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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Linear independence constraint qualification</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>Die <b>Linear independence constraint qualification</b> oder kurz <b>LICQ</b> ist eine wichtige Voraussetzung, dass <a href="Optimalit%C3%A4tskriterium" title="Optimalitätskriterium">notwendige Optimalitätskriterien</a> in der <a href="Nichtlineare_Optimierung" title="Nichtlineare Optimierung">nichtlinearen Optimierung</a> gelten. Sie ist eine Bedingung an die Regularität eines zulässigen Punktes. Ist die LICQ in einem Punkt <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 {\tilde {x}}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f5c5435030c952a58a756e691ea64f60c1bd240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}}" loading="lazy"></span> erfüllt und ist dieser Punkt ein <a href="Lokales_Minimum" class="mw-redirect" title="Lokales Minimum">lokales Minimum</a>, so sind auch die <a href="Karush-Kuhn-Tucker-Bedingungen" title="Karush-Kuhn-Tucker-Bedingungen">Karush-Kuhn-Tucker-Bedingungen</a> an diesem Punkt erfüllt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Gegeben ist ein Optimierungsproblem in der Form
</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 \min _{x\in X}f(x)}">
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<annotation encoding="application/x-tex">{\displaystyle \min _{x\in X}f(x)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a65dc2ab7b5b9b7fda6c4c357857fe603159ceac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.68ex; height:4.009ex;" alt="{\displaystyle \min _{x\in X}f(x)}" loading="lazy"></span>,</dd></dl>
<p>wobei
</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=\{x\in \mathbb {R} ^{n}\,|\,g_{i}(x)\leq 0,h_{j}(x)=0,\;i=1,\dots ,k;\;j=1,\dots ,l\}}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle X=\{x\in \mathbb {R} ^{n}\,|\,g_{i}(x)\leq 0,h_{j}(x)=0,\;i=1,\dots ,k;\;j=1,\dots ,l\}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b66ed770435ad842e18b312f91ea0235e58a26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.784ex; height:3.009ex;" alt="{\displaystyle X=\{x\in \mathbb {R} ^{n}\,|\,g_{i}(x)\leq 0,h_{j}(x)=0,\;i=1,\dots ,k;\;j=1,\dots ,l\}}" loading="lazy"></span></dd></dl>
<p>die Restriktionsmenge ist und alle Funktionen <a href="Stetig_differenzierbar" class="mw-redirect" title="Stetig differenzierbar">stetig differenzierbar</a> sein sollen. Es sei <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 K(x)=\{i\,|\,g_{i}(x)=0\}}">
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<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>i</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
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<annotation encoding="application/x-tex">{\displaystyle K(x)=\{i\,|\,g_{i}(x)=0\}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77083b1b3062d88126cec8f4b94152b8350a1629.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.16ex; height:2.843ex;" alt="{\displaystyle K(x)=\{i\,|\,g_{i}(x)=0\}}" loading="lazy"></span> die Menge der Indizes, bei denen die Ungleichungsrestriktionen mit Gleichheit erfüllt sind, d. h. die Ungleichungsrestriktion <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 g_{i}(x)}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e0c405bd84a5e6361d1d36aa625891bb16b02e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.048ex; height:2.843ex;" alt="{\displaystyle g_{i}(x)}" loading="lazy"></span> ist aktiv. Dann erfüllt ein zulässiger Punkt <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 {\tilde {x}}\in X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
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<mo>∈<!-- ∈ --></mo>
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}\in X}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ff20f022851c08f26f149533038e3b85465b922.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}\in X}" loading="lazy"></span> des restringierten Optimierungsproblems die LICQ, wenn die <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</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 \nabla h_{j}({\tilde {x}})}">
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<annotation encoding="application/x-tex">{\displaystyle \nabla h_{j}({\tilde {x}})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c4ada5ce1053ee42edfb5c232138dcf9815b654.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.324ex; height:3.009ex;" alt="{\displaystyle \nabla h_{j}({\tilde {x}})}" 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 \nabla g_{i}({\tilde {x}})}">
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<annotation encoding="application/x-tex">{\displaystyle \nabla g_{i}({\tilde {x}})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/090b60c4a57b2198faa088d6d24c4a8937005c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.984ex; height:2.843ex;" alt="{\displaystyle \nabla g_{i}({\tilde {x}})}" loading="lazy"></span> mit <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\in K({\tilde {x}})}">
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<annotation encoding="application/x-tex">{\displaystyle i\in K({\tilde {x}})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2062d96e9b924bc1786cfb4dd56bb5f5286ac674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.848ex; height:2.843ex;" alt="{\displaystyle i\in K({\tilde {x}})}" loading="lazy"></span> <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängig</a> sind.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<div class="mw-heading mw-heading3"><h3 id="LICQ">LICQ</h3></div>
<p>Betrachten wir als Beispiel die Restriktionsfunktionen <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 g_{1}(x)=x_{1}+x_{2}-1\leq 0}">
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<annotation encoding="application/x-tex">{\displaystyle g_{1}(x)=x_{1}+x_{2}-1\leq 0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc6ff96c983d23c113b181704db9b7dcc7b48d8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.273ex; height:2.843ex;" alt="{\displaystyle g_{1}(x)=x_{1}+x_{2}-1\leq 0}" 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 g_{2}(x)=x_{1}^{2}+x_{2}^{2}-1\leq 0}">
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<annotation encoding="application/x-tex">{\displaystyle g_{2}(x)=x_{1}^{2}+x_{2}^{2}-1\leq 0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd7a63e1265cb91dbaf6d61ab08786e36d33fec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.273ex; height:3.176ex;" alt="{\displaystyle g_{2}(x)=x_{1}^{2}+x_{2}^{2}-1\leq 0}" loading="lazy"></span>. Wir untersuchen, ob der Punkt <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 {\tilde {x}}=(0,1)}">
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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}=(0,1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd3f8c6a4fb710033f9310fc482f18e8f2061cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.596ex; height:2.843ex;" alt="{\displaystyle {\tilde {x}}=(0,1)}" loading="lazy"></span> die LICQ erfüllt. Es 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 K({\tilde {x}})=\{1,2\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K({\tilde {x}})=\{1,2\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d26929cafbebd37527b8391e8f645188a721e61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.987ex; height:2.843ex;" alt="{\displaystyle K({\tilde {x}})=\{1,2\}}" loading="lazy"></span>, da beide Ungleichungen 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 {\tilde {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f5c5435030c952a58a756e691ea64f60c1bd240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}}" loading="lazy"></span> aktiv sind. Die Gradienten sind <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 \nabla g_{1}({\tilde {x}})=(1,1)^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla g_{1}({\tilde {x}})=(1,1)^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fd044bd4e213a06a3022fc9b4aaf8410f61e487.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.894ex; height:3.176ex;" alt="{\displaystyle \nabla g_{1}({\tilde {x}})=(1,1)^{T}}" 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 \nabla g_{2}({\tilde {x}})=(0,2)^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla g_{2}({\tilde {x}})=(0,2)^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/073049819b029f63ebcb35f27ab20b2ffd30762c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.894ex; height:3.176ex;" alt="{\displaystyle \nabla g_{2}({\tilde {x}})=(0,2)^{T}}" loading="lazy"></span>. Beide Ungleichungsrestriktionen sind im untersuchten Punkt aktiv und die Gradienten sind linear unabhängig. Daher erfüllt der Punkt die LICQ.
</p>
<div class="mw-heading mw-heading3"><h3 id="MFCQ_ohne_LICQ">MFCQ ohne LICQ</h3></div>
<p>Betrachtet man die Restriktionsfunktionen <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 g_{1}(x)=-x_{2}\leq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{1}(x)=-x_{2}\leq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e643714e88d7a29bfdc03ce3118ef4b4e9286cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.854ex; height:2.843ex;" alt="{\displaystyle g_{1}(x)=-x_{2}\leq 0}" 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 g_{2}(x)=x_{1}^{4}-x_{2}\leq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{2}(x)=x_{1}^{4}-x_{2}\leq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee5bdc2bbed36a236c23eb94557e37ed15ec7cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.27ex; height:3.176ex;" alt="{\displaystyle g_{2}(x)=x_{1}^{4}-x_{2}\leq 0}" loading="lazy"></span> und untersucht diese im Punkt <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 {\tilde {x}}=(0,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}=(0,0)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c4abca41141cce562036fa3c78d113452c1c10c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.596ex; height:2.843ex;" alt="{\displaystyle {\tilde {x}}=(0,0)}" loading="lazy"></span>, so ist die LICQ nicht erfüllt. Die Gradienten <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 \nabla g_{1}({\tilde {x}})=(0,-1)^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla g_{1}({\tilde {x}})=(0,-1)^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c66a9bdc620c1a9159d4d71a7c30e4471cad4cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.702ex; height:3.176ex;" alt="{\displaystyle \nabla g_{1}({\tilde {x}})=(0,-1)^{T}}" 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 \nabla g_{2}({\tilde {x}})=(0,-1)^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla g_{2}({\tilde {x}})=(0,-1)^{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/337b67e6e22a19801af997b0bb388479d799ff40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.702ex; height:3.176ex;" alt="{\displaystyle \nabla g_{2}({\tilde {x}})=(0,-1)^{T}}" loading="lazy"></span> sind linear abhängig und beide Ungleichungen sind im untersuchten Punkt aktiv.
Die <a href="MFCQ" class="mw-redirect" title="MFCQ">MFCQ</a> sind aber erfüllt, da für den Vektor <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=(0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=(0,1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21addf2886c2c92be79d7d2f8b61b04f8cdaf6db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.482ex; height:2.843ex;" alt="{\displaystyle d=(0,1)}" loading="lazy"></span> gilt, dass <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 \nabla g_{i}({\tilde {x}})^{T}d<0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>d</mi>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla g_{i}({\tilde {x}})^{T}d<0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/775e5011728ec20c18b12f760139a678b8fe0ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.85ex; height:3.176ex;" alt="{\displaystyle \nabla g_{i}({\tilde {x}})^{T}d<0}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Vergleich_mit_anderen_constraint_qualifications">Vergleich mit anderen <i>constraint qualifications</i></h2></div>
<p>Gilt die LICQ, so ist auch die <a href="MFCQ" class="mw-redirect" title="MFCQ">MFCQ</a> und daher die <a href="Abadie_CQ" class="mw-redirect" title="Abadie CQ">Abadie CQ</a> automatisch erfüllt. Die LICQ hat im Gegensatz zur MFCQ und zur Abadie CQ den Vorteil, dass sie leicht zu überprüfen ist. Ein Nachteil ist, dass sie nicht so allgemein gültig ist wie die anderen <i><a href="Karush-Kuhn-Tucker-Bedingungen#Regularitätsvoraussetzungen" title="Karush-Kuhn-Tucker-Bedingungen">constraint qualifications</a></i>. Dies wird durch das obige Beispiel illustriert. Es gelten die Implikationen
</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 {\text{LICQ}}\implies {\text{MFCQ}}\implies {\text{Abadie CQ}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>LICQ</mtext>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo stretchy="false">⟹<!-- ⟹ --></mo>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>MFCQ</mtext>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo stretchy="false">⟹<!-- ⟹ --></mo>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Abadie CQ</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{LICQ}}\implies {\text{MFCQ}}\implies {\text{Abadie CQ}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de3ddbff45904af7ef4fb34c1683ab9a3ccb1dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.923ex; height:2.509ex;" alt="{\displaystyle {\text{LICQ}}\implies {\text{MFCQ}}\implies {\text{Abadie CQ}}}" loading="lazy"></span>.</dd></dl>
<p>Die Umkehrungen gelten aber nicht.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>C. Geiger, C. Kanzow: <i>Theorie und Numerik restringierter Optimierungsaufgaben</i>. Springer, 2002. ISBN 3-540-42790-2. <a rel="nofollow" class="external text" href="https://books.google.de/books?id=spmzFyso_b8C&hl=de">https://books.google.de/books?id=spmzFyso_b8C&hl=de</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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