Semidefinite programming

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definition subsection</span> </button> <ul id="toc-Motivation_and_definition-sublist" class="vector-toc-list"> <li id="toc-Initial_motivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Initial_motivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Initial motivation</span> </div> </a> <ul id="toc-Initial_motivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalent_formulations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_formulations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Equivalent formulations</span> </div> </a> <ul id="toc-Equivalent_formulations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relations_to_other_optimization_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relations_to_other_optimization_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relations to other optimization problems</span> </div> </a> <ul id="toc-Relations_to_other_optimization_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Duality_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Duality_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Duality theory</span> </div> </a> <button aria-controls="toc-Duality_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Duality theory subsection</span> </button> <ul id="toc-Duality_theory-sublist" class="vector-toc-list"> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_duality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Weak duality</span> </div> </a> <ul id="toc-Weak_duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strong_duality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strong_duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Strong duality</span> </div> </a> <ul id="toc-Strong_duality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Example_1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Example 1</span> </div> </a> <ul id="toc-Example_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Example 2</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_3_(Goemans–Williamson_max_cut_approximation_algorithm)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_3_(Goemans–Williamson_max_cut_approximation_algorithm)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Example 3 (Goemans–Williamson max cut approximation algorithm)</span> </div> </a> <ul id="toc-Example_3_(Goemans–Williamson_max_cut_approximation_algorithm)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Other applications</span> </div> </a> <ul id="toc-Other_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Run-time_complexity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Run-time_complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Run-time complexity</span> </div> </a> <ul id="toc-Run-time_complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithms_for_solving_SDPs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithms_for_solving_SDPs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Algorithms for solving SDPs</span> </div> </a> <button aria-controls="toc-Algorithms_for_solving_SDPs-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algorithms for solving SDPs subsection</span> </button> <ul id="toc-Algorithms_for_solving_SDPs-sublist" class="vector-toc-list"> <li id="toc-Ellipsoid_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ellipsoid_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Ellipsoid method</span> </div> </a> <ul id="toc-Ellipsoid_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interior_point_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interior_point_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Interior point methods</span> </div> </a> <ul id="toc-Interior_point_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First-order_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First-order_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>First-order methods</span> </div> </a> <ul id="toc-First-order_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bundle_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bundle_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Bundle method</span> </div> </a> <ul id="toc-Bundle_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_solving_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_solving_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Other solving methods</span> </div> </a> <ul id="toc-Other_solving_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximate_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximate_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Approximate methods</span> </div> </a> <ul id="toc-Approximate_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Preprocessing_algorithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Preprocessing_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Preprocessing algorithms</span> </div> </a> <ul id="toc-Preprocessing_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Subfield of convex optimization</div> <p><b>Semidefinite programming</b> (<b>SDP</b>) is a subfield of <a href="/wiki/Mathematical_programming" class="mw-redirect" title="Mathematical programming">mathematical programming</a> concerned with the optimization of a linear <a href="/wiki/Objective_function" class="mw-redirect" title="Objective function">objective function</a> (a user-specified function that the user wants to minimize or maximize) over the intersection of the <a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">cone</a> of <a href="/wiki/Positive-definite_matrix#Negative-definite,_semidefinite_and_indefinite_matrices" class="mw-redirect" title="Positive-definite matrix">positive semidefinite</a> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> with an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, i.e., a <a href="/wiki/Spectrahedron" title="Spectrahedron">spectrahedron</a>.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in <a href="/wiki/Operations_research" title="Operations research">operations research</a> and <a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> can be modeled or approximated as semidefinite programming problems. In automatic control theory, SDPs are used in the context of <a href="/wiki/Linear_matrix_inequality" title="Linear matrix inequality">linear matrix inequalities</a>. SDPs are in fact a special case of <a href="/wiki/Conic_optimization" title="Conic optimization">cone programming</a> and can be efficiently solved by <a href="/wiki/Interior_point_methods" class="mw-redirect" title="Interior point methods">interior point methods</a>. All <a href="/wiki/Linear_programming" title="Linear programming">linear programs</a> and (convex) <a href="/wiki/Quadratic_programming" title="Quadratic programming">quadratic programs</a> can be expressed as SDPs, and <a href="/wiki/Sum-of-squares_optimization" title="Sum-of-squares optimization">via hierarchies of SDPs</a> the solutions of polynomial optimization problems can be approximated. Semidefinite programming has been used in the <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimization</a> of complex systems. In recent years, some quantum query complexity problems have been formulated in terms of semidefinite programs. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation_and_definition">Motivation and definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=1" title="Edit section: Motivation and definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Initial_motivation">Initial motivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=2" title="Edit section: Initial motivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a> problem is one in which we wish to maximize or minimize a linear objective function of real variables over a <a href="/wiki/Polytope" title="Polytope">polytope</a>. In semidefinite programming, we instead use real-valued vectors and are allowed to take the dot product of vectors; nonnegativity constraints on real variables in LP (<i>linear programming</i>) are replaced by semidefiniteness constraints on matrix variables in SDP (<i>semidefinite programming</i>). Specifically, a general semidefinite programming problem can be defined as any mathematical programming problem of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{x^{1},\ldots ,x^{n}\in \mathbb {R} ^{n}}}&{\displaystyle \sum _{i,j\in [n]}c_{i,j}(x^{i}\cdot x^{j})}\\{\text{subject to}}&{\displaystyle \sum _{i,j\in [n]}a_{i,j,k}(x^{i}\cdot x^{j})\leq b_{k}}{\text{ for all }}k\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </munder> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \min _{x^{1},\ldots ,x^{n}\in \mathbb {R} ^{n}}}&{\displaystyle \sum _{i,j\in [n]}c_{i,j}(x^{i}\cdot x^{j})}\\{\text{subject to}}&{\displaystyle \sum _{i,j\in [n]}a_{i,j,k}(x^{i}\cdot x^{j})\leq b_{k}}{\text{ for all }}k\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a6aca1dab770d073eff7e64ba7591a4249b297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.706ex; height:12.509ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{x^{1},\ldots ,x^{n}\in \mathbb {R} ^{n}}}&{\displaystyle \sum _{i,j\in [n]}c_{i,j}(x^{i}\cdot x^{j})}\\{\text{subject to}}&{\displaystyle \sum _{i,j\in [n]}a_{i,j,k}(x^{i}\cdot x^{j})\leq b_{k}}{\text{ for all }}k\\\end{array}}}" /></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i,j},a_{i,j,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i,j},a_{i,j,k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4a513404c347c5507f04f2730556818439b431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.453ex; height:2.343ex;" alt="{\displaystyle c_{i,j},a_{i,j,k}}" /></span>, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca33a19a52bbe58090767bcfb0aae551c925dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.086ex; height:2.509ex;" alt="{\displaystyle b_{k}}" /></span> are real numbers and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{i}\cdot x^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}\cdot x^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc795ce41c12413be801e2cae53254fa9d49f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.048ex; height:2.676ex;" alt="{\displaystyle x^{i}\cdot x^{j}}" /></span> is the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8cd88951c7c0e3b181f956531b0a878bbed203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle x^{i}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720ef99822043e29664d4eccb8f866c897fcab09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.239ex; height:2.676ex;" alt="{\displaystyle x^{j}}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_formulations">Equivalent formulations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=3" title="Edit section: Equivalent formulations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /></span> matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> is said to be <a href="/wiki/Positive-definite_matrix#Positive-semidefinite" class="mw-redirect" title="Positive-definite matrix">positive semidefinite</a> if it is the <a href="/wiki/Gram_matrix" title="Gram matrix">Gram matrix</a> of some vectors (i.e. if there exist vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1},\ldots ,x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1},\ldots ,x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ba4766c77e384fcf3fa5a11063d993cd794355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:3.009ex;" alt="{\displaystyle x^{1},\ldots ,x^{n}}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i,j}=x^{i}\cdot x^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i,j}=x^{i}\cdot x^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645652b0bc80c2ea1cc0fe5b3e8031c8d09fe95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.121ex; height:3.343ex;" alt="{\displaystyle m_{i,j}=x^{i}\cdot x^{j}}" /></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cbf8bbc622154cda8208d6e339495fe16a1f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" /></span>). If this is the case, we denote this as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf2842f260245474bb8f1ef747654ef83b4d3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.703ex; height:2.343ex;" alt="{\displaystyle M\succeq 0}" /></span>. Note that there are several other equivalent definitions of being positive semidefinite, for example, positive semidefinite matrices are <a href="/wiki/Self-adjoint" title="Self-adjoint">self-adjoint</a> matrices that have only non-negative <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>. </p><p> Denote by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c00c9d3635f5230e6ac11902a50f2323794ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.343ex;" alt="{\displaystyle \mathbb {S} ^{n}}" /></span> the space of all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /></span> real symmetric matrices. The space is equipped with the <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {trace}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {trace}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a49b758d9c0cece6b4cd2f0e0babf66de12f71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.044ex; height:2.009ex;" alt="{\displaystyle {\rm {trace}}}" /></span> denotes the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a>):</p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A,B\rangle :={\rm {trace}}(A^{T}B)=\sum _{i=1,j=1}^{n}A_{ij}B_{ij}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A,B\rangle :={\rm {trace}}(A^{T}B)=\sum _{i=1,j=1}^{n}A_{ij}B_{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0369185982b8b9db3844b3a7af5d676cc1b2acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.342ex; height:7.176ex;" alt="{\displaystyle \langle A,B\rangle :={\rm {trace}}(A^{T}B)=\sum _{i=1,j=1}^{n}A_{ij}B_{ij}.}" /></span></p></blockquote><p>We can rewrite the mathematical program given in the previous section equivalently as </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle \leq b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </munder> </mstyle> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em"></mspace> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>X</mi> <mo>⪰</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle \leq b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6453385cd37f8a528f0c983eab7b93df9a3d08e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:41.792ex; height:10.843ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle \leq b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}" /></span></dd></dl> <p>where entry <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cbf8bbc622154cda8208d6e339495fe16a1f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" /></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c_{i,j}+c_{j,i}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c_{i,j}+c_{j,i}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ab41f1d344fa8a637433eead3def5b1e58639d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.559ex; height:5.343ex;" alt="{\displaystyle {\frac {c_{i,j}+c_{j,i}}{2}}}" /></span> from the previous section and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72095229db907e86eb4343cb4736429fcc56507d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle A_{k}}" /></span> is a symmetric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" /></span> matrix having <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cbf8bbc622154cda8208d6e339495fe16a1f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" /></span>th entry <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a_{i,j,k}+a_{j,i,k}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a_{i,j,k}+a_{j,i,k}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1db430bf0224d6ac7c53e19fcb5e5188dd7aaf79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.633ex; height:5.343ex;" alt="{\displaystyle {\frac {a_{i,j,k}+a_{j,i,k}}{2}}}" /></span> from the previous section. Thus, the matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72095229db907e86eb4343cb4736429fcc56507d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle A_{k}}" /></span> are symmetric and the above inner products are well-defined. </p><p>Note that if we add <a href="/wiki/Slack_variable" title="Slack variable">slack variables</a> appropriately, this SDP can be converted to an <i>equational form</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle =b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </munder> </mstyle> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em"></mspace> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>X</mi> <mo>⪰</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle =b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6aaa54b9729f3747e1603763f5d2c3076594c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:41.792ex; height:10.843ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle =b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}" /></span></dd></dl> <p>For convenience, an SDP may be specified in a slightly different, but equivalent form. For example, linear expressions involving nonnegative <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> variables may be added to the program specification. This remains an SDP because each variable can be incorporated into the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> as a diagonal entry (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{ii}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{ii}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864f66cf9bc819512778139d0094970689a541b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.291ex; height:2.509ex;" alt="{\displaystyle X_{ii}}" /></span> for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span>). To ensure that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{ii}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{ii}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb49ed4907d5792f019fc0ee409ab98c07348545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.552ex; height:2.509ex;" alt="{\displaystyle X_{ii}\geq 0}" /></span>, constraints <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{ij}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{ij}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9dcb2e9f4541c9090ef159a9319168a28b7eaa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.662ex; height:2.843ex;" alt="{\displaystyle X_{ij}=0}" /></span> can be added for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\neq i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\neq i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b7538abe3f69348eb371e48d7b33a8994610e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:4.886ex; height:2.676ex;" alt="{\displaystyle j\neq i}" /></span>. As another example, note that for any positive semidefinite matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span>, there exists a set of vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbcad592f3bd6588754054c8f46ea693cbe3d8a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.252ex; height:2.843ex;" alt="{\displaystyle \{v_{i}\}}" /></span> such that the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" /></span> entry of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{ij}=(v_{i},v_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{ij}=(v_{i},v_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de7bb32daac8405b103d19d2eefe3842c8976af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.308ex; height:3.009ex;" alt="{\displaystyle X_{ij}=(v_{i},v_{j})}" /></span> the <a href="/wiki/Dot_product" title="Dot product">scalar product</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73fffa4919c0d6268f6a8d9f38c04dd3296fd0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.037ex; height:2.343ex;" alt="{\displaystyle v_{j}}" /></span>. Therefore, SDPs are often formulated in terms of linear expressions on scalar products of vectors. Given the solution to the SDP in the standard form, the vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbcad592f3bd6588754054c8f46ea693cbe3d8a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.252ex; height:2.843ex;" alt="{\displaystyle \{v_{i}\}}" /></span> can be recovered in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b04f5c5cfea38f43406d9442387ad28555e2609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{3})}" /></span> time (e.g., by using an incomplete <a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a> of X). </p> <div class="mw-heading mw-heading2"><h2 id="Relations_to_other_optimization_problems">Relations to other optimization problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=4" title="Edit section: Relations to other optimization problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The space of semidefinite matrices is a <a href="/wiki/Convex_cone" title="Convex cone">convex cone</a>. Therefore, SDP is a special case of <a href="/wiki/Conic_optimization" title="Conic optimization">conic optimization</a>, which is a special case of convex optimization. </p><p>When the matrix <i>C</i> is diagonal, the inner products <<i>C</i>,<i>X</i>> is equivalent to a vector product of the diagonal of <i>C</i> and the diagonal of <i>X</i>. Analogously, when the matrices <i>A<sub>k</sub></i> are diagonal, the corresponding inner products are equivalent to vector products. In these vector products, only the diagonal elements of <i>X</i> are used, so we can add constraints equating the non-diagonal elements of <i>X</i> to 0. The condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1155afdd8f7cecf8636811cab513ced6cc597a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.241ex; height:2.343ex;" alt="{\displaystyle X\succeq 0}" /></span> is then equivalent to the condition that all diagonal elements of <i>X</i> are non-negative. Then, the resulting SDP becomes a <a href="/wiki/Linear_program" class="mw-redirect" title="Linear program">linear program</a> in which the variables are the diagonal elements of <i>X</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Duality_theory">Duality theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=5" title="Edit section: Duality theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=6" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analogously to linear programming, given a general SDP of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{i},X\rangle =b_{i},\quad i=1,\ldots ,m\\&X\succeq 0\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </munder> </mstyle> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em"></mspace> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>X</mi> <mo>⪰</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{i},X\rangle =b_{i},\quad i=1,\ldots ,m\\&X\succeq 0\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e8d6b7beccfb5a3b0cbca98f30af1ded2f3d104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:40.805ex; height:10.843ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \min _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{i},X\rangle =b_{i},\quad i=1,\ldots ,m\\&X\succeq 0\end{array}}}" /></span></dd></dl> <p>(the primal problem or P-SDP), we define the <i><a href="/wiki/Dual_problem" class="mw-redirect" title="Dual problem">dual</a></i> semidefinite program (D-SDP) as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \max _{y\in \mathbb {R} ^{m}}}&b^{T}y\\{\text{subject to}}&{\displaystyle \sum _{i=1}^{m}}y_{i}A_{i}\preceq C\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </munder> </mstyle> </mrow> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> </mstyle> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⪯</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \max _{y\in \mathbb {R} ^{m}}}&b^{T}y\\{\text{subject to}}&{\displaystyle \sum _{i=1}^{m}}y_{i}A_{i}\preceq C\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf1c39f6545a009cd366a1e34bed105ae1b8e3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:25.606ex; height:11.843ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \max _{y\in \mathbb {R} ^{m}}}&b^{T}y\\{\text{subject to}}&{\displaystyle \sum _{i=1}^{m}}y_{i}A_{i}\preceq C\end{array}}}" /></span></dd></dl> <p>where for any two matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\succeq Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>⪰</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\succeq Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929d0eecf10f2b4fd620ae54a2aaedb7b759ead0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P\succeq Q}" /></span> means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P-Q\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>−</mo> <mi>Q</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P-Q\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734b73e0199057fd06b3511a01dc85735f7fcfa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.685ex; height:2.509ex;" alt="{\displaystyle P-Q\succeq 0}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Weak_duality">Weak duality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=7" title="Edit section: Weak duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Weak_duality" title="Weak duality">weak duality</a> theorem states that the value of the primal SDP is at least the value of the dual SDP. Therefore, any feasible solution to the dual SDP lower-bounds the primal SDP value, and conversely, any feasible solution to the primal SDP upper-bounds the dual SDP value. This is because </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle C,X\rangle -b^{T}y=\langle C,X\rangle -\sum _{i=1}^{m}y_{i}b_{i}=\langle C,X\rangle -\sum _{i=1}^{m}y_{i}\langle A_{i},X\rangle =\langle C-\sum _{i=1}^{m}y_{i}A_{i},X\rangle \geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>y</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle C,X\rangle -b^{T}y=\langle C,X\rangle -\sum _{i=1}^{m}y_{i}b_{i}=\langle C,X\rangle -\sum _{i=1}^{m}y_{i}\langle A_{i},X\rangle =\langle C-\sum _{i=1}^{m}y_{i}A_{i},X\rangle \geq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5c925608db5d5be0a49416c83fbaf146717a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:84.214ex; height:6.843ex;" alt="{\displaystyle \langle C,X\rangle -b^{T}y=\langle C,X\rangle -\sum _{i=1}^{m}y_{i}b_{i}=\langle C,X\rangle -\sum _{i=1}^{m}y_{i}\langle A_{i},X\rangle =\langle C-\sum _{i=1}^{m}y_{i}A_{i},X\rangle \geq 0,}" /></span></dd></dl> <p>where the last inequality is because both matrices are positive semidefinite, and the result of this function is sometimes referred to as duality gap. </p> <div class="mw-heading mw-heading3"><h3 id="Strong_duality">Strong duality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=8" title="Edit section: Strong duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the value of the primal and dual SDPs are equal, the SDP is said to satisfy the <a href="/wiki/Strong_duality" title="Strong duality">strong duality</a> property. Unlike <a href="/wiki/Linear_programming" title="Linear programming">linear programs</a>, where every dual linear program has optimal objective equal to the primal objective, not every SDP satisfies strong duality; in general, the value of the dual SDP may lie strictly below the value of the primal, and the P-SDP and D-SDP satisfy the following properties: </p><p>(i) Suppose the primal problem (P-SDP) is bounded below and strictly feasible (i.e., there exists <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}\in \mathbb {S} ^{n},X_{0}\succ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≻</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}\in \mathbb {S} ^{n},X_{0}\succ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fa36bd75a86b645e91f7e2be97cb9b80a5e6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.604ex; height:2.676ex;" alt="{\displaystyle X_{0}\in \mathbb {S} ^{n},X_{0}\succ 0}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A_{i},X_{0}\rangle =b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A_{i},X_{0}\rangle =b_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63e4ca89324d7fb6f72640005b6010fcafb34f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.26ex; height:2.843ex;" alt="{\displaystyle \langle A_{i},X_{0}\rangle =b_{i}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74690f54a3c93a332ecb2935e900178b9a555483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.282ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,m}" /></span>). Then there is an optimal solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcfcfa0fbced647ea73759c68ffd7a028729d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.215ex; height:2.676ex;" alt="{\displaystyle y^{*}}" /></span> to (D-SDP) and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle C,X^{*}\rangle =b^{T}y^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle C,X^{*}\rangle =b^{T}y^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666dec6c6f286893a01e4451622f2236d5f67b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.007ex; height:3.176ex;" alt="{\displaystyle \langle C,X^{*}\rangle =b^{T}y^{*}.}" /></span></dd></dl> <p>(ii) Suppose the dual problem (D-SDP) is bounded above and strictly feasible (i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{m}(y_{0})_{i}A_{i}\prec C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≺</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{m}(y_{0})_{i}A_{i}\prec C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426496ab28ebdf2dfbdc688368d6a07f55aeb084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.565ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{m}(y_{0})_{i}A_{i}\prec C}" /></span> for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}\in \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867165284591f4b9d2ce0abab35ab599ad47258d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.387ex; height:2.676ex;" alt="{\displaystyle y_{0}\in \mathbb {R} ^{m}}" /></span>). Then there is an optimal solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}" /></span> to (P-SDP) and the equality from (i) holds. </p><p>A sufficient condition for strong duality to hold for a SDP problem (and in general, for any convex optimization problem) is the <a href="/wiki/Slater%27s_condition" title="Slater's condition">Slater's condition</a>. It is also possible to attain strong duality for SDPs without additional regularity conditions by using an extended dual problem proposed by Ramana.<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Example_1">Example 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=10" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider three random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>. A given set of <a href="/wiki/Correlation" title="Correlation">correlation coefficients</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{AB},\ \rho _{AC},\rho _{BC}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{AB},\ \rho _{AC},\rho _{BC}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d792e486fa944dbaf3fb96a96a6fb5a5777d3c6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.409ex; height:2.176ex;" alt="{\displaystyle \rho _{AB},\ \rho _{AC},\rho _{BC}}" /></span> are possible if and only if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&\rho _{AB}&\rho _{AC}\\\rho _{AB}&1&\rho _{BC}\\\rho _{AC}&\rho _{BC}&1\end{pmatrix}}\succeq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⪰</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&\rho _{AB}&\rho _{AC}\\\rho _{AB}&1&\rho _{BC}\\\rho _{AC}&\rho _{BC}&1\end{pmatrix}}\succeq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c120a50f0b6d59134f7832433be5bc115175064d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.148ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}1&\rho _{AB}&\rho _{AC}\\\rho _{AB}&1&\rho _{BC}\\\rho _{AC}&\rho _{BC}&1\end{pmatrix}}\succeq 0.}" /></span></dd></dl> <p>This matrix is called the <a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">correlation matrix</a>. Suppose that we know from some prior knowledge (empirical results of an experiment, for example) that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -0.2\leq \rho _{AB}\leq -0.1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0.2</mn> <mo>≤</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>≤</mo> <mo>−</mo> <mn>0.1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0.2\leq \rho _{AB}\leq -0.1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f920ab99dfd0dbcff34f7bd91271f9d4f0118be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.671ex; height:2.676ex;" alt="{\displaystyle -0.2\leq \rho _{AB}\leq -0.1}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.4\leq \rho _{BC}\leq 0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.4</mn> <mo>≤</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> <mo>≤</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.4\leq \rho _{BC}\leq 0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5d4020689bc8dd1170546f47e4ca23922a43f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.071ex; height:2.676ex;" alt="{\displaystyle 0.4\leq \rho _{BC}\leq 0.5}" /></span>. The problem of determining the smallest and largest values that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{AC}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{AC}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fa03bd86f6b4bc874f910508a17690671b7694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.496ex; height:2.176ex;" alt="{\displaystyle \rho _{AC}\ }" /></span> can take is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \min /\max }&x_{13}\\{\text{subject to}}&-0.2\leq x_{12}\leq -0.1\\&0.4\leq x_{23}\leq 0.5\\&{\begin{pmatrix}1&x_{12}&x_{13}\\x_{12}&1&x_{23}\\x_{13}&x_{23}&1\end{pmatrix}}\succeq 0\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo movablelimits="true" form="prefix">max</mo> </mstyle> </mrow> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mo>−</mo> <mn>0.2</mn> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>≤</mo> <mo>−</mo> <mn>0.1</mn> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mn>0.4</mn> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>≤</mo> <mn>0.5</mn> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⪰</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \min /\max }&x_{13}\\{\text{subject to}}&-0.2\leq x_{12}\leq -0.1\\&0.4\leq x_{23}\leq 0.5\\&{\begin{pmatrix}1&x_{12}&x_{13}\\x_{12}&1&x_{23}\\x_{13}&x_{23}&1\end{pmatrix}}\succeq 0\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3502bbf161995435f536d8b54610ece632e58cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:36.554ex; height:19.176ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \min /\max }&x_{13}\\{\text{subject to}}&-0.2\leq x_{12}\leq -0.1\\&0.4\leq x_{23}\leq 0.5\\&{\begin{pmatrix}1&x_{12}&x_{13}\\x_{12}&1&x_{23}\\x_{13}&x_{23}&1\end{pmatrix}}\succeq 0\end{array}}}" /></span></dd></dl> <p>We set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{AB}=x_{12},\ \rho _{AC}=x_{13},\ \rho _{BC}=x_{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{AB}=x_{12},\ \rho _{AC}=x_{13},\ \rho _{BC}=x_{23}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505d5d0ba73ca7a7ec23221014c7f4a171ad4b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.903ex; height:2.176ex;" alt="{\displaystyle \rho _{AB}=x_{12},\ \rho _{AC}=x_{13},\ \rho _{BC}=x_{23}}" /></span> to obtain the answer. This can be formulated by an SDP. We handle the inequality constraints by augmenting the variable matrix and introducing <a href="/wiki/Slack_variable" title="Slack variable">slack variables</a>, for example </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} \left(\left({\begin{array}{cccccc}0&1&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&1&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\end{array}}\right)\cdot \left({\begin{array}{cccccc}1&x_{12}&x_{13}&0&0&0\\x_{12}&1&x_{23}&0&0&0\\x_{13}&x_{23}&1&0&0&0\\0&0&0&s_{1}&0&0\\0&0&0&0&s_{2}&0\\0&0&0&0&0&s_{3}\end{array}}\right)\right)=x_{12}+s_{1}=-0.1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} \left(\left({\begin{array}{cccccc}0&1&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&1&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\end{array}}\right)\cdot \left({\begin{array}{cccccc}1&x_{12}&x_{13}&0&0&0\\x_{12}&1&x_{23}&0&0&0\\x_{13}&x_{23}&1&0&0&0\\0&0&0&s_{1}&0&0\\0&0&0&0&s_{2}&0\\0&0&0&0&0&s_{3}\end{array}}\right)\right)=x_{12}+s_{1}=-0.1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc2fb4997d80ea0c5ba7832065f030ab7a6d73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:83.006ex; height:19.176ex;" alt="{\displaystyle \mathrm {tr} \left(\left({\begin{array}{cccccc}0&1&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&1&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\end{array}}\right)\cdot \left({\begin{array}{cccccc}1&x_{12}&x_{13}&0&0&0\\x_{12}&1&x_{23}&0&0&0\\x_{13}&x_{23}&1&0&0&0\\0&0&0&s_{1}&0&0\\0&0&0&0&s_{2}&0\\0&0&0&0&0&s_{3}\end{array}}\right)\right)=x_{12}+s_{1}=-0.1}" /></span> </p><p>Solving this SDP gives the minimum and maximum values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{AC}=x_{13}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{AC}=x_{13}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6462a47579ffc08b1e6a2d9bf7b6b3c091754f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.801ex; height:2.176ex;" alt="{\displaystyle \rho _{AC}=x_{13}\ }" /></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -0.978}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0.978</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0.978}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfb7acab56698f01bbf57ef8942a68bc87d78e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.105ex; height:2.343ex;" alt="{\displaystyle -0.978}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.872}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.872</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.872}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c546328e85affa0a1dac82798bcc46d95483b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.297ex; height:2.176ex;" alt="{\displaystyle 0.872}" /></span> respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=11" title="Edit section: Example 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the problem </p> <dl><dd>minimize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(c^{T}x)^{2}}{d^{T}x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(c^{T}x)^{2}}{d^{T}x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69ec09f2a59ad662c711bf3444bb43bb79d1a449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.425ex; height:6.176ex;" alt="{\displaystyle {\frac {(c^{T}x)^{2}}{d^{T}x}}}" /></span></dd> <dd>subject to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+b\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+b\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4c5b3a88d9a58dfe863955d8c41437b99b969e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.172ex; height:2.343ex;" alt="{\displaystyle Ax+b\geq 0}" /></span></dd></dl> <p>where we assume that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{T}x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{T}x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5efa2a8b82507ab46f51ded61b1db03ff129966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.198ex; height:2.676ex;" alt="{\displaystyle d^{T}x>0}" /></span> whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+b\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+b\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4c5b3a88d9a58dfe863955d8c41437b99b969e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.172ex; height:2.343ex;" alt="{\displaystyle Ax+b\geq 0}" /></span>. </p><p>Introducing an auxiliary variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span> the problem can be reformulated: </p> <dl><dd>minimize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span></dd> <dd>subject to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+b\geq 0,\,{\frac {(c^{T}x)^{2}}{d^{T}x}}\leq t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>≤</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+b\geq 0,\,{\frac {(c^{T}x)^{2}}{d^{T}x}}\leq t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f8be8b8d137fe01a5e0bac8ce8f8acb263e3e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.956ex; height:6.176ex;" alt="{\displaystyle Ax+b\geq 0,\,{\frac {(c^{T}x)^{2}}{d^{T}x}}\leq t}" /></span></dd></dl> <p>In this formulation, the objective is a linear function of the variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef10a0c06f8a5238d439b9a7bde431605db5190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.203ex; height:2.343ex;" alt="{\displaystyle x,t}" /></span>. </p><p>The first restriction can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {diag}}(Ax+b)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">diag</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {diag}}(Ax+b)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd7e7e0ea4b2a17fc88b58d7dd6609770424b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.844ex; height:2.843ex;" alt="{\displaystyle {\textbf {diag}}(Ax+b)\geq 0}" /></span></dd></dl> <p>where the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {diag}}(Ax+b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">diag</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {diag}}(Ax+b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c32e261cfb192d0c2c1240d1e5aaedc39cdf33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.584ex; height:2.843ex;" alt="{\displaystyle {\textbf {diag}}(Ax+b)}" /></span> is the square matrix with values in the diagonal equal to the elements of the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97bf01130cd91babee50cc269c22025c6673654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.911ex; height:2.343ex;" alt="{\displaystyle Ax+b}" /></span>. </p><p>The second restriction can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle td^{T}x-(c^{T}x)^{2}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle td^{T}x-(c^{T}x)^{2}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b577c0d31c3051d87a59df91005fc052902b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.467ex; height:3.176ex;" alt="{\displaystyle td^{T}x-(c^{T}x)^{2}\geq 0}" /></span></dd></dl> <p>Defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /></span> as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\left[{\begin{array}{cc}t&c^{T}x\\c^{T}x&d^{T}x\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mtd> <mtd> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\left[{\begin{array}{cc}t&c^{T}x\\c^{T}x&d^{T}x\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8c58bc318a5b533c7fc909081ed4bb2420db39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.215ex; height:6.176ex;" alt="{\displaystyle D=\left[{\begin{array}{cc}t&c^{T}x\\c^{T}x&d^{T}x\end{array}}\right]}" /></span></dd></dl> <p>We can use the theory of Schur Complements to see that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c27d715b408af3b2d47a3580b76b1eef88c709c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.185ex; height:2.343ex;" alt="{\displaystyle D\succeq 0}" /></span></dd></dl> <p>(Boyd and Vandenberghe, 1996) </p><p>The semidefinite program associated with this problem is </p> <dl><dd>minimize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span></dd> <dd>subject to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{array}{ccc}{\textbf {diag}}(Ax+b)&0&0\\0&t&c^{T}x\\0&c^{T}x&d^{T}x\end{array}}\right]\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">diag</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>t</mi> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mtd> <mtd> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{ccc}{\textbf {diag}}(Ax+b)&0&0\\0&t&c^{T}x\\0&c^{T}x&d^{T}x\end{array}}\right]\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9221c0488d143a6a9173540acdf8ce1b909c84d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:34.005ex; height:9.509ex;" alt="{\displaystyle \left[{\begin{array}{ccc}{\textbf {diag}}(Ax+b)&0&0\\0&t&c^{T}x\\0&c^{T}x&d^{T}x\end{array}}\right]\succeq 0}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Example_3_(Goemans–Williamson_max_cut_approximation_algorithm)"><span id="Example_3_.28Goemans.E2.80.93Williamson_max_cut_approximation_algorithm.29"></span>Example 3 (Goemans–Williamson max cut approximation algorithm)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=12" title="Edit section: Example 3 (Goemans–Williamson max cut approximation algorithm)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Semidefinite programs are important tools for developing approximation algorithms for NP-hard maximization problems. The first approximation algorithm based on an SDP is due to <a href="/wiki/Michel_Goemans" title="Michel Goemans">Michel Goemans</a> and <a href="/wiki/David_P._Williamson" title="David P. Williamson">David P. Williamson</a> (JACM, 1995).<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Chap.1">: Chap.1 </span></sup> They studied the <a href="/wiki/Maximum_cut" title="Maximum cut">max cut problem</a>: Given a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <i>G</i> = (<i>V</i>, <i>E</i>), output a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of the vertices <i>V</i> so as to maximize the number of edges crossing from one side to the other. This problem can be expressed as an <a href="/wiki/Quadratic_programming" title="Quadratic programming">integer quadratic program</a>: </p> <dl><dd>Maximize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{(i,j)\in E}{\frac {1-v_{i}v_{j}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{(i,j)\in E}{\frac {1-v_{i}v_{j}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b9ccbffe42e2593e53a1a98f986fefa090ed11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.171ex; height:7.176ex;" alt="{\displaystyle \sum _{(i,j)\in E}{\frac {1-v_{i}v_{j}}{2}},}" /></span> such that each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}\in \{1,-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}\in \{1,-1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8795e72f7c025f474ff1e0d7b9574a3ee8031dac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.26ex; height:2.843ex;" alt="{\displaystyle v_{i}\in \{1,-1\}}" /></span>.</dd></dl> <p>Unless <a href="/wiki/P_%3D_NP" class="mw-redirect" title="P = NP">P = NP</a>, we cannot solve this maximization problem efficiently. However, Goemans and Williamson observed a general three-step procedure for attacking this sort of problem: </p> <ol><li><i>Relax</i> the integer quadratic program into an SDP.</li> <li>Solve the SDP (to within an arbitrarily small additive error <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" /></span>).</li> <li><i>Round</i> the SDP solution to obtain an approximate solution to the original integer quadratic program.</li></ol> <p>For max cut, the most natural relaxation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max \sum _{(i,j)\in E}{\frac {1-\langle v_{i},v_{j}\rangle }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max \sum _{(i,j)\in E}{\frac {1-\langle v_{i},v_{j}\rangle }{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a491024a168bb778f03af89c105e258e0240ed14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.727ex; height:7.343ex;" alt="{\displaystyle \max \sum _{(i,j)\in E}{\frac {1-\langle v_{i},v_{j}\rangle }{2}},}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert v_{i}\rVert ^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert v_{i}\rVert ^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29753338b6b335b6bef9aa263e6f5728c4786550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.567ex; height:3.176ex;" alt="{\displaystyle \lVert v_{i}\rVert ^{2}=1}" /></span>, where the maximization is over vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbcad592f3bd6588754054c8f46ea693cbe3d8a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.252ex; height:2.843ex;" alt="{\displaystyle \{v_{i}\}}" /></span> instead of integer scalars.</dd></dl> <p>This is an SDP because the objective function and constraints are all linear functions of vector inner products. Solving the SDP gives a set of unit vectors in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R^{n}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R^{n}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c191fa5e67baf8cbaa9285ed4268645f53a66f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.286ex; height:2.343ex;" alt="{\displaystyle \mathbf {R^{n}} }" /></span>; since the vectors are not required to be collinear, the value of this relaxed program can only be higher than the value of the original quadratic integer program. Finally, a rounding procedure is needed to obtain a partition. Goemans and Williamson simply choose a uniformly random hyperplane through the origin and divide the vertices according to which side of the hyperplane the corresponding vectors lie. Straightforward analysis shows that this procedure achieves an expected <i>approximation ratio</i> (performance guarantee) of 0.87856 - ε. (The expected value of the cut is the sum over edges of the probability that the edge is cut, which is proportional to the angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{-1}\langle v_{i},v_{j}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{-1}\langle v_{i},v_{j}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc368071c186c161d40bbab5ba5461d128eccbb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.252ex; height:3.343ex;" alt="{\displaystyle \cos ^{-1}\langle v_{i},v_{j}\rangle }" /></span> between the vectors at the endpoints of the edge over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span>. Comparing this probability to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\langle v_{i},v_{j}\rangle )/{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\langle v_{i},v_{j}\rangle )/{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004b81de8a07cdd825fe18d97dab9ce7281e11d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.945ex; height:3.009ex;" alt="{\displaystyle (1-\langle v_{i},v_{j}\rangle )/{2}}" /></span>, in expectation the ratio is always at least 0.87856.) Assuming the <a href="/wiki/Unique_games_conjecture" title="Unique games conjecture">unique games conjecture</a>, it can be shown that this approximation ratio is essentially optimal. </p><p>Since the original paper of Goemans and Williamson, SDPs have been applied to develop numerous approximation algorithms. Subsequently, Prasad Raghavendra has developed a general framework for constraint satisfaction problems based on the <a href="/wiki/Unique_games_conjecture" title="Unique games conjecture">unique games conjecture</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_applications">Other applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=13" title="Edit section: Other applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Semidefinite programming has been applied to find approximate solutions to combinatorial optimization problems, such as the solution of the <a href="/wiki/Max_cut" class="mw-redirect" title="Max cut">max cut</a> problem with an <a href="/wiki/Approximation_ratio" class="mw-redirect" title="Approximation ratio">approximation ratio</a> of 0.87856. SDPs are also used in geometry to determine tensegrity graphs, and arise in control theory as <a href="/wiki/Linear_matrix_inequality" title="Linear matrix inequality">LMIs</a>, and in inverse elliptic coefficient problems as convex, non-linear, semidefiniteness constraints.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It is also widely used in physics to constrain <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theories</a> with the <a href="/wiki/Conformal_bootstrap" title="Conformal bootstrap">conformal bootstrap</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Run-time_complexity">Run-time complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=14" title="Edit section: Run-time complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>semidefinite feasibility problem</b> (SDF) is the following <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a>: given an SDP, decide whether it has at least one feasible solution. The exact run-time complexity of this problem is unknown (as of 1997). However, Ramana proved the following:<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li>In the <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> model, SDF is in NP iff it is in co-NP. Therefore, SDF is not NP-complete unless NP=coNP.</li> <li>In the <a href="/wiki/Blum%E2%80%93Shub%E2%80%93Smale_machine" title="Blum–Shub–Smale machine">Blum–Shub–Smale machine</a> model, SDF is in the intersection of NP and co-NP.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Algorithms_for_solving_SDPs">Algorithms for solving SDPs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=15" title="Edit section: Algorithms for solving SDPs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several types of algorithms for solving SDPs. These algorithms output the value of the SDP up to an additive error <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" /></span> in time that is polynomial in the program description size and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(1/\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(1/\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1554f0addb7dd2200260f2401e11a503ae7eeedf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.05ex; height:2.843ex;" alt="{\displaystyle \log(1/\epsilon )}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ellipsoid_method">Ellipsoid method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=16" title="Edit section: Ellipsoid method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> The <a href="/wiki/Ellipsoid_method" title="Ellipsoid method">ellipsoid method</a> is a general method for convex programming, and can be used in particular to solve SDPs. In the context of SDPs, the ellipsoid method provides the following guarantee.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Thm.2.6.1">: Thm.2.6.1 </span></sup>Consider an SDP in the following equational form:</p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}{\displaystyle \max _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle =b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </munder> </mstyle> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to</mtext> </mrow> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em"></mspace> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi>X</mi> <mo>⪰</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}{\displaystyle \max _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle =b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e96fc52864c4c9571a5a21779d2453a6a46d705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:41.792ex; height:10.843ex;" alt="{\displaystyle {\begin{array}{rl}{\displaystyle \max _{X\in \mathbb {S} ^{n}}}&\langle C,X\rangle \\{\text{subject to}}&\langle A_{k},X\rangle =b_{k},\quad k=1,\ldots ,m\\&X\succeq 0.\end{array}}}" /></span></p></blockquote><p>Let <i>L</i> be the affine subspace of matrices in S<sup>n</sup> satisfying the <i>m</i> equational constraints; so the SDP can be written as: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{X\in L}\langle C,X\rangle {\text{ subject to }}X\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>∈</mo> <mi>L</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> subject to </mtext> </mrow> <mi>X</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{X\in L}\langle C,X\rangle {\text{ subject to }}X\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709f7808aa5ebafe9bc6cd4eb2a83c9065890397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.148ex; height:4.176ex;" alt="{\displaystyle \max _{X\in L}\langle C,X\rangle {\text{ subject to }}X\succeq 0}" /></span>. Suppose all coefficients in the SDP are rational numbers. Let <i>R</i> be an explicitly given upper bound on the maximum <a href="/wiki/Frobenius_norm" class="mw-redirect" title="Frobenius norm">Frobenius norm</a> of a feasible solution, and <i>ε></i>0 a constant. A matrix <i>X</i> in S<sup>n</sup> is called <i>ε-deep</i> if every matrix <i>Y</i> in <i>L</i> with <a href="/wiki/Frobenius_distance" class="mw-redirect" title="Frobenius distance">Frobenius distance</a> at most <i>ε</i> from <i>X</i> satisfies the feasibility condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e03c568f6014780cfd2704442a884001a3aaa71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.034ex; height:2.343ex;" alt="{\displaystyle Y\succeq 0}" /></span>. Denote <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{deep}:=\sup\{\langle C,X\rangle :X{\text{ is }}\epsilon {\text{-deep}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>e</mi> <mi>p</mi> </mrow> </msub> <mo>:=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is </mtext> </mrow> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-deep</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{deep}:=\sup\{\langle C,X\rangle :X{\text{ is }}\epsilon {\text{-deep}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f26cc39665245003d5d927b17ee6846087dc974f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.75ex; height:3.009ex;" alt="{\displaystyle v_{deep}:=\sup\{\langle C,X\rangle :X{\text{ is }}\epsilon {\text{-deep}}\}}" /></span>. The ellipsoid returns one of the following outputs: </p><ul><li>A matrix X* in L (that is, satisfying all linear equality constraints exactly), such that the Frobenius distance between X* and some feasible solution is at most <i>ε</i> (that is, approximately satisfying the inequality constraint <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\succeq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⪰</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\succeq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1155afdd8f7cecf8636811cab513ced6cc597a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.241ex; height:2.343ex;" alt="{\displaystyle X\succeq 0}" /></span>), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle C,X^{*}\rangle \geq v_{deep}-\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>C</mi> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>e</mi> <mi>p</mi> </mrow> </msub> <mo>−</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle C,X^{*}\rangle \geq v_{deep}-\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c92318a2c0dce96e456c3244a23a710c01138d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.122ex; height:3.009ex;" alt="{\displaystyle \langle C,X^{*}\rangle \geq v_{deep}-\epsilon }" /></span> (that is, approximately optimal objective value).</li> <li>A certificate that the problem has no <i>ε-deep</i> solutions (that is, the problem is approximately infeasible).</li></ul> <p>The run-time is polynomial in the binary encodings of the inputs and in log(R/<i>ε</i>), in the <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> model. </p><p>Note that, in general, <i>R</i> may be doubly-exponential in <i>n.</i> In that case, the run-time guarantee of the ellipsoid method is exponential in <i>n</i>. But in most applications, <i>R</i> is not so huge. In these cases, the ellipsoid method is the only known method that guarantees polynomial runtime in the Turing machine model.<i><sup id="cite_ref-:0_1-3" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></i><sup class="reference nowrap"><span title="Page: 23">: 23 </span></sup> But in practice, its performance is not so good. </p> <div class="mw-heading mw-heading3"><h3 id="Interior_point_methods">Interior point methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=17" title="Edit section: Interior point methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most codes are based on <a href="/wiki/Interior_point_methods" class="mw-redirect" title="Interior point methods">interior point methods</a> (CSDP, <a href="/wiki/MOSEK" title="MOSEK">MOSEK</a>, SeDuMi, <a rel="nofollow" class="external text" href="https://www.math.cmu.edu/~reha/sdpt3.html">SDPT3</a>, DSDP, SDPA). These are robust and efficient for general linear SDP problems, but restricted by the fact that the algorithms are second-order methods and need to store and factorize a large (and often dense) matrix. Theoretically, the state-of-the-art high-accuracy SDP algorithms<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> are based on this approach. </p> <div class="mw-heading mw-heading3"><h3 id="First-order_methods">First-order methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=18" title="Edit section: First-order methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>First-order methods for <a href="/wiki/Conic_optimization" title="Conic optimization">conic optimization</a> avoid computing, storing and factorizing a large Hessian matrix and scale to much larger problems than interior point methods, at some cost in accuracy. A first-order method is implemented in the Splitting Cone Solver (SCS).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Another first-order method is the <a href="/wiki/Alternating_direction_method_of_multipliers" class="mw-redirect" title="Alternating direction method of multipliers">alternating direction method of multipliers</a> (ADMM).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This method requires in every step projection on the cone of semidefinite matrices. </p> <div class="mw-heading mw-heading3"><h3 id="Bundle_method">Bundle method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=19" title="Edit section: Bundle method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The code ConicBundle formulates the SDP problem as a <a href="/w/index.php?title=Nonsmooth_optimization&action=edit&redlink=1" class="new" title="Nonsmooth optimization (page does not exist)">nonsmooth optimization</a> problem and solves it by the Spectral Bundle method of nonsmooth optimization. This approach is very efficient for a special class of linear SDP problems. </p> <div class="mw-heading mw-heading3"><h3 id="Other_solving_methods">Other solving methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=20" title="Edit section: Other solving methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algorithms based on <a href="/wiki/Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian method</a> (PENSDP) are similar in behavior to the interior point methods and can be specialized to some very large scale problems. Other algorithms use low-rank information and reformulation of the SDP as a <a href="/wiki/Nonlinear_programming" title="Nonlinear programming">nonlinear programming</a> problem (SDPLR, ManiSDP).<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Approximate_methods">Approximate methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=21" title="Edit section: Approximate methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algorithms that solve SDPs approximately have been proposed as well. The main goal of such methods is to achieve lower complexity in applications where approximate solutions are sufficient and complexity must be minimal. A prominent method that has been used for data detection in multiple-input multiple-output (MIMO) wireless systems is Triangular Approximate SEmidefinite Relaxation (TASER),<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> which operates on the Cholesky decomposition factors of the semidefinite matrix instead of the semidefinite matrix. This method calculates approximate solutions for a max-cut-like problem that are often comparable to solutions from exact solvers but in only 10-20 algorithm iterations. <a href="/wiki/Elad_Hazan" title="Elad Hazan">Hazan</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> has developed an approximate algorithm for solving SDPs with the additional constraint that the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of the variables matrix must be 1. </p> <div class="mw-heading mw-heading2"><h2 id="Preprocessing_algorithms">Preprocessing algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=22" title="Edit section: Preprocessing algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Facial reduction algorithms</b> are algorithms used to preprocess SDPs problems by inspecting the constraints of the problem. These can be used to </p> <ul><li>Detect lack of strict feasibility;</li> <li>Delete redundant rows and columns;</li> <li>Reduce the size of the variable matrix.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Square-root_sum_problem" title="Square-root sum problem">Square-root sum problem</a> - a special case of an SDP feasibility problem.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semidefinite_programming&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGärtnerMatoušek2012" class="citation cs2">Gärtner, Bernd; 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href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a> on Semidefinite Programming</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output 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template">e</abbr></a></li></ul></div><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="/wiki/Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="/wiki/Iterative_method" title="Iterative method">methods</a>, and <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Golden-section_search" title="Golden-section search">Golden-section search</a></li> <li><a href="/wiki/Powell%27s_method" title="Powell's method">Powell's method</a></li> <li><a href="/wiki/Line_search" title="Line search">Line search</a></li> <li><a href="/wiki/Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li> <li><a href="/wiki/Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Trust_region" title="Trust region">Trust region</a></li> <li><a href="/wiki/Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li> <li><a href="/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="/wiki/Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li> <li><a href="/wiki/Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li> <li><a href="/wiki/Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient</a></li> <li><a href="/wiki/Mirror_descent" title="Mirror descent">Mirror</a></li> <li><a href="/wiki/Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li> <li><a href="/wiki/Powell%27s_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Max_paraboloid.svg" class="mw-file-description" title="Optimization computes maxima and minima."><img alt="Graph of a strictly concave quadratic function with unique maximum." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/150px-Max_paraboloid.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/225px-Max_paraboloid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/300px-Max_paraboloid.svg.png 2x" data-file-width="700" data-file-height="560" /></a><figcaption>Optimization computes maxima and minima.</figcaption></figure></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrier_function" title="Barrier function">Barrier methods</a></li> <li><a href="/wiki/Penalty_method" title="Penalty method">Penalty methods</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li> <li><a href="/wiki/Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li> <li><a href="/wiki/Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization381" style="font-size:114%;margin:0 4em"><a href="/wiki/Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Convex_minimization" class="mw-redirect" title="Convex minimization">Convex<br /> minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li> <li><a href="/wiki/Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li> <li><a href="/wiki/Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming" title="Linear programming">Linear</a> and<br /><a href="/wiki/Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_scaling" title="Affine scaling">Affine scaling</a></li> <li><a href="/wiki/Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li> <li><a href="/wiki/Karmarkar%27s_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matroid" title="Matroid">Basis-</a><a href="/wiki/Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li> <li><a href="/wiki/Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li> <li><a href="/wiki/Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li> <li><a href="/wiki/Lemke%27s_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li> <li><a href="/wiki/Active-set_method" title="Active-set method">Active-set method</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial381" style="font-size:114%;margin:0 4em"><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li> <li><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li> <li><a href="/wiki/Integer_programming" title="Integer programming">Integer programming</a> <ul><li><a href="/wiki/Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="/wiki/Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph<br /> algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree52" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Minimum_spanning_tree" title="Minimum spanning tree">Minimum<br /> spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bor%C5%AFvka%27s_algorithm" title="Borůvka's algorithm">Borůvka</a></li> <li><a href="/wiki/Prim%27s_algorithm" title="Prim's algorithm">Prim</a></li> <li><a href="/wiki/Kruskal%27s_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul> </div></td></tr></tbody></table><div> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path39" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a> <ul><li><a href="/wiki/Shortest_Path_Faster_Algorithm" class="mw-redirect" title="Shortest Path Faster Algorithm">SPFA</a></li></ul></li> <li><a href="/wiki/Dijkstra%27s_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li> <li><a href="/wiki/Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dinic%27s_algorithm" title="Dinic's algorithm">Dinic</a></li> <li><a href="/wiki/Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li> <li><a href="/wiki/Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li> <li><a href="/wiki/Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Metaheuristics381" style="font-size:114%;margin:0 4em"><a href="/wiki/Metaheuristic" title="Metaheuristic">Metaheuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li> <li><a href="/wiki/Hill_climbing" title="Hill climbing">Hill climbing</a></li> <li><a href="/wiki/Local_search_(optimization)" title="Local search (optimization)">Local search</a></li> <li><a href="/wiki/Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li> <li><a href="/wiki/Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li> <li><a href="/wiki/Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li> <li><a href="/wiki/Tabu_search" title="Tabu search">Tabu search</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_optimization_software220" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_optimization_software" title="Template:Mathematical optimization software"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_optimization_software" title="Template talk:Mathematical optimization software"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_optimization_software" title="Special:EditPage/Template:Mathematical optimization software"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_optimization_software220" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_optimization_software" title="List of optimization software">Mathematical optimization software</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Data formats</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MPS_(format)" title="MPS (format)">MPS</a></li> <li><a href="/wiki/Nl_(format)" title="Nl (format)">nl</a></li> <li><a href="/wiki/Sol_(format)" title="Sol (format)">sol</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Mathematical_modeling" class="mw-redirect" title="Mathematical modeling">Modeling</a> tools</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AIMMS" title="AIMMS">AIMMS</a></li> <li><a href="/wiki/AMPL" title="AMPL">AMPL</a></li> <li><a href="/wiki/APMonitor" title="APMonitor">APMonitor</a></li> <li><a href="/wiki/ECLiPSe" title="ECLiPSe">ECLiPSe</a>-CLP</li> <li><a href="/wiki/Gekko_(optimization_software)" title="Gekko (optimization software)">Gekko</a></li> <li><a href="/wiki/General_Algebraic_Modeling_System" class="mw-redirect" title="General Algebraic Modeling System">GAMS</a></li> <li><a href="/wiki/GNU_Linear_Programming_Kit" title="GNU Linear Programming Kit">GNU MathProg</a></li> <li><a href="/wiki/JuMP" title="JuMP">JuMP</a></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a></li> <li><a href="/wiki/CPLEX" title="CPLEX">OPL</a></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MiniZinc" title="MiniZinc">MiniZinc</a></li> <li><a href="/wiki/OptimJ" title="OptimJ">OptimJ</a></li> <li><a href="/wiki/COIN-OR#PuLP" title="COIN-OR">PuLP</a></li> <li><a href="/wiki/Pyomo" title="Pyomo">Pyomo</a></li> <li><a href="/wiki/TOMLAB" title="TOMLAB">TOMLAB</a></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Mosel</a></li> <li><a href="/wiki/SCIP_(optimization_software)#SCIP_Optimization_Suite" class="mw-redirect" title="SCIP (optimization software)">ZIMPL</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Solver" title="Solver">Solvers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming" title="Linear programming">LP</a>, <a href="/wiki/Linear_programming#Integer_unknowns" title="Linear programming">MILP</a><sup style="font-weight:normal;">∗</sup></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/APOPT" title="APOPT">APOPT</a><sup>∗</sup></li> <li><a href="/wiki/ANTIGONE" title="ANTIGONE">ANTIGONE</a><sup>∗</sup></li> <li><a href="/wiki/Artelys_Knitro" title="Artelys Knitro">Artelys Knitro</a><sup>∗</sup></li> <li><a href="/wiki/COIN-OR" title="COIN-OR">BCP</a><sup>∗</sup></li> <li><a href="/wiki/COIN-OR#CLP" title="COIN-OR">CLP</a></li> <li><a href="/wiki/COIN-OR#CBC" title="COIN-OR">CBC</a><sup>∗</sup></li> <li><a href="/wiki/CPLEX" title="CPLEX">CPLEX</a><sup>∗</sup></li> <li><a href="/wiki/FortMP" title="FortMP">FortMP</a><sup>∗</sup></li> <li><a href="/wiki/SCIP_(optimization_software)#SCIP_Optimization_Suite" class="mw-redirect" title="SCIP (optimization software)">GCG</a><sup>∗</sup></li> <li><a href="/wiki/GLOP" title="GLOP">GLOP</a><sup>∗</sup></li> <li><a href="/wiki/GNU_Linear_Programming_Kit" title="GNU Linear Programming Kit">GLPK/GLPSOL</a><sup>∗</sup></li> <li><a href="/wiki/Gurobi_Optimizer" title="Gurobi Optimizer">Gurobi Optimizer</a><sup>∗</sup></li> <li><a href="/wiki/HiGHS_optimization_solver" title="HiGHS optimization solver">HiGHS</a><sup>∗</sup></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a><sup>∗</sup></li> <li><a href="/wiki/Lp_solve" title="Lp solve">Lp_solve</a></li> <li><a href="/wiki/Robert_J._Vanderbei" title="Robert J. Vanderbei">LOQO</a></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MINOS_(optimization_software)" title="MINOS (optimization software)">MINOS</a></li> <li><a href="/wiki/MINTO" title="MINTO">MINTO</a><sup>∗</sup></li> <li><a href="/wiki/MOSEK" title="MOSEK">MOSEK</a><sup>∗</sup></li> <li><a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a></li> <li><a href="/wiki/SCIP_(optimization_software)" class="mw-redirect" title="SCIP (optimization software)">SCIP</a><sup>∗</sup></li> <li><a href="/wiki/SCIP_(optimization_software)#SCIP_Optimization_Suite" class="mw-redirect" title="SCIP (optimization software)">SoPlex</a></li> <li><a href="/wiki/Octeract_Engine" title="Octeract Engine">Octeract Engine</a><sup>∗</sup></li> <li><a href="/wiki/COIN-OR#SYMPHONY" title="COIN-OR">SYMPHONY</a><sup>∗</sup></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Optimizer</a><sup>∗</sup></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadratic_programming" title="Quadratic programming">QP</a>, MIQP<sup style="font-weight:normal;">∗</sup></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/APOPT" title="APOPT">APOPT</a><sup>∗</sup></li> <li><a href="/wiki/ANTIGONE" title="ANTIGONE">ANTIGONE</a><sup>∗</sup></li> <li><a href="/wiki/Artelys_Knitro" title="Artelys Knitro">Artelys Knitro</a><sup>∗</sup></li> <li><a href="/wiki/COIN-OR#CBC" title="COIN-OR">CBC</a><sup>∗</sup></li> <li><a href="/wiki/COIN-OR#CLP" title="COIN-OR">CLP</a></li> <li><a href="/wiki/CPLEX" title="CPLEX">CPLEX</a><sup>∗</sup></li> <li><a href="/wiki/FortMP" title="FortMP">FortMP</a><sup>∗</sup></li> <li><a href="/wiki/HiGHS_optimization_solver" title="HiGHS optimization solver">HiGHS</a></li> <li><a href="/wiki/Gurobi_Optimizer" title="Gurobi Optimizer">Gurobi Optimizer</a><sup>∗</sup></li> <li><a href="/wiki/IPOPT" title="IPOPT">IPOPT</a></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a><sup>∗</sup></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MINOS_(optimization_software)" title="MINOS (optimization software)">MINOS</a></li> <li><a href="/wiki/MOSEK" title="MOSEK">MOSEK</a><sup>∗</sup></li> <li><a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a></li> <li><a href="/wiki/Octeract_Engine" title="Octeract Engine">Octeract Engine</a><sup>∗</sup></li> <li><a href="/wiki/SCIP_(optimization_software)" class="mw-redirect" title="SCIP (optimization software)">SCIP</a><sup>∗</sup></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Optimizer</a><sup>∗</sup></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadratically_constrained_quadratic_program" title="Quadratically constrained quadratic program">QCP</a>, MIQCP<sup style="font-weight:normal;">∗</sup></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/APOPT" title="APOPT">APOPT</a><sup>∗</sup></li> <li><a href="/wiki/ANTIGONE" title="ANTIGONE">ANTIGONE</a><sup>∗</sup></li> <li><a href="/wiki/Artelys_Knitro" title="Artelys Knitro">Artelys Knitro</a><sup>∗</sup></li> <li><a href="/wiki/CPLEX" title="CPLEX">CPLEX</a><sup>∗</sup></li> <li><a href="/wiki/Gurobi_Optimizer" title="Gurobi Optimizer">Gurobi Optimizer</a><sup>∗</sup></li> <li><a href="/wiki/IPOPT" title="IPOPT">IPOPT</a></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a><sup>∗</sup></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MINOS_(optimization_software)" title="MINOS (optimization software)">MINOS</a></li> <li><a href="/wiki/MOSEK" title="MOSEK">MOSEK</a><sup>∗</sup></li> <li><a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a></li> <li><a href="/wiki/SCIP_(optimization_software)" class="mw-redirect" title="SCIP (optimization software)">SCIP</a><sup>∗</sup></li> <li><a href="/wiki/Octeract_Engine" title="Octeract Engine">Octeract Engine</a><sup>∗</sup></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Optimizer</a><sup>∗</sup></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress NonLinear</a><sup>∗</sup></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Second-order_cone_programming" title="Second-order cone programming">SOCP</a>, MISOCP<sup style="font-weight:normal;">∗</sup></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Artelys_Knitro" title="Artelys Knitro">Artelys Knitro</a><sup>∗</sup></li> <li><a href="/wiki/CPLEX" title="CPLEX">CPLEX</a><sup>∗</sup></li> <li><a href="/wiki/Gurobi_Optimizer" title="Gurobi Optimizer">Gurobi Optimizer</a><sup>∗</sup></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a><sup>∗</sup></li> <li><a href="/wiki/Robert_J._Vanderbei" title="Robert J. Vanderbei">LOQO</a></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MOSEK" title="MOSEK">MOSEK</a><sup>∗</sup></li> <li><a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a></li> <li><a href="/wiki/SCIP_(optimization_software)" class="mw-redirect" title="SCIP (optimization software)">SCIP</a><sup>∗</sup></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Optimizer</a><sup>∗</sup></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">SDP</a>, MISDP<sup style="font-weight:normal;">∗</sup></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/MOSEK" title="MOSEK">MOSEK</a></li> <li><a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">NLP</a>, MINLP<sup style="font-weight:normal;">∗</sup></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AIMMS_Outer_Approximation_(optimization_software)" class="mw-redirect" title="AIMMS Outer Approximation (optimization software)">AOA</a><sup>∗</sup></li> <li><a href="/wiki/APOPT" title="APOPT">APOPT</a><sup>∗</sup></li> <li><a href="/wiki/ANTIGONE" title="ANTIGONE">ANTIGONE</a><sup>∗</sup></li> <li><a href="/wiki/Artelys_Knitro" title="Artelys Knitro">Artelys Knitro</a><sup>∗</sup></li> <li><a href="/wiki/BARON" title="BARON">BARON</a><sup>∗</sup></li> <li><a href="/wiki/Couenne" title="Couenne">Couenne</a><sup>∗</sup></li> <li><a href="/wiki/Galahad_library" title="Galahad library">Galahad library</a></li> <li><a href="/wiki/Gurobi_Optimizer" title="Gurobi Optimizer">Gurobi Optimizer</a><sup>∗</sup></li> <li><a href="/wiki/IPOPT" title="IPOPT">IPOPT</a></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a><sup>∗</sup></li> <li><a href="/wiki/Robert_J._Vanderbei" title="Robert J. Vanderbei">LOQO</a></li> <li><a href="/wiki/MIDACO" title="MIDACO">MIDACO</a><sup>∗</sup></li> <li><a href="/wiki/MINOS_(optimization_software)" title="MINOS (optimization software)">MINOS</a></li> <li><a href="/wiki/Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">NAG</a></li> <li><a href="/wiki/NLPQLP" title="NLPQLP">NLPQLP</a></li> <li><a href="/wiki/NPSOL" title="NPSOL">NPSOL</a></li> <li><a href="/wiki/SCIP_(optimization_software)" class="mw-redirect" title="SCIP (optimization software)">SCIP</a><sup>∗</sup></li> <li><a href="/wiki/SNOPT" title="SNOPT">SNOPT</a><sup>∗</sup></li> <li><a href="/wiki/Octeract_Engine" title="Octeract Engine">Octeract Engine</a><sup>∗</sup></li> <li><a href="/wiki/WORHP" title="WORHP">WORHP</a></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress NonLinear</a><sup>∗</sup></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Global_optimization" title="Global optimization">GO</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ANTIGONE" title="ANTIGONE">ANTIGONE</a><sup>∗</sup></li> <li><a href="/wiki/BARON" title="BARON">BARON</a></li> <li><a href="/wiki/Couenne" title="Couenne">Couenne</a><sup>∗</sup></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Global</a></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/LINDO" title="LINDO">LINDO</a></li> <li><a href="/wiki/SCIP_(optimization_software)" class="mw-redirect" title="SCIP (optimization software)">SCIP</a></li> <li><a href="/wiki/Octeract_Engine" title="Octeract Engine">Octeract Engine</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Constraint_programming" title="Constraint programming">CP</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Artelys Kalis</li> <li>Comet</li> <li><a href="/wiki/CPLEX" title="CPLEX">CPLEX</a> CP Optimizer</li> <li><a href="/wiki/Gecode" title="Gecode">Gecode</a></li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></li> <li><a href="/wiki/JaCoP_(solver)" title="JaCoP (solver)">JaCoP</a></li> <li><a href="/wiki/FICO_Xpress" title="FICO Xpress">Xpress Kalis</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><a href="/wiki/List_of_optimization_software" title="List of optimization software">List of optimization software</a></b></li> <li><b><a href="/wiki/Comparison_of_optimization_software" title="Comparison of optimization software">Comparison of optimization software</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox600" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q2269096#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="semidefinitní programování"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph705676&CON_LNG=ENG">Czech Republic</a></span></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Semidefinite programming - Wikipedia