Generalized semi-infinite programming

In mathematics, a semi-infinite programming (SIP) 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 (Multilevel programming).

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

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

== See also == * optimization * Semi-Infinite Programming (SIP)

== 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