# Generalized semi-infinite programming

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{{Refimprove|date=May 2008}}

In [mathematics](/source/mathematics), a [semi-infinite programming (SIP)](/source/Semi-Infinite_Programming) problem is an optimization problem with a finite number of variables and an infinite number of constraints. The constraints are typically parameterized. In a '''generalized semi-infinite programming''' ('''GSIP''') problem, the feasible set of the parameters depends on the variables.<ref>O. Stein and G. Still, ''[https://pdfs.semanticscholar.org/ce4f/c65e0dddd2c24580f0f3e05f5bf9b42ad723.pdf On generalized semi-infinite optimization and bilevel optimization]'', European J. Oper. Res., 142 (2002), pp. 444-462</ref>

== Mathematical formulation of the problem ==
The problem can be stated simply as:
:<math> \min\limits_{x \in X}\;\; f(x) </math>

:<math> \mbox{subject to: }\ </math>

::<math> g(x,y) \le 0, \;\;  \forall y \in Y(x) </math>

where
:<math>f: R^n \to R</math>
:<math>g: R^n \times R^m \to R</math>
:<math>X \subseteq R^n</math>
:<math>Y \subseteq R^m.</math>

In the special case that the set :<math>Y(x)</math> is nonempty for all <math>x \in X</math> GSIP can be cast as [bilevel programs](/source/Bilevel_optimization) ([Multilevel programming](/source/Multilevel_programming)).

== Methods for solving the problem ==
{{Empty section|date=July 2010}}

==Examples==
{{Empty section|date=July 2010}}

== See also ==
* [optimization](/source/Optimization_(mathematics))
* [Semi-Infinite Programming (SIP)](/source/Semi-Infinite_Programming)

== References==
{{Reflist}}

== External links==
*[http://glossary.computing.society.informs.org/ Mathematical Programming Glossary] {{Webarchive|url=https://web.archive.org/web/20100328165516/http://glossary.computing.society.informs.org/ |date=2010-03-28 }}

Category:Optimization in vector spaces

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Adapted from the Wikipedia article [Generalized semi-infinite programming](https://en.wikipedia.org/wiki/Generalized_semi-infinite_programming) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Generalized_semi-infinite_programming?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.