In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized.
Mathematical formulation of the problem The problem can be stated simply as:
min x ∈ X f ( x ) {\displaystyle \min _{x\in X}\;\;f(x)}
subject to: {\displaystyle {\text{subject to: }}}
g ( x , y ) ≤ 0 , ∀ y ∈ Y {\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y}
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}.}
SIP can be seen as a special case of bilevel programs in which the lower-level variables do not participate in the objective function.
Methods for solving the problem
In the meantime, see external links below for a complete tutorial.
Examples
In the meantime, see external links below for a complete tutorial.
See also Optimization Generalized semi-infinite programming (GSIP)
References
External links Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science).
