Semi-infinite programming

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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.[1]

Mathematical formulation of the problem

The problem can be stated simply as:

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

where

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

References