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
- Anderson, Edward J.; Nash, Peter (1987). Linear Programming in Infinite-Dimensional Spaces. Wiley. ISBN 0-471-91250-6. OCLC 15053031.
- Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer. pp. 496–526, 581. ISBN 978-0-387-98705-7. MR 1756264.
- Goberna, M.A.; López, M.A. (1998). Linear Semi-Infinite Optimization. Wiley.
- Goberna, M.A.; López, M.A. (2014). Post-Optimal Analysis in Linear Semi-Infinite Optimization. SpringerBriefs in Optimization. Springer. doi:10.1007/978-1-4899-8044-1. ISBN 978-1-4899-8044-1.
- Hettich, R.; Kortanek, K.O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3): 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.
- Luenberger, David G. (1997). Optimization by Vector Space Methods. Wiley. ISBN 0-471-18117-X. OCLC 52405793.
- Reemtsen and, Rembert; Rückmann, Jan-J., eds. (1998). Semi-Infinite Programming. Nonconvex Optimization and Its Applications. Vol. 25. Springer. doi:10.1007/978-1-4757-2868-2. ISBN 978-1-4757-2868-2.
- Guerra Vázquez, F.; Rückmann, J.-J.; Stein, O.; Still, G. (1 August 2008). "Generalized semi-infinite programming: A tutorial". Journal of Computational and Applied Mathematics. 217 (2): 394–419. Bibcode:2008JCoAM.217..394G. doi:10.1016/j.cam.2007.02.012.
External links
References
Bonnans & Shapiro 2000, pp. 496–526, 581 Goberna & López 1998 Hettich & Kortanek 1993, pp. 380–429 - Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer. pp. 496–526, 581. ISBN 978-0-387-98705-7. MR 1756264. https://mathscinet.ams.org/mathscinet-getitem?mr=1756264 ↩