Semi-infinite programming explained

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} f(x) subjectto: g(x,y) \le 0 , \forall y \in Y

where

f: R ^{n} \to R g: R ^{n} x R ^{m} \to R X \subseteq R ^{n} 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

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

External links

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