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<title>Generalized semi-infinite programming</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalized semi-infinite programming</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <a href="Semi-Infinite_Programming" class="mw-redirect" title="Semi-Infinite Programming">semi-infinite programming (SIP)</a> problem is an optimization problem with a finite number of variables and an infinite number of constraints. The constraints are typically parameterized. In a <b>generalized semi-infinite programming</b> (<b>GSIP</b>) problem, the feasible set of the parameters depends on the variables.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
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<div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation_of_the_problem">Mathematical formulation of the problem</h2></div>
<p>The problem can be stated simply as:
</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 \limits _{x\in X}\;\;f(x)}">
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<annotation encoding="application/x-tex">{\displaystyle \min \limits _{x\in X}\;\;f(x)}</annotation>
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</math></span><img src="./ca3894ab942944b2be1fb33ad2790f57ccf22fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.97ex; height:4.009ex;" alt="{\displaystyle \min \limits _{x\in X}\;\;f(x)}" loading="lazy"></span></dd></dl>
<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 {\mbox{subject to: }}\ }">
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<annotation encoding="application/x-tex">{\displaystyle {\mbox{subject to: }}\ }</annotation>
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</math></span><img src="./509659a592e8e2a2124341f947c61c971aa4c807.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.638ex; height:2.509ex;" alt="{\displaystyle {\mbox{subject to: }}\ }" loading="lazy"></span></dd></dl>
<dl><dd><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 g(x,y)\leq 0,\;\;\forall y\in Y(x)}">
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<annotation encoding="application/x-tex">{\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y(x)}</annotation>
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</math></span><img src="./49c84cee80d1b5c597eb0cf44be667c1595367e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.231ex; height:2.843ex;" alt="{\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y(x)}" loading="lazy"></span></dd></dl></dd></dl>
<p>where
</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 f:R^{n}\to R}">
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<annotation encoding="application/x-tex">{\displaystyle f:R^{n}\to R}</annotation>
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</math></span><img src="./2ff44474c6224ad756092e078f89097fac134c31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.576ex; height:2.676ex;" alt="{\displaystyle f:R^{n}\to R}" loading="lazy"></span></dd>
<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 g:R^{n}\times R^{m}\to R}">
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<annotation encoding="application/x-tex">{\displaystyle g:R^{n}\times R^{m}\to R}</annotation>
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</math></span><img src="./eb6e4a9ae7c8f5274e1a41d2ecf254773562552f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.693ex; height:2.676ex;" alt="{\displaystyle g:R^{n}\times R^{m}\to R}" loading="lazy"></span></dd>
<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\subseteq R^{n}}">
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<annotation encoding="application/x-tex">{\displaystyle X\subseteq R^{n}}</annotation>
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</math></span><img src="./079114f0861b4433abe9c25f3ea80f7a50b425fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.061ex; height:2.509ex;" alt="{\displaystyle X\subseteq R^{n}}" loading="lazy"></span></dd>
<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 Y\subseteq R^{m}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle Y\subseteq R^{m}.}</annotation>
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</math></span><img src="./39dfd85c84dc602db603fb99f7d615f91c6e0a05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.958ex; height:2.509ex;" alt="{\displaystyle Y\subseteq R^{m}.}" loading="lazy"></span></dd></dl>
<p>In the special case that the set :<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 Y(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle Y(x)}</annotation>
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</math></span><img src="./fca9de77ea29879f48f3f08c469fb73b8945a094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.912ex; height:2.843ex;" alt="{\displaystyle Y(x)}" loading="lazy"></span> is nonempty for all <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\in X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>
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</math></span><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.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 x\in X}" loading="lazy"></span> GSIP can be cast as <a href="Bilevel_optimization" title="Bilevel optimization">bilevel programs</a> (Multilevel programming).
</p>
<div class="mw-heading mw-heading2"><h2 id="Methods_for_solving_the_problem">Methods for solving the problem</h2></div>
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization</a></li>
<li><a href="Semi-Infinite_Programming" class="mw-redirect" title="Semi-Infinite Programming">Semi-Infinite Programming (SIP)</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">O. Stein and G. Still, <i><a rel="nofollow" class="external text" href="https://pdfs.semanticscholar.org/ce4f/c65e0dddd2c24580f0f3e05f5bf9b42ad723.pdf">On generalized semi-infinite optimization and bilevel optimization</a></i>, European J. Oper. Res., 142 (2002), pp. 444-462</span>
</li>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://glossary.computing.society.informs.org/">Mathematical Programming Glossary</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100328165516/http://glossary.computing.society.informs.org/">Archived</a> 2010-03-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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