# Generalized semi-infinite programming

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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.[1]

## Mathematical formulation of the problem

The problem can be stated simply as:

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

where

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

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