Semi-infinite programming

In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized.

Mathematical formulation of the problem
The problem can be stated simply as:
 * $$ \min_{x \in X}\;\; f(x) $$


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


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

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

SIP can be seen as a special case of bilevel programs in which the lower-level variables do not participate in the objective function.

Methods for solving the problem
In the meantime, see external links below for a complete tutorial.

Examples
In the meantime, see external links below for a complete tutorial.