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:
- [math]\displaystyle{ \min_{x \in X}\;\; f(x) }[/math]
- [math]\displaystyle{ \text{subject to: } }[/math]
- [math]\displaystyle{ g(x,y) \le 0, \;\; \forall y \in Y }[/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]
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
- ↑
- Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4 and 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. New York: Springer-Verlag. pp. 496–526 and 581. ISBN 978-0-387-98705-7.
- M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
- Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review 35 (3): 380–429. doi:10.1137/1035089.
- Edward J. Anderson and Peter Nash, Linear Programming in Infinite-Dimensional Spaces, Wiley, 1987.
- Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4 and 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. New York: Springer-Verlag. pp. 496–526 and 581. ISBN 978-0-387-98705-7.
- M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
- Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review 35 (3): 380–429. doi:10.1137/1035089.
- David Luenberger (1997). Optimization by Vector Space Methods. John Wiley & Sons. ISBN 0-471-18117-X.
- Rembert Reemtsen and Jan-J. Rückmann (Editors), Semi-Infinite Programming (Nonconvex Optimization and Its Applications). Springer, 1998, ISBN 0-7923-5054-5, 1998
External links
- Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science).
- A complete, free, open source Semi Infinite Programming Tutorial is available here from Elsevier as a pdf download from their Journal of Computational and Applied Mathematics, Volume 217, Issue 2, 1 August 2008, Pages 394–419
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