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.

Mathematical formulation of the problem

The problem can be stated simply as:

[math]\displaystyle{ \min\limits_{x \in X}\;\; f(x) }[/math]

[math]\displaystyle{ \mbox{subject to: }\ }[/math]

[math]\displaystyle{ g(x,y) \le 0, \;\; \forall y \in Y(x) }[/math]

where

[math]\displaystyle{ f: R^n \to R }[/math]

[math]\displaystyle{ g: R^n \times R^m \to R }[/math]

[math]\displaystyle{ X \subseteq R^n }[/math]

[math]\displaystyle{ Y \subseteq R^m. }[/math]

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

See also

• optimization
• Semi-Infinite Programming (SIP)

External links

• Mathematical Programming Glossary