Generalized semi-infinite programming

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

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).