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]

Mathematical formulation of the problem

The problem can be stated simply as:

${\displaystyle \min \limits _{x\in X}\;\;f(x)}$

$\mbox{subject to: }\$

${\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y(x)}$

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

See also

• optimization
• Semi-Infinite Programming (SIP)

References

1. O. Stein and G. Still, On generalized semi-infinite optimization and bilevel optimization, European J. Oper. Res., 142 (2002), pp. 444-462

External links

• Mathematical Programming Glossary