Definitions

# Generalized semi-infinite programming

In mathematics, generalized semi-infinite programming (GSIP) is an optimization problem with a finite number of variables and an infinite number of constraints. The constraints are parameterized by parameters and the feasible set of the parameters depends on the variables.

## Mathematical formulation of the problem

The problem can be stated simply as:
$minlimits_\left\{x in X\right\};; f\left(x\right)$

$mbox\left\{subject to: \right\}$

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

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 :$Y\left(x\right)$ is nonempty for all $x in X$ GSIP can be cast as bilevel programs (Multilevel programming).

## References

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