ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Invex function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Vector_calculus" title="Vector calculus">vector calculus</a>, an invex function is a <a href="Differentiable_function" title="Differentiable function">differentiable function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> for which there exists a vector valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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

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</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,}</annotation>

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</math><img src="./27048c660525cd401cdb80b9a9b200e604419c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.512ex; height:2.843ex;" alt="{\displaystyle f(x)-f(u)\geq \eta (x,u)\cdot \nabla f(u),\,}" loading="lazy"></dd></dl>

for all x and u.

Invex functions were introduced by Hanson as a generalization of <a href="Convex_function" title="Convex function">convex functions</a>.<a href="#cite_note-1">[1]</a> Ben-Israel and Mond provided a simple proof that a function is invex if and only if every <a href="Stationary_point" title="Stationary point">stationary point</a> is a <a href="Global_minimum" class="mw-redirect" title="Global minimum">global minimum</a>, a theorem first stated by Craven and Glover.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

Hanson also showed that if the objective and the constraints of an <a href="Optimization_problem" title="Optimization problem">optimization problem</a> are invex with respect to the same function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (x,u)}">

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

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</math><img src="./99ffab60ee8cb27f37567deabf76f253ff93837f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle \eta (x,u)}" loading="lazy">, then the <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a> are sufficient for a global minimum.

<div class="mw-heading mw-heading2"><h2 id="Type_I_invex_functions">Type I invex functions</h2></div>

A slight generalization of invex functions called Type I invex functions are the most general class of functions for which the <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a> are necessary and sufficient for a global minimum.<a href="#cite_note-:0-4">[4]</a> Consider a mathematical program of the form

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}\min &f(x)\\{\text{s.t.}}&g(x)\leq 0\end{array}}}</annotation>

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</math><img src="./b6c6d1dba6c9a703afda770368f6f9fc94a8ea65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.465ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rl}\min &f(x)\\{\text{s.t.}}&g(x)\leq 0\end{array}}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>

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</math><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>

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</math><img src="./3681774451066e3f7705b73cedee75bc619fcabe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.917ex; height:2.676ex;" alt="{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"> are differentiable functions. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}}">

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<annotation encoding="application/x-tex">{\displaystyle F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}}</annotation>

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</math><img src="./c76ddc1040a4e31cd8fe602481d9c8e1b3eb0f1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.684ex; height:2.843ex;" alt="{\displaystyle F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}}" loading="lazy"> denote the feasible region of this program. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a Type I objective function and the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is a Type I constraint function at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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

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</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> if there exists a vector-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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

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</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle f(x)-f(x_{0})\geq \eta (x)\cdot \nabla {f(x_{0})}}</annotation>

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</math><img src="./ad862d764d68c06db87b1482d9b0d29917063599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.223ex; height:2.843ex;" alt="{\displaystyle f(x)-f(x_{0})\geq \eta (x)\cdot \nabla {f(x_{0})}}" loading="lazy">

and

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<annotation encoding="application/x-tex">{\displaystyle -g(x_{0})\geq \eta (x)\cdot \nabla {g(x_{0})}}</annotation>

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</math><img src="./12359175d1b03046efc179c5ddae55b934575ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.448ex; height:2.843ex;" alt="{\displaystyle -g(x_{0})\geq \eta (x)\cdot \nabla {g(x_{0})}}" loading="lazy">

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {F}}">

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

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</math><img src="./f02ade94555fdefcf03ae59e4e61d4b28807252c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle x\in {F}}" loading="lazy">.<a href="#cite_note-5">[5]</a> Note that, unlike invexity, Type I invexity is defined relative to a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">.

Theorem (Theorem 2.1 in<a href="#cite_note-:0-4">[4]</a>): If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> are Type I invex at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}}">

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

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</math><img src="./e5be23ee5d433f8b576e63bcb47518128ee0b6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle x^{*}}" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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

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</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy">, and the <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a> are satisfied at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}}">

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

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</math><img src="./e5be23ee5d433f8b576e63bcb47518128ee0b6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle x^{*}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}}">

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

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</math><img src="./e5be23ee5d433f8b576e63bcb47518128ee0b6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle x^{*}}" loading="lazy"> is a global minimizer of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="E-invex_function">E-invex function</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {M} }">

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

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</math><img src="./0c466bc0f750f74236f81922b9bff54b6d50452a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.194ex; height:2.176ex;" alt="{\displaystyle \mathbb {M} }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> be an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">-differentiable function on a nonempty open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {M} \subset \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {M} \subset \mathbb {R} ^{n}}</annotation>

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</math><img src="./3ab04d2410613e74e26f6db7532266471c95bb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {M} \subset \mathbb {R} ^{n}}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is said to be an E-invex function at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> if there exists a vector valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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

</semantics>

</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> such that

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (f\circ E)(x)-(f\circ E)(u)\geq \nabla (f\circ E)(u)\cdot \eta (E(x),E(u)),\,}</annotation>

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</math><img src="./4f4a1ab4ab6ec2086cb33353adc0b6f7c6826446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.021ex; height:2.843ex;" alt="{\displaystyle (f\circ E)(x)-(f\circ E)(u)\geq \nabla (f\circ E)(u)\cdot \eta (E(x),E(u)),\,}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {M} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0c466bc0f750f74236f81922b9bff54b6d50452a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.194ex; height:2.176ex;" alt="{\displaystyle \mathbb {M} }" loading="lazy">.

E-invex functions were introduced by Abdulaleem as a generalization of differentiable <a href="Convex_function" title="Convex function">convex functions</a>.<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="E-type_I_Functions">E-type I Functions</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./30624f7e1866d10798b729e028e0b001e00e9ec1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.12ex; height:2.343ex;" alt="{\displaystyle E:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c8e2ebac5aaed569dacf365f3365b1cd07a1b4a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.437ex; height:2.343ex;" alt="{\displaystyle M\subset \mathbb {R} ^{n}}" loading="lazy">be an open E-invex set. A vector-valued pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,g)}">

</mstyle>

</mrow>

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

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</math><img src="./a230d05990654b0fdcc68c1987f0cc6a27ead28c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.238ex; height:2.843ex;" alt="{\displaystyle (f,g)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> represent objective and constraint functions respectively, is said to be E-type I with respect to a vector-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta :M\times M\to \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta :M\times M\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./979bfc59e368d43219e51400c7de4236b0bd5519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.342ex; height:2.843ex;" alt="{\displaystyle \eta :M\times M\to \mathbb {R} ^{n}}" loading="lazy">, at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6a60d3681dd9eeb4419fa13726d21d4c8afdfb83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle u\in M}" loading="lazy">, if the following inequalities hold for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in F_{E}=\{x\in \mathbb {R} ^{n}\;|\;g(E(x))\leq 0\}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in F_{E}=\{x\in \mathbb {R} ^{n}\;|\;g(E(x))\leq 0\}}</annotation>

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</math><img src="./d0af88cffd6cb1a36a73aa520fe08444494e0d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.681ex; height:2.843ex;" alt="{\displaystyle x\in F_{E}=\{x\in \mathbb {R} ^{n}\;|\;g(E(x))\leq 0\}}" loading="lazy">:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}(E(x))-f_{i}(E(u))\geq \nabla f_{i}(E(u))\cdot \eta (E(x),E(u)),}</annotation>

</semantics>

</math><img src="./7e3005b038a58b3b136a089895e0e49944746976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.031ex; height:2.843ex;" alt="{\displaystyle f_{i}(E(x))-f_{i}(E(u))\geq \nabla f_{i}(E(u))\cdot \eta (E(x),E(u)),}" loading="lazy">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -g_{j}(E(u))\geq \nabla g_{j}(E(u))\cdot \eta (E(x),E(u)).}</annotation>

</semantics>

</math><img src="./9b41310eae4c3fee7023f45754e5a07106cf9dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.496ex; height:3.009ex;" alt="{\displaystyle -g_{j}(E(u))\geq \nabla g_{j}(E(u))\cdot \eta (E(x),E(u)).}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Remark_1.">Remark 1.</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> are differentiable functions and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(x)=x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0274c832a1dccb235d1cf5b8dadc072f6c6533a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.343ex; height:2.843ex;" alt="{\displaystyle E(x)=x}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is an identity map), then the definition of E-type I functions<a href="#cite_note-7">[7]</a> reduces to the definition of type I functions introduced by Rueda and Hanson.<a href="#cite_note-8">[8]</a>

<ul><li><a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Pseudoconvex_function" title="Pseudoconvex function">Pseudoconvex function</a></li>

<li><a href="Quasiconvex_function" title="Quasiconvex function">Quasiconvex function</a></li></ul>

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

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFHanson1981" class="citation journal cs1">Hanson, Morgan A. (1981). "On sufficiency of the Kuhn-Tucker conditions". Journal of Mathematical Analysis and Applications. 80 (2): 545–550. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-247X%2881%2990123-2">10.1016/0022-247X(81)90123-2</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F141569">10338.dmlcz/141569</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-247X">0022-247X</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBen-IsraelMond1986" class="citation journal cs1">Ben-Israel, A.; Mond, B. (1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0334270000005142">"What is invexity?"</a>. The ANZIAM Journal. 28 (1): 1–9. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0334270000005142">10.1017/S0334270000005142</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1839-4078">1839-4078</a>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFCravenGlover1985" class="citation journal cs1">Craven, B. D.; Glover, B. M. (1985). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1446788700022126">"Invex functions and duality"</a>. Journal of the Australian Mathematical Society. 39 (1): 1–20. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1446788700022126">10.1017/S1446788700022126</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0263-6115">0263-6115</a>.</cite>

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<li id="cite_note-:0-4">^ <a href="#cite_ref-:0_4-0">a</a> <a href="#cite_ref-:0_4-1">b</a> <cite id="CITEREFHanson1999" class="citation journal cs1">Hanson, Morgan A. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1999.6484">"Invexity and the Kuhn–Tucker Theorem"</a>. Journal of Mathematical Analysis and Applications. 236 (2): 594–604. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1999.6484">10.1006/jmaa.1999.6484</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-247X">0022-247X</a>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFHansonMond1987" class="citation journal cs1">Hanson, M. A.; Mond, B. (1987). "Necessary and sufficient conditions in constrained optimization". Mathematical Programming. 37 (1): 51–58. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02591683">10.1007/BF02591683</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1436-4646">1436-4646</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206818360">206818360</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFAbdulaleem2019" class="citation journal cs1">Abdulaleem, Najeeb (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fitmconf%2F20192401002">"E-invexity and generalized E-invexity in E-differentiable multiobjective programming"</a>. ITM Web of Conferences. 24 (1) 01002. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fitmconf%2F20192401002">10.1051/itmconf/20192401002</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFAbdulaleem2023" class="citation journal cs1">Abdulaleem, Najeeb (2023). <a rel="nofollow" class="external text" href="https://www.aimsciences.org/article/doi/10.3934/jimo.2022004">"Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions"</a>. Journal of Industrial and Management Optimization. 19 (2): 1513. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3934%2Fjimo.2022004">10.3934/jimo.2022004</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1547-5816">1547-5816</a>.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFRuedaHanson1988" class="citation journal cs1">Rueda, Norma G; Hanson, Morgan A (1988-03-01). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/0022247X88903137">"Optimality criteria in mathematical programming involving generalized invexity"</a>. Journal of Mathematical Analysis and Applications. 130 (2): 375–385. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-247X%2888%2990313-7">10.1016/0022-247X(88)90313-7</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-247X">0022-247X</a>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex Optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008.</li>

<li>S. K. Mishra, S.-Y. Wang and K. K. Lai, Generalized Convexity and Vector Optimization, Springer, New York, 2009.</li></ul>

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</style><div id="Convex_analysis_and_variational_analysis174" style="font-size:114%;margin:0 4em"><a href="Convex_analysis" title="Convex analysis">Convex analysis</a> and <a href="Variational_analysis" title="Variational analysis">variational analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</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="Convex_combination" title="Convex combination">Convex combination</a></li>

<li><a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Convex_set" title="Convex set">Convex set</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_convexity_topics" title="List of convexity topics">Topics (list)</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="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Convex_geometry" title="Convex geometry">Convex geometry</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex metric space</a></li>

<li><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></li>

<li><a href="Duality_(optimization)" title="Duality (optimization)">Duality</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a></li>

<li><a href="Simplex" title="Simplex">Simplex</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</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="Convex_conjugate" title="Convex conjugate">Convex conjugate</a></li>

<li><a href="Concave_function" title="Concave function">Concave</a></li>

<li>(<a href="Closed_convex_function" title="Closed convex function">Closed</a></li>

<li><a href="Logarithmically_convex_function" title="Logarithmically convex function">Logarithmically</a></li>

<li><a href="Proper_convex_function" title="Proper convex function">Proper</a></li>

<li><a href="Pseudoconvex_function" title="Pseudoconvex function">Pseudo-</a></li>

<li><a href="Quasiconvex_function" title="Quasiconvex function">Quasi-</a>) <a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Semi-continuity" title="Semi-continuity">Semi-continuity</a></li>

<li><a href="Subderivative" title="Subderivative">Subderivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</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="Carath%C3%A9odory's_theorem_(convex_hull)" title="Carathéodory's theorem (convex hull)">Carathéodory's theorem</a></li>

<li><a href="Ekeland's_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li>

<li><a href="Fenchel%E2%80%93Moreau_theorem" title="Fenchel–Moreau theorem">Fenchel–Moreau theorem</a></li>

<li><a href="Fenchel-Young_inequality" class="mw-redirect" title="Fenchel-Young inequality">Fenchel-Young inequality</a></li>

<li><a href="Jensen's_inequality" title="Jensen's inequality">Jensen's inequality</a></li>

<li><a href="Hermite%E2%80%93Hadamard_inequality" title="Hermite–Hadamard inequality">Hermite–Hadamard inequality</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman theorem</a></li>

<li><a href="Mazur's_lemma" title="Mazur's lemma">Mazur's lemma</a></li>

<li><a href="Shapley%E2%80%93Folkman_lemma" title="Shapley–Folkman lemma">Shapley–Folkman lemma</a></li>

<li><a href="Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson–Ursescu</a></li>

<li><a href="Ursescu_theorem#Simons'_theorem" title="Ursescu theorem">Simons</a></li>

<li><a href="Ursescu_theorem" title="Ursescu theorem">Ursescu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</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="Convex_hull" title="Convex hull">Convex hull</a></li>

<li>(<a href="Orthogonally_convex_set" class="mw-redirect" title="Orthogonally convex set">Orthogonally</a>, <a href="Pseudoconvexity" title="Pseudoconvexity">Pseudo-</a>) <a href="Convex_set" title="Convex set">Convex set</a></li>

<li><a href="Effective_domain" title="Effective domain">Effective domain</a></li>

<li><a href="Epigraph_(mathematics)" title="Epigraph (mathematics)">Epigraph</a></li>

<li><a href="Hypograph_(mathematics)" title="Hypograph (mathematics)">Hypograph</a></li>

<li><a href="John_ellipsoid" title="John ellipsoid">John ellipsoid</a></li>

<li><a href="Radial_set" title="Radial set">Radial set</a>/<a href="Algebraic_interior" title="Algebraic interior">Algebraic interior</a></li>

<li><a href="Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Series</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="Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> (<a href="Convex_series#Types_of_subsets" title="Convex series">(cs, lcs)-closed</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(cs, bcs)-complete</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(lower) ideally convex</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(Hx)</a>, and <a href="Convex_series#Types_of_subsets" title="Convex series">(Hwx)</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Duality_(optimization)" title="Duality (optimization)">Duality</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="Dual_system" title="Dual system">Dual system</a></li>

<li><a href="Duality_gap" title="Duality gap">Duality gap</a></li>

<li><a href="Strong_duality" title="Strong duality">Strong duality</a></li>

<li><a href="Weak_duality" title="Weak duality">Weak duality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications and related</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="Convexity_in_economics" title="Convexity in economics">Convexity in economics</a></li></ul>

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