# 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.[1]

## 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 (multilevel programming) 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.