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:

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

where

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

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

Methods for solving the problem

Examples

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
  • Mathematical Programming Glossary Archived 2010-03-28 at the Wayback Machine