href=\"#E376-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,476)\"><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E376-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMAIN-3D\" x=\"412\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMAIN-31\" x=\"1189\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(14041,0)\"><use xlink:href=\"#E376-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-2207\" x=\"15148\" y=\"0\"><\/use><g transform=\"translate(15981,0)\"><use xlink:href=\"#E376-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E376-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-28\" x=\"16949\" y=\"0\"><\/use><g transform=\"translate(17338,0)\"><use xlink:href=\"#E376-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E376-MJMAIN-29\" x=\"17910\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-3D\" x=\"18577\" y=\"0\"><\/use><use xlink:href=\"#E376-MJMAIN-30\" x=\"19632\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\"> \\nabla f(\\bar{x}) + \\sum ^{m_1} _{i=1} \\bar{v}_i \\nabla g_i(\\bar{x}) + \\sum ^{m_2} _{j=1} \\bar{w}_j\\nabla h_j (\\bar{x}) = 0 <\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.283ex\" height=\"2.344ex\" viewBox=\"0 -755.9 2705.3 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E377-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E377-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E377-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E377-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E377-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E377-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E377-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E377-MJMATHI-69\" x=\"746\" y=\"-213\"><\/use><use xlink:href=\"#E377-MJMAIN-2265\" x=\"1149\" y=\"0\"><\/use><use xlink:href=\"#E377-MJMAIN-30\" x=\"2205\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{v}_i \\ge 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.326ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4876.3 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E379-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E379-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E379-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E379-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E379-MJMATHI-69\" x=\"746\" y=\"-213\"><\/use><g transform=\"translate(871,0)\"><use xlink:href=\"#E379-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E379-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E379-MJMAIN-28\" x=\"1692\" y=\"0\"><\/use><g transform=\"translate(2081,0)\"><use xlink:href=\"#E379-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E379-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E379-MJMAIN-29\" x=\"2653\" y=\"0\"><\/use><use xlink:href=\"#E379-MJMAIN-3D\" x=\"3320\" y=\"0\"><\/use><use xlink:href=\"#E379-MJMAIN-30\" x=\"4376\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{v}_ig_i(\\bar{x}) = 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.301ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4004.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E381-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E381-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E381-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E381-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><use xlink:href=\"#E381-MJMAIN-28\" x=\"820\" y=\"0\"><\/use><g transform=\"translate(1209,0)\"><use xlink:href=\"#E381-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E381-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E381-MJMAIN-29\" x=\"1781\" y=\"0\"><\/use><use xlink:href=\"#E381-MJMAIN-2264\" x=\"2448\" y=\"0\"><\/use><use xlink:href=\"#E381-MJMAIN-30\" x=\"3504\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g_i(\\bar{x}) \\le 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.641ex\" height=\"2.694ex\" viewBox=\"0 -806.1 4150.9 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E383-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E383-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E383-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E383-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><use xlink:href=\"#E383-MJMAIN-28\" x=\"967\" y=\"0\"><\/use><g transform=\"translate(1356,0)\"><use xlink:href=\"#E383-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E383-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E383-MJMAIN-29\" x=\"1928\" y=\"0\"><\/use><use xlink:href=\"#E383-MJMAIN-3D\" x=\"2595\" y=\"0\"><\/use><use xlink:href=\"#E383-MJMAIN-30\" x=\"3650\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">h_j(\\bar{x}) = 0<\/script><span>, for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E384-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E384-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E384-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E384-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E384-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E384-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E384-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E384-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E384-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E384-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E384-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_2<\/script><\/p><h3><a name=\"lagrangian-duality\" class=\"md-header-anchor\"><\/a><span>Lagrangian Duality<\/span><\/h3><p><strong><span>Primal Problem:<\/span><\/strong><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.185ex\" height=\"2.461ex\" viewBox=\"0 -705.6 940.7 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E385-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E385-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E385-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E385-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E385-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E385-MJMATHI-70\" x=\"685\" y=\"-211\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v^* _p<\/script><span>\t<\/span><span>=<\/span><span>\t\t\t<\/span><span>inf<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"4.413ex\" height=\"2.577ex\" viewBox=\"0 -806.1 1900 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E386-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E386-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E386-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMAIN-28\" x=\"550\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMATHI-78\" x=\"939\" y=\"0\"><\/use><use xlink:href=\"#E386-MJMAIN-29\" x=\"1511\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">f(x)<\/script><\/p><p><span>\t\t\t\t<\/span><span>subject to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.301ex\" height=\"2.577ex\" viewBox=\"0 -806.1 4004.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E387-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E387-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E387-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E387-MJMATHI-69\" x=\"674\" y=\"-213\"><\/use><use xlink:href=\"#E387-MJMAIN-28\" x=\"820\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMATHI-78\" x=\"1209\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMAIN-29\" x=\"1781\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMAIN-2264\" x=\"2448\" y=\"0\"><\/use><use xlink:href=\"#E387-MJMAIN-30\" x=\"3504\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g_i(x) \\le 0<\/script><span>,<\/span><span>\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.316ex\" height=\"2.344ex\" viewBox=\"0 -755.9 5733.4 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E388-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E388-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E388-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-3D\" x=\"622\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-31\" x=\"1678\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"2178\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"2623\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3067\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2E\" x=\"3512\" y=\"0\"><\/use><use xlink:href=\"#E388-MJMAIN-2C\" x=\"3957\" y=\"0\"><\/use><g transform=\"translate(4401,0)\"><use xlink:href=\"#E388-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E388-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">i = 1,...,m_1<\/script><\/p><p><span>\t\t\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"9.641ex\" height=\"2.694ex\" viewBox=\"0 -806.1 4150.9 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E389-MJMATHI-68\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E389-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E389-MJMATHI-68\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E389-MJMATHI-6A\" x=\"814\" y=\"-213\"><\/use><use xlink:href=\"#E389-MJMAIN-28\" x=\"967\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMATHI-78\" x=\"1356\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMAIN-29\" x=\"1928\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMAIN-3D\" x=\"2595\" y=\"0\"><\/use><use xlink:href=\"#E389-MJMAIN-30\" x=\"3650\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">h_j(x) = 0<\/script><span>,<\/span><span>\t<\/span><span>for <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"13.5ex\" height=\"2.461ex\" viewBox=\"-12 -755.9 5812.4 1059.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex; margin-left: -0.028ex;\"><defs><path stroke-width=\"0\" id=\"E390-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E390-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E390-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E390-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E390-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E390-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E390-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E390-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-3D\" x=\"689\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-31\" x=\"1745\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-2C\" x=\"2245\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-2E\" x=\"2690\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-2E\" x=\"3134\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-2E\" x=\"3579\" y=\"0\"><\/use><use xlink:href=\"#E390-MJMAIN-2C\" x=\"4024\" y=\"0\"><\/use><g transform=\"translate(4468,0)\"><use xlink:href=\"#E390-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E390-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">j = 1,...,m_2<\/script><\/p><p><span>\t\t\t\t\t\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.147ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2646.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E406-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E406-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E406-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E406-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E406-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E406-MJMATHI-58\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in X<\/script><\/p><p><strong><span>Dual Problem:<\/span><\/strong><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.218ex\" height=\"2.577ex\" viewBox=\"0 -705.6 954.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E392-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E392-MJMAIN-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path stroke-width=\"0\" id=\"E392-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E392-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E392-MJMAIN-2217\" x=\"685\" y=\"452\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E392-MJMATHI-64\" x=\"685\" y=\"-462\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">v^*_d<\/script><span>\t<\/span><span>=<\/span><span>\t\t\t<\/span><span>sup <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.106ex\" height=\"2.227ex\" viewBox=\"0 -404.1 4781.9 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.289ex;\"><defs><path stroke-width=\"0\" id=\"E393-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E393-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><g transform=\"translate(0,-282)\"><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMAIN-2208\" x=\"485\" y=\"0\"><\/use><g transform=\"translate(814,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(536,404)\"><use transform=\"scale(0.5)\" xlink:href=\"#E393-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E393-MJMAIN-31\" x=\"878\" y=\"-322\"><\/use><\/g><use transform=\"scale(0.5)\" xlink:href=\"#E393-MJMAIN-2B\" x=\"1073\" y=\"-351\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMAIN-2C\" x=\"3056\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMATHI-77\" x=\"3334\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMAIN-2208\" x=\"4050\" y=\"0\"><\/use><g transform=\"translate(3335,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E393-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(536,289)\"><use transform=\"scale(0.5)\" xlink:href=\"#E393-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E393-MJMAIN-32\" x=\"878\" y=\"-322\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">_{v \\in R^{m_1}_+, w\\in R^{m_2}}<\/script><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.718ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2892.7 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E394-MJMATHI-3B8\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path stroke-width=\"0\" id=\"E394-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E394-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E394-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E394-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E394-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E394-MJMATHI-3B8\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E394-MJMAIN-28\" x=\"469\" y=\"0\"><\/use><use xlink:href=\"#E394-MJMATHI-76\" x=\"858\" y=\"0\"><\/use><use xlink:href=\"#E394-MJMAIN-2C\" x=\"1343\" y=\"0\"><\/use><use xlink:href=\"#E394-MJMATHI-77\" x=\"1787\" y=\"0\"><\/use><use xlink:href=\"#E394-MJMAIN-29\" x=\"2503\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\theta(v,w)<\/script><\/p><p><span>where<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"26.387ex\" height=\"2.577ex\" viewBox=\"0 -806.1 11361.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E395-MJMATHI-3B8\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E395-MJMATHI-4C\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E395-MJMATHI-3B8\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-28\" x=\"469\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-76\" x=\"858\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-2C\" x=\"1343\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-77\" x=\"1787\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-29\" x=\"2503\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-3D\" x=\"3170\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-69\" x=\"4226\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-6E\" x=\"4571\" y=\"0\"><\/use><g transform=\"translate(5171,0)\"><use xlink:href=\"#E395-MJMATHI-66\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(490,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E395-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E395-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E395-MJMATHI-58\" x=\"1239\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E395-MJMATHI-4C\" x=\"7239\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-28\" x=\"7920\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-78\" x=\"8309\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-2C\" x=\"8881\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-76\" x=\"9326\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-2C\" x=\"9811\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMATHI-77\" x=\"10256\" y=\"0\"><\/use><use xlink:href=\"#E395-MJMAIN-29\" x=\"10972\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\theta(v,w) = inf _{x \\in X} L(x,v,w)<\/script><\/p><p><span>\t\t\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"38.486ex\" height=\"2.811ex\" viewBox=\"0 -906.7 16570.4 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E396-MJMATHI-4C\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-47\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E396-MJMATHI-48\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E396-MJMATHI-4C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-28\" x=\"681\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-78\" x=\"1070\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-2C\" x=\"1642\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-76\" x=\"2086\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-2C\" x=\"2571\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-77\" x=\"3016\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-29\" x=\"3732\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-3D\" x=\"4399\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-66\" x=\"5454\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-28\" x=\"6004\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-78\" x=\"6393\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-29\" x=\"6965\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-2B\" x=\"7577\" y=\"0\"><\/use><g transform=\"translate(8577,0)\"><use xlink:href=\"#E396-MJMATHI-76\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E396-MJMATHI-54\" x=\"685\" y=\"513\"><\/use><\/g><use xlink:href=\"#E396-MJMATHI-47\" x=\"9660\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-28\" x=\"10446\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-78\" x=\"10835\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-29\" x=\"11407\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-2B\" x=\"12018\" y=\"0\"><\/use><g transform=\"translate(13018,0)\"><use xlink:href=\"#E396-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E396-MJMATHI-54\" x=\"1012\" y=\"513\"><\/use><\/g><use xlink:href=\"#E396-MJMATHI-48\" x=\"14332\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-28\" x=\"15220\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMATHI-78\" x=\"15609\" y=\"0\"><\/use><use xlink:href=\"#E396-MJMAIN-29\" x=\"16181\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">L(x,v,w) = f(x) + v^TG(x) + w^TH(x)<\/script><\/p><p><strong><span>Theorem 7: (Weak Duality Theorem)<\/span><\/strong><span>\t<\/span><span>Let <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.149ex\" height=\"1.994ex\" viewBox=\"0 -755.9 3077.8 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E397-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E397-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E397-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E397-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E397-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><g transform=\"translate(1794,0)\"><use xlink:href=\"#E397-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E397-MJMATHI-6E\" x=\"1073\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in R^n<\/script><span> be feasible for (P) and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"19.771ex\" height=\"2.577ex\" viewBox=\"0 -806.1 8512.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E398-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E398-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E398-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E398-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E398-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E398-MJMAIN-2C\" x=\"916\" y=\"0\"><\/use><g transform=\"translate(1361,0)\"><use xlink:href=\"#E398-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E398-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E398-MJMAIN-29\" x=\"2077\" y=\"0\"><\/use><use xlink:href=\"#E398-MJMAIN-2208\" x=\"2744\" y=\"0\"><\/use><g transform=\"translate(3689,0)\"><use xlink:href=\"#E398-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E398-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E398-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><use xlink:href=\"#E398-MJMAIN-D7\" x=\"5711\" y=\"0\"><\/use><g transform=\"translate(6712,0)\"><use xlink:href=\"#E398-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E398-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E398-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{v},\\bar{w}) \\in R^{m_1} \\times R^{m_2}<\/script><span> be feasible for (D). Then we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.328ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6169 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E399-MJMATHI-3B8\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMATHI-66\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path stroke-width=\"0\" id=\"E399-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E399-MJMATHI-3B8\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMAIN-28\" x=\"469\" y=\"0\"><\/use><g transform=\"translate(858,0)\"><use xlink:href=\"#E399-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E399-MJMAIN-2C\" x=\"1385\" y=\"0\"><\/use><g transform=\"translate(1830,0)\"><use xlink:href=\"#E399-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E399-MJMAIN-29\" x=\"2546\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMAIN-2264\" x=\"3213\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMATHI-66\" x=\"4269\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMAIN-28\" x=\"4819\" y=\"0\"><\/use><g transform=\"translate(5208,0)\"><use xlink:href=\"#E399-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E399-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E399-MJMAIN-29\" x=\"5780\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\theta(\\bar{v},\\bar{w}) \\le f(\\bar{x})<\/script><\/p><p><strong><span>Theorem 8: (Strong Duality Theorem)<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.09ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3483.1 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E400-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E400-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E400-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E400-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E400-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E400-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E400-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E400-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E400-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E400-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E400-MJMAIN-2C\" x=\"961\" y=\"0\"><\/use><g transform=\"translate(1405,0)\"><use xlink:href=\"#E400-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E400-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E400-MJMAIN-2C\" x=\"1933\" y=\"0\"><\/use><g transform=\"translate(2378,0)\"><use xlink:href=\"#E400-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E400-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E400-MJMAIN-29\" x=\"3094\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{x}, \\bar{v}, \\bar{w})<\/script><span> is a saddle point of the Lagrangian function <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.582ex\" height=\"1.994ex\" viewBox=\"0 -755.9 681 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E401-MJMATHI-4C\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E401-MJMATHI-4C\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">L<\/script><span> associated with (P) iff the duality gap between (P) and (D) is zero, <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.76ex\" viewBox=\"0 -655.4 572 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.026ex; margin-bottom: -0.213ex;\"><defs><path stroke-width=\"0\" id=\"E402-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E402-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E402-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E402-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x}<\/script><span> is optimal for (P), <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.729ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2466.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E403-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E403-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E403-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E403-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E403-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E403-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E403-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,0)\"><use xlink:href=\"#E403-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E403-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E403-MJMAIN-2C\" x=\"916\" y=\"0\"><\/use><g transform=\"translate(1361,0)\"><use xlink:href=\"#E403-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E403-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E403-MJMAIN-29\" x=\"2077\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">(\\bar{v}, \\bar{w})<\/script><span> is optimal for (D)<\/span><\/p><p><strong><span>Saddle Point:<\/span><\/strong><\/p><p><span>1)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.147ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2646.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E404-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E404-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E404-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E404-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E404-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E404-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><use xlink:href=\"#E404-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E404-MJMATHI-58\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{x} \\in X<\/script><\/p><p><span>2)<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.484ex\" height=\"2.227ex\" viewBox=\"0 -755.9 2361.4 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E405-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E405-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E405-MJMAIN-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path stroke-width=\"0\" id=\"E405-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E405-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E405-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><use xlink:href=\"#E405-MJMAIN-2265\" x=\"805\" y=\"0\"><\/use><use xlink:href=\"#E405-MJMAIN-30\" x=\"1861\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\bar{v} \\ge 0<\/script><\/p><p><span>3)<\/span><span>\t<\/span><span>For all <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.147ex\" height=\"1.994ex\" viewBox=\"0 -755.9 2646.6 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E406-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E406-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E406-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E406-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E406-MJMAIN-2208\" x=\"849\" y=\"0\"><\/use><use xlink:href=\"#E406-MJMATHI-58\" x=\"1794\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x \\in X<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"19.672ex\" height=\"2.928ex\" viewBox=\"0 -856.4 8469.8 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E407-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMATHI-52\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path stroke-width=\"0\" id=\"E407-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E407-MJMAIN-28\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E407-MJMATHI-76\" x=\"389\" y=\"0\"><\/use><use xlink:href=\"#E407-MJMAIN-2C\" x=\"874\" y=\"0\"><\/use><use xlink:href=\"#E407-MJMATHI-77\" x=\"1318\" y=\"0\"><\/use><use xlink:href=\"#E407-MJMAIN-29\" x=\"2034\" y=\"0\"><\/use><use xlink:href=\"#E407-MJMAIN-2208\" x=\"2701\" y=\"0\"><\/use><g transform=\"translate(3646,0)\"><use xlink:href=\"#E407-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,450)\"><use transform=\"scale(0.707)\" xlink:href=\"#E407-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E407-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E407-MJMAIN-2B\" x=\"1073\" y=\"-351\"><\/use><\/g><use xlink:href=\"#E407-MJMAIN-D7\" x=\"5668\" y=\"0\"><\/use><g transform=\"translate(6669,0)\"><use xlink:href=\"#E407-MJMATHI-52\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(759,362)\"><use transform=\"scale(0.707)\" xlink:href=\"#E407-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E407-MJMAIN-32\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">(v,w) \\in R^{m_1}_+ \\times R^{m_2}<\/script><span>, we have <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"35.11ex\" height=\"2.577ex\" viewBox=\"0 -806.1 15116.7 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E408-MJMATHI-4C\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMATHI-76\" d=\"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E408-MJMAIN-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E408-MJMATHI-4C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-28\" x=\"681\" y=\"0\"><\/use><g transform=\"translate(1070,0)\"><use xlink:href=\"#E408-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E408-MJMAIN-2C\" x=\"1642\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMATHI-76\" x=\"2086\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-2C\" x=\"2571\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMATHI-77\" x=\"3016\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-29\" x=\"3732\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-2264\" x=\"4399\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMATHI-4C\" x=\"5454\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-28\" x=\"6135\" y=\"0\"><\/use><g transform=\"translate(6524,0)\"><use xlink:href=\"#E408-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-AF\" x=\"63\" y=\"-27\"><\/use><\/g><use xlink:href=\"#E408-MJMAIN-2C\" x=\"7096\" y=\"0\"><\/use><g transform=\"translate(7541,0)\"><use xlink:href=\"#E408-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E408-MJMAIN-2C\" x=\"8069\" y=\"0\"><\/use><g transform=\"translate(8514,0)\"><use xlink:href=\"#E408-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E408-MJMAIN-29\" x=\"9230\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-2264\" x=\"9896\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMATHI-4C\" x=\"10952\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-28\" x=\"11633\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMATHI-78\" x=\"12022\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-2C\" x=\"12594\" y=\"0\"><\/use><g transform=\"translate(13039,0)\"><use xlink:href=\"#E408-MJMATHI-76\" x=\"7\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-AF\" x=\"27\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E408-MJMAIN-2C\" x=\"13567\" y=\"0\"><\/use><g transform=\"translate(14011,0)\"><use xlink:href=\"#E408-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E408-MJMAIN-AF\" x=\"191\" y=\"-26\"><\/use><\/g><use xlink:href=\"#E408-MJMAIN-29\" x=\"14727\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">L(\\bar{x}, v, w) \\le L(\\bar{x}, \\bar{v}, \\bar{w}) \\le L(x, \\bar{v}, \\bar{w})<\/script><\/p><p><span> <\/span><\/p><\/div>\n<\/body>\n<\/html>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Optimization Cheatsheet Handout 1 Introduction Basic Notions Optimal Value: Is defined to be the greatest lower bound or infimum of… <\/p>\n<div class=\"art-el-more\"><a href=\"https:\/\/yunpeng.site\/index.php\/2021\/04\/16\/optsum\/\" class=\"art-link art-color-link art-w-chevron\">Read more<\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[3],"tags":[5],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/119"}],"collection":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/comments?post=119"}],"version-history":[{"count":3,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/119\/revisions"}],"predecessor-version":[{"id":224,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/119\/revisions\/224"}],"wp:attachment":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/media?parent=119"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/categories?post=119"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/tags?post=119"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}