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<title>Fast-konvexe Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Fast-konvexe Funktion</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">Die fast konvexen Funktionen (englisch convex-like functions) bilden eine Verallgemeinerung der <a href="Konvexe_Funktion" class="mw-redirect" title="Konvexe Funktion">konvexen Funktionen</a> und werden in der <a href="Optimierung_(Mathematik)" class="mw-redirect" title="Optimierung (Mathematik)">mathematischen Optimierung</a> verwendet, da für sie einfache Regularitätsvoraussetzungen wie die <a href="Slater-Bedingung" title="Slater-Bedingung">Slater-Bedingung</a> gelten, unter denen <a href="Starke_Dualit%C3%A4t" class="mw-redirect" title="Starke Dualität">starke Dualität</a> gilt und damit auch die <a href="Karush-Kuhn-Tucker-Bedingungen" title="Karush-Kuhn-Tucker-Bedingungen">Karush-Kuhn-Tucker-Bedingungen</a> gelten.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1},V_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{1},V_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50193d5201b7ff4d46487f5d5e58ee9c1e6f29fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.853ex; height:2.509ex;" alt="{\displaystyle V_{1},V_{2}}" loading="lazy"> reelle Vektorräume und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\,}">

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<annotation encoding="application/x-tex">{\displaystyle K\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/492094dd0f8aec54c51b011cf9a1ae0aeaf89e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.453ex; height:2.176ex;" alt="{\displaystyle K\,}" loading="lazy"> ein <a href="Ordnungskegel" title="Ordnungskegel">Ordnungskegel</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceaa689a894f5020a7b46177d201cbce2d41122b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{2}}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7714158afb63a98e7dbc6e885f70d9141c6923a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{1}}" loading="lazy"> eine nichtleere Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adfdbc929f16cb00bb43289c223651b41f7b9f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}" loading="lazy">. Dann heißt eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon D_{1}\mapsto V_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon D_{1}\mapsto V_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9759ed00626d6a624b7cf8cfd8a9ec0c1c7d1d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.315ex; height:2.509ex;" alt="{\displaystyle f\colon D_{1}\mapsto V_{2}}" loading="lazy"> fast konvex, wenn die Menge

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<annotation encoding="application/x-tex">{\displaystyle M:=f(D_{1})+K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a78622041d1d4a415f16e37bed522c6f8ffd75b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.16ex; height:2.843ex;" alt="{\displaystyle M:=f(D_{1})+K}" loading="lazy"></dd></dl>

konvex ist. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" 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}" loading="lazy"> lässt sich äquivalent beschreiben als

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<annotation encoding="application/x-tex">{\displaystyle M=\{y\in V_{2}\,|\,\exists x\in D_{1}{\text{ so dass }}f(x)-y\in -K\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad69fcfb7ed56e158776711cd3c23bdb3813712b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.37ex; height:2.843ex;" alt="{\displaystyle M=\{y\in V_{2}\,|\,\exists x\in D_{1}{\text{ so dass }}f(x)-y\in -K\}}" loading="lazy"></dd></dl>

Ist der Kegel ein <a href="Echter_Kegel" class="mw-redirect" title="Echter Kegel">echter Kegel</a> und definiert damit eine <a href="Verallgemeinerte_Ungleichung" title="Verallgemeinerte Ungleichung">verallgemeinerte Ungleichung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \preccurlyeq _{K}}">

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<annotation encoding="application/x-tex">{\displaystyle \preccurlyeq _{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bef9e870b49c8958943bdcc35e59a25b31466b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.501ex; height:2.343ex;" alt="{\displaystyle \preccurlyeq _{K}}" loading="lazy">, so lautet diese Menge

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<annotation encoding="application/x-tex">{\displaystyle M=\{y\in V_{2}\,|\,\exists x\in D_{1}{\text{ so dass }}f(x)\preccurlyeq _{K}y\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b74918afceac7b39fa377df622b25a788cd4c391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.606ex; height:2.843ex;" alt="{\displaystyle M=\{y\in V_{2}\,|\,\exists x\in D_{1}{\text{ so dass }}f(x)\preccurlyeq _{K}y\}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

Betrachtet man die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \mapsto \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \mapsto \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4241a9499e8eac534dfcccfcb7aaa6d70f43ae30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \mapsto \mathbb {R} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sin(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\sin(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fb1266b7f7718442e31e45eef3d81bef6a8b9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.511ex; height:2.843ex;" alt="{\displaystyle f(x)=\sin(x)}" loading="lazy"> und den echten Kegel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} _{+}=\{x\in \mathbb {R} \,|\,x\geq 0\}}">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} _{+}=\{x\in \mathbb {R} \,|\,x\geq 0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0a695ee312f8310094c3ed99adbdd527dbb3d46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.637ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {R} _{+}=\{x\in \mathbb {R} \,|\,x\geq 0\}}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1}=[0,2\pi ]}">

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<annotation encoding="application/x-tex">{\displaystyle D_{1}=[0,2\pi ]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4047867182cbbe3ddd6b51db74369b70f1c06833.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.062ex; height:2.843ex;" alt="{\displaystyle D_{1}=[0,2\pi ]}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(D_{1})=[-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle \sin(D_{1})=[-1,1]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a28edbc07a39f86296086ab74c180b20265c1ccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.203ex; height:2.843ex;" alt="{\displaystyle \sin(D_{1})=[-1,1]}" loading="lazy">. Damit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=[-1;1]+\mathbb {R} _{+}=\{x\in \mathbb {R} \,|\,x\geq -1\}}">

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<annotation encoding="application/x-tex">{\displaystyle M=[-1;1]+\mathbb {R} _{+}=\{x\in \mathbb {R} \,|\,x\geq -1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cff578da396690459c53c57f3ac12e3441f4f4e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.122ex; height:2.843ex;" alt="{\displaystyle M=[-1;1]+\mathbb {R} _{+}=\{x\in \mathbb {R} \,|\,x\geq -1\}}" loading="lazy">. Diese Menge ist konvex und damit ist die Sinusfunktion fast konvex.

Betrachtet man die Funktion

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}-1&{\text{ falls }}x\leq 0\\1&{\text{ falls }}x>0\end{cases}}}">

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<mtext> falls </mtext>

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<mtext> falls </mtext>

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}-1&{\text{ falls }}x\leq 0\\1&{\text{ falls }}x>0\end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc423dcc7345f937e1a3d56db9e26aee62cb3c37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.14ex; height:6.176ex;" alt="{\displaystyle f(x)={\begin{cases}-1&{\text{ falls }}x\leq 0\\1&{\text{ falls }}x>0\end{cases}}}" loading="lazy">

und definiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} \mapsto \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} \mapsto \mathbb {R} ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71a0122e37a10ea46172cd14d18c4131d4b22406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.174ex; height:3.009ex;" alt="{\displaystyle g\colon \mathbb {R} \mapsto \mathbb {R} ^{2}}" loading="lazy"> durch

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<annotation encoding="application/x-tex">{\displaystyle g(x)={\begin{pmatrix}f(x)\\5x\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa6675b49cfbd702ce0e8fe147668cc7584bc6f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.944ex; height:6.176ex;" alt="{\displaystyle g(x)={\begin{pmatrix}f(x)\\5x\end{pmatrix}}}" loading="lazy"></dd></dl>

auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle D_{0}=\mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5a8fcc0547706c5bc6009fdae847afcbc04df3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.755ex; height:2.509ex;" alt="{\displaystyle D_{0}=\mathbb {R} }" loading="lazy"> mit dem Ordnungskegel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} _{+}\times \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} _{+}\times \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d35d60ab52b2565bc156eff37a783f24546aad0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.681ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {R} _{+}\times \{0\}}" loading="lazy">. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in D_{1}=\{x\in \mathbb {R} \,|\,x\leq 0\}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in D_{1}=\{x\in \mathbb {R} \,|\,x\leq 0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/385eb6a20a1fd2ad863f477cd76c78969580a0b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.432ex; height:2.843ex;" alt="{\displaystyle x\in D_{1}=\{x\in \mathbb {R} \,|\,x\leq 0\}}" loading="lazy"> ist jeder Punkt der Bildmenge von der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,5x)}">

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<annotation encoding="application/x-tex">{\displaystyle (-1,5x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c14f4b704086e9bf86919043d72c5744092da40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.306ex; height:2.843ex;" alt="{\displaystyle (-1,5x)}" loading="lazy"> und damit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(D_{1})=\{y\in \mathbb {R} ^{2}\,|\,y_{1}=-1,y_{2}\leq 0\}}">

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<annotation encoding="application/x-tex">{\displaystyle g(D_{1})=\{y\in \mathbb {R} ^{2}\,|\,y_{1}=-1,y_{2}\leq 0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a38c781d20d194fda5bfa59fd2880237689d728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.228ex; height:3.176ex;" alt="{\displaystyle g(D_{1})=\{y\in \mathbb {R} ^{2}\,|\,y_{1}=-1,y_{2}\leq 0\}}" loading="lazy">. Analog folgt mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}=\{x\in \mathbb {R} \,|\,x>0\}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{2}=\{x\in \mathbb {R} \,|\,x>0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef1ef0ab1ac8f02ba387f66213a4cb3fcb0411a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.262ex; height:2.843ex;" alt="{\displaystyle D_{2}=\{x\in \mathbb {R} \,|\,x>0\}}" loading="lazy">, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(D_{2})=\{y\in \mathbb {R} ^{2}\,|\,y_{1}=1,y_{2}>0\}}">

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<annotation encoding="application/x-tex">{\displaystyle g(D_{2})=\{y\in \mathbb {R} ^{2}\,|\,y_{1}=1,y_{2}>0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba07831a11808b9ec336ea2c94dfef02c013b1a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.419ex; height:3.176ex;" alt="{\displaystyle g(D_{2})=\{y\in \mathbb {R} ^{2}\,|\,y_{1}=1,y_{2}>0\}}" loading="lazy">. Somit ist

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g(D_{1})+K=\{y\in \mathbb {R} ^{2}\,|\,y_{1}\geq -1,y_{2}\leq 0\}\\g(D_{2})+K=\{y\in \mathbb {R} ^{2}\,|\,y_{1}\geq 1,y_{2}>0\}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(D_{1})+K=\{y\in \mathbb {R} ^{2}\,|\,y_{1}\geq -1,y_{2}\leq 0\}\\g(D_{2})+K=\{y\in \mathbb {R} ^{2}\,|\,y_{1}\geq 1,y_{2}>0\}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/151df4d219995f9a16ec13ba426da8f30d2fbbb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.885ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}g(D_{1})+K=\{y\in \mathbb {R} ^{2}\,|\,y_{1}\geq -1,y_{2}\leq 0\}\\g(D_{2})+K=\{y\in \mathbb {R} ^{2}\,|\,y_{1}\geq 1,y_{2}>0\}\end{aligned}}}" loading="lazy"></dd></dl>

Da aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}=D_{1}\cup D_{2}}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle D_{0}=D_{1}\cup D_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4964813f9c2838e953f809c30f356b0bf10cf135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.617ex; height:2.509ex;" alt="{\displaystyle D_{0}=D_{1}\cup D_{2}}" loading="lazy"> ist, kann die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(D_{0})+K=g(D_{1}\cup D_{2})+K=\left(g(D_{1})+K\right)\cup \left(g(D_{2})+K\right)}">

<msub>

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</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

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</msub>

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<mrow>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(D_{0})+K=g(D_{1}\cup D_{2})+K=\left(g(D_{1})+K\right)\cup \left(g(D_{2})+K\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4df95656b10c91bc4e70988b119911bc23df8178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.2ex; height:2.843ex;" alt="{\displaystyle g(D_{0})+K=g(D_{1}\cup D_{2})+K=\left(g(D_{1})+K\right)\cup \left(g(D_{2})+K\right)}" loading="lazy"> nicht konvex sein, da zum Beispiel die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)^{T}}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle (-1,0)^{T}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/331da4c59959ad28ad5fe67cf5af4127d0a45096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.366ex; height:3.176ex;" alt="{\displaystyle (-1,0)^{T}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1)^{T}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (1,1)^{T}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e94ff99366ee7ab8252d7cdb21a580e62a84f87f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.557ex; height:3.176ex;" alt="{\displaystyle (1,1)^{T}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(D_{0})+K}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g(D_{0})+K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a175c36f9f08a6b7602581e0602b822dbcfce646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.81ex; height:2.843ex;" alt="{\displaystyle g(D_{0})+K}" loading="lazy"> enthalten sind, aber keiner der Punkte auf der Strecke zwischen ihnen. Zum Beispiel ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0{,}5)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,0{,}5)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c5d123468d506d754ac3b5f2674ac6817932335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle (0,0{,}5)}" loading="lazy"> der Mittelpunkt dieser Strecke, aber nicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(D_{0})+K}">

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<annotation encoding="application/x-tex">{\displaystyle g(D_{0})+K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a175c36f9f08a6b7602581e0602b822dbcfce646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.81ex; height:2.843ex;" alt="{\displaystyle g(D_{0})+K}" loading="lazy"> enthalten.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

Jede konvexe Funktion ist fast konvex bezüglich des natürlichen Kegels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} _{+}}">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} _{+}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48719ebed414821be4a950cd14ffc17842c2b518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.353ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {R} _{+}}" loading="lazy">. Dies folgt direkt aus der Konvexität des <a href="Epigraph_(Mathematik)" title="Epigraph (Mathematik)">Epigraphs</a>. Genauso ist auch jede <a href="K-konvexe_Funktion" title="K-konvexe Funktion">K-konvexe Funktion</a> fast konvex bezüglich ihres Kegels.

<div class="mw-heading mw-heading2"><h2 id="Verwendung">Verwendung</h2></div>

Die fast konvexen Funktionen sind eine Funktionenklasse, die so definiert ist, dass wenn sie die <a href="Slater-Bedingung" title="Slater-Bedingung">Slater-Bedingung</a> erfüllt, die starke Dualität gilt. Sei also ein Optimierungsproblem der Form

<mtr>

<mtd>

<mtext>Minimiere </mtext>

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<mtd>

</mtd>

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<mtd>

<mtext>unter den Nebenbedingungen </mtext>

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<mtr>

<mtd>

</mtd>

</mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Minimiere }}&f(x)\\{\text{unter den Nebenbedingungen }}&g(x)\in -K\\&x\in R\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f259bc60a4cd09a0ce62b0a8a248e2e12bcb36b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.699ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}{\text{Minimiere }}&f(x)\\{\text{unter den Nebenbedingungen }}&g(x)\in -K\\&x\in R\end{aligned}}}" loading="lazy"></dd></dl>

gegeben für einen Ordnungskegel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\,}">

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<annotation encoding="application/x-tex">{\displaystyle K\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/492094dd0f8aec54c51b011cf9a1ae0aeaf89e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.453ex; height:2.176ex;" alt="{\displaystyle K\,}" loading="lazy"> mit nichtleerem Inneren und Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon V\mapsto \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f\colon V\mapsto \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c642c2b9d42e5ea5e12c914ddaec8403dd6db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.392ex; height:2.509ex;" alt="{\displaystyle f\colon V\mapsto \mathbb {R} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon V\mapsto Y}">

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<annotation encoding="application/x-tex">{\displaystyle g\colon V\mapsto Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f845abc927fac19111bbf21e184193bd31bf166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.325ex; height:2.509ex;" alt="{\displaystyle g\colon V\mapsto Y}" loading="lazy">. Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,Y}">

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<annotation encoding="application/x-tex">{\displaystyle V,Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/242e9bf9f800bc6e848cfdf3b23028a3d490cc17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.594ex; height:2.509ex;" alt="{\displaystyle V,Y}" loading="lazy"> normierte reelle Vektorräume und die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\colon V\mapsto \mathbb {R} \times Y}">

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<annotation encoding="application/x-tex">{\displaystyle K\colon V\mapsto \mathbb {R} \times Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96ba0721026f1098bd0d9bd3d0dd1f57c91c98ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.793ex; height:2.176ex;" alt="{\displaystyle K\colon V\mapsto \mathbb {R} \times Y}" loading="lazy"> definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x)=(f(x),g(x))}">

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<annotation encoding="application/x-tex">{\displaystyle K(x)=(f(x),g(x))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ab611650b686d2c1a99c74c697dee91a8ae49d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.819ex; height:2.843ex;" alt="{\displaystyle K(x)=(f(x),g(x))}" loading="lazy"> ist fast konvex bezüglich des Kegels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{+}\times K}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}\times K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9791e58fa0538cba1126ea2da8acc426fccfffb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.095ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{+}\times K}" loading="lazy">. Weiter sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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 R}" loading="lazy"> eine beliebige nichtleere Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">.

Das Problem erfüllt nun die Slater-Bedingung, wenn es einen zulässigen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}}</annotation>

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</math><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"> gibt. Das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}\in R,\,g({\tilde {x}})\in -K}">

<mover>

</mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}\in R,\,g({\tilde {x}})\in -K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d07ee394ae065822a6845b1ebb4a1d590fe44a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.325ex; height:2.843ex;" alt="{\displaystyle {\tilde {x}}\in R,\,g({\tilde {x}})\in -K}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g({\tilde {x}})\in \operatorname {int} (-K)}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle g({\tilde {x}})\in \operatorname {int} (-K)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f872bc73d3df8be44400e7d6443e309e825107c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.623ex; height:2.843ex;" alt="{\displaystyle g({\tilde {x}})\in \operatorname {int} (-K)}" loading="lazy"> ist. Dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} (M)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} (M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/052eea3c84c6277a78f5742b48d9adadee693da5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.096ex; height:2.843ex;" alt="{\displaystyle \operatorname {int} (M)}" loading="lazy"> das Innere einer Menge.

Erfüllt solch ein Problem mit fast konvexen Funktionen nun die Slater-Bedingung, so gilt <a href="Starke_Dualit%C3%A4t" class="mw-redirect" title="Starke Dualität">starke Dualität</a> und damit zum Beispiel auch die <a href="Karush-Kuhn-Tucker-Bedingungen" title="Karush-Kuhn-Tucker-Bedingungen">Karush-Kuhn-Tucker-Bedingungen</a>. Der Begriff der fast konvexen Funktion erweitert also die Dualitätstheorie der konvexen Funktionen auf Probleme, die nicht notwendigerweise konvex sein müssen. Dies hat den Vorteil, dass die Slater-Bedingung im Gegensatz zu vielen anderen Regularitätsbedingungen oder „constraint qualifications“ die Regularität des gesamten Problemes liefert, und nicht nur die Regularität in einem Punkt.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Johannes Jahn: Introduction to the Theory of Nonlinear Optimization. 3. Auflage. Springer, Berlin 2007, ISBN 978-3-540-49378-5.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2017-04-05" href="https://de.wikipedia.org/wiki/?title=Fast-konvexe_Funktion&oldid=164261111">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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