# 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:

${\displaystyle \min _{x\in X}\;\;f(x)}$
${\displaystyle {\text{subject to: }}}$
${\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y}$

where

${\displaystyle f:R^{n}\to R}$
${\displaystyle g:R^{n}\times R^{m}\to R}$
${\displaystyle X\subseteq R^{n}}$
${\displaystyle Y\subseteq R^{m}.}$

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.