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:

$\min_{x \in X}\;\; f(x)$

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

$g(x,y) \le 0, \;\; \forall y \in Y$

where

$f: R^n \to R$

$g: R^n \times R^m \to R$

$X \subseteq R^n$

$Y \subseteq R^m.$

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

1. • 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 0-387-98705-3. MR 1756264.
   • 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). pp. 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.

• 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 0-387-98705-3. MR 1756264.
• 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). pp. 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.
• 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