Generalized semi-infinite programming
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Revision as of 18:58, 6 March 2021 by imported>NBrush (add)
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.[1]
Mathematical formulation of the problem
The problem can be stated simply as:
- [math]\displaystyle{ \min\limits_{x \in X}\;\; f(x) }[/math]
- [math]\displaystyle{ \mbox{subject to: }\ }[/math]
- [math]\displaystyle{ g(x,y) \le 0, \;\; \forall y \in Y(x) }[/math]
where
- [math]\displaystyle{ f: R^n \to R }[/math]
- [math]\displaystyle{ g: R^n \times R^m \to R }[/math]
- [math]\displaystyle{ X \subseteq R^n }[/math]
- [math]\displaystyle{ Y \subseteq R^m. }[/math]
In the special case that the set :[math]\displaystyle{ Y(x) }[/math] is nonempty for all [math]\displaystyle{ x \in X }[/math] GSIP can be cast as bilevel programs (Multilevel programming).
Methods for solving the problem
Examples
See also
- optimization
- Semi-Infinite Programming (SIP)
References
- ↑ O. Stein and G. Still, On generalized semi-infinite optimization and bilevel optimization, European J. Oper. Res., 142 (2002), pp. 444-462
External links
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