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:
 * $$ \min\limits_{x \in X}\;\; f(x) $$


 * $$ \mbox{subject to: }\ $$


 * $$ g(x,y) \le 0, \;\; \forall y \in Y(x) $$

where
 * $$f: R^n \to R$$
 * $$g: R^n \times R^m \to R$$
 * $$X \subseteq R^n$$
 * $$Y \subseteq R^m.$$

In the special case that the set :$$Y(x)$$ is nonempty for all $$x \in X$$ GSIP can be cast as bilevel programs (Multilevel programming).