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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">SOCP</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>SOCP</b> (oder <b>Second Order Cone Program</b>) ist ein Problem in der <a href="Optimierung_(Mathematik)" class="mw-redirect" title="Optimierung (Mathematik)">mathematischen Optimierung</a>, bei dem die Lösung des Problems nicht nur linearen Restriktionen unterliegt, sondern auch noch in einem bestimmten <a href="Kegel_(Lineare_Algebra)" title="Kegel (Lineare Algebra)">Kegel</a> liegen soll. Dieser Kegel wird im Englischen der second-order cone genannt, woraus sich der Name des Programms herleitet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Gegeben sei der <span class="mwe-math-element mwe-math-element-inline"><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 {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> versehen mit dem <a href="Standardskalarprodukt" title="Standardskalarprodukt">Standardskalarprodukt</a> <span class="mwe-math-element mwe-math-element-inline"><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 {\cdot },{\cdot }\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>⋅<!-- ⋅ --></mo>
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<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>⋅<!-- ⋅ --></mo>
</mrow>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle {\cdot },{\cdot }\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/468a71a5490f2d634b4878381aadb2f9ccfc3f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle {\cdot },{\cdot }\rangle }" loading="lazy"></span> und der Second-Order-Kegel (auch <a href="Lorentz-Kegel" class="mw-redirect" title="Lorentz-Kegel">Lorentz-Kegel</a> genannt) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:=\{x\in \mathbb {R} ^{n+1}\,|\,\Vert (x_{1},\dots ,x_{n})\Vert _{2}\leq x_{n+1}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>:=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msub>
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S:=\{x\in \mathbb {R} ^{n+1}\,|\,\Vert (x_{1},\dots ,x_{n})\Vert _{2}\leq x_{n+1}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e760234e5882908d4a79828bee93bfac377ecff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.204ex; height:3.176ex;" alt="{\displaystyle S:=\{x\in \mathbb {R} ^{n+1}\,|\,\Vert (x_{1},\dots ,x_{n})\Vert _{2}\leq x_{n+1}\}}" loading="lazy"></span> der die <a href="Verallgemeinerte_Ungleichung" title="Verallgemeinerte Ungleichung">verallgemeinerte Ungleichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \preccurlyeq _{S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>≼<!-- ≼ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \preccurlyeq _{S}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55fa238837297f637d149965627e3a3f275dbd36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.101ex; height:2.343ex;" alt="{\displaystyle \preccurlyeq _{S}}" loading="lazy"></span> definiert. Dann heißt das <a href="Optimierungsproblem" title="Optimierungsproblem">Optimierungsproblem</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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{aligned}{\text{Minimiere }}&s(x)=\langle {c},{x}\rangle &\\{\text{unter den Nebenbedingungen }}&(A_{i}x+b_{i},c_{i}^{T}x+d_{i})\succcurlyeq _{S}0&i=1,\dots ,m\\&Fx=g&\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Minimiere </mtext>
</mrow>
</mtd>
<mtd>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>c</mi>
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<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mtd>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>unter den Nebenbedingungen </mtext>
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</mtd>
<mtd>
<mi></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mi>x</mi>
<mo>+</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msubsup>
<mi>x</mi>
<mo>+</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msub>
<mo>≽<!-- ≽ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
<mn>0</mn>
</mtd>
<mtd>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>m</mi>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi>F</mi>
<mi>x</mi>
<mo>=</mo>
<mi>g</mi>
</mtd>
<mtd></mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Minimiere }}&s(x)=\langle {c},{x}\rangle &\\{\text{unter den Nebenbedingungen }}&(A_{i}x+b_{i},c_{i}^{T}x+d_{i})\succcurlyeq _{S}0&i=1,\dots ,m\\&Fx=g&\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c14a3e1b71ef30bc34dbc499b8a39f141d5737f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.825ex; margin-bottom: -0.18ex; width:73.138ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}{\text{Minimiere }}&s(x)=\langle {c},{x}\rangle &\\{\text{unter den Nebenbedingungen }}&(A_{i}x+b_{i},c_{i}^{T}x+d_{i})\succcurlyeq _{S}0&i=1,\dots ,m\\&Fx=g&\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Dabei ist <span class="mwe-math-element mwe-math-element-inline"><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_{i}\in \mathbb {R} ^{n\times n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{i}\in \mathbb {R} ^{n\times n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29ca5860d73aaadf69da1c641bcfd7d696841180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.545ex; height:2.676ex;" alt="{\displaystyle A_{i}\in \mathbb {R} ^{n\times n}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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_{i},c_{i},x\in \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>x</mi>
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{i},c_{i},x\in \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29d20c460cf6c7aa1cdbacb86c5bb634b87c7d70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.739ex; height:2.676ex;" alt="{\displaystyle b_{i},c_{i},x\in \mathbb {R} ^{n}}" loading="lazy"></span> sowie <span class="mwe-math-element mwe-math-element-inline"><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_{i}\in \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{i}\in \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e7ff6571c19869105ee22398583d5d914b1ef93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.527ex; height:2.509ex;" alt="{\displaystyle d_{i}\in \mathbb {R} }" loading="lazy"></span>
</p><p>Alternativ lässt sich die Ungleichungsrestriktion auch als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vert A_{i}x+b_{i}\Vert _{2}\leq c_{i}^{T}x+d_{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>
<mi>x</mi>
<mo>+</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msubsup>
<mi>x</mi>
<mo>+</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Vert A_{i}x+b_{i}\Vert _{2}\leq c_{i}^{T}x+d_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0280fabb9587f3f97ea51e06068b030b4dc66ae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.562ex; height:3.176ex;" alt="{\displaystyle \Vert A_{i}x+b_{i}\Vert _{2}\leq c_{i}^{T}x+d_{i}}" loading="lazy"></span> formulieren.
</p>
<div class="mw-heading mw-heading2"><h2 id="Klassifikation_und_Spezialfälle"><span id="Klassifikation_und_Spezialf.C3.A4lle"></span>Klassifikation und Spezialfälle</h2></div>
<p>Ein SOCP ist ein <a href="Konisches_Programm" title="Konisches Programm">Konisches Programm</a>, wie die obige Formulierung mittels des Kegels zeigt. Damit ist es auch immer ein <a href="Konvexe_Optimierung" title="Konvexe Optimierung">konvexes Optimierungsproblem</a>.
</p><p>Sind alle <span class="mwe-math-element mwe-math-element-inline"><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}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{i}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06dbc33495a748ad5067cb6ff07cba653cffaf2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.067ex; height:2.509ex;" alt="{\displaystyle c_{i}=0}" loading="lazy"></span>, so lässt sich ein SOCP als ein spezielles <a href="Quadratisches_Programm_mit_quadratischen_Nebenbedingungen" class="mw-redirect" title="Quadratisches Programm mit quadratischen Nebenbedingungen">Quadratisches Programm mit quadratischen Nebenbedingungen</a> formulieren. Dazu nutzt man aus, dass
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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)\succcurlyeq _{S}0\iff {\begin{bmatrix}x\\t\end{bmatrix}}^{T}{\begin{bmatrix}E_{n}&0\\0&-1\end{bmatrix}}{\begin{bmatrix}x\\t\end{bmatrix}}\leq 0\,{\text{ und }}\,-t\leq 0}">
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<annotation encoding="application/x-tex">{\displaystyle (x,t)\succcurlyeq _{S}0\iff {\begin{bmatrix}x\\t\end{bmatrix}}^{T}{\begin{bmatrix}E_{n}&0\\0&-1\end{bmatrix}}{\begin{bmatrix}x\\t\end{bmatrix}}\leq 0\,{\text{ und }}\,-t\leq 0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e961d8609912fbbec5cd7beeec6626e96ec2818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.373ex; height:6.676ex;" alt="{\displaystyle (x,t)\succcurlyeq _{S}0\iff {\begin{bmatrix}x\\t\end{bmatrix}}^{T}{\begin{bmatrix}E_{n}&0\\0&-1\end{bmatrix}}{\begin{bmatrix}x\\t\end{bmatrix}}\leq 0\,{\text{ und }}\,-t\leq 0}" loading="lazy"></span></dd></dl>
<p>ist. Hierbei ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}">
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<mi>E</mi>
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy"></span> die n-dimensionale <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>. Jede Kegeleinschränkung lässt sich in diesem Fall also durch eine quadratische und eine lineare Restriktion ersetzen.
</p><p>Sind alle <span class="mwe-math-element mwe-math-element-inline"><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_{i}}">
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<mi>A</mi>
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<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"></span> gleich Null, so lassen sich die Ungleichungsrestriktionen umformulieren in <span class="mwe-math-element mwe-math-element-inline"><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_{i}-\Vert b_{i}\Vert _{2}\geq -c_{i}^{T}x}">
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<annotation encoding="application/x-tex">{\displaystyle d_{i}-\Vert b_{i}\Vert _{2}\geq -c_{i}^{T}x}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6c165b0089bd29e830a9df213b53528f892cc76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.658ex; height:3.176ex;" alt="{\displaystyle d_{i}-\Vert b_{i}\Vert _{2}\geq -c_{i}^{T}x}" loading="lazy"></span>. Fasst man nun die linke Seite der Ungleichung für alle <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" 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}" loading="lazy"></span> zu einem Vektor zusammen und die rechte Zeile zu einer Matrix, die zeilenweise aus den Vektoren <span class="mwe-math-element mwe-math-element-inline"><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}}">
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<annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy"></span> besteht, so lässt sich das SOCP als <a href="Lineare_Optimierung" title="Lineare Optimierung">Lineares Optimierungsproblem</a> formulieren.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Stephen Boyd, Lieven Vandenberghe: <i>Convex Optimization</i>. Cambridge University Press, 2004, ISBN 978-0-521-83378-3 (<a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/">online</a>).</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-12-11" href="https://de.wikipedia.org/wiki/?title=SOCP&oldid=240098711">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
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