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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.cite-ref-1[1]

• Mathematical formulation of the problem

• Methods for solving the problem

• Examples

• See also

• References

• External links

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).

• optimization

• Semi-Infinite Programming (SIP)

cite-note-11. ↑ 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