Semi-infinite programming

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:

${\displaystyle \min _{x\in X}\;\;f(x)}$

${\displaystyle {\text{subject to: }}}$

${\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y}$

where

${\displaystyle f:R^{n}\to R}$

${\displaystyle g:R^{n}\times R^{m}\to R}$

${\displaystyle X\subseteq R^{n}}$

${\displaystyle Y\subseteq R^{m}.}$

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

1. Bonnans & Shapiro 2000, pp. 496–526, 581 Goberna & López 1998 Hettich & Kortanek 1993, pp. 380–429

• 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

• Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science).