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.
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
External links
- Mathematical Programming Glossary Archived 2010-03-28 at the Wayback Machine
References
O. Stein and G. Still, On generalized semi-infinite optimization and bilevel optimization, European J. Oper. Res., 142 (2002), pp. 444-462 https://pdfs.semanticscholar.org/ce4f/c65e0dddd2c24580f0f3e05f5bf9b42ad723.pdf ↩