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 optimization Semi-Infinite Programming (SIP)
References
External links Mathematical Programming Glossary Archived 2010-03-28 at the Wayback Machine
