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.^[1]

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

1. • Book: Bonnans. J. Frédéric. Shapiro. Alexander. 5.4 and 7.4.4 Semi-infinite programming. Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer-Verlag. New York. 2000. 496–526 and 581. 978-0-387-98705-7. 1756264.
   • M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
   • Hettich. R.. Kortanek. K. O.. Semi-infinite programming: Theory, methods, and applications. SIAM Review. 35. 1993. 3. 380–429. 10.1137/1035089. 1234637 . 2132425.

• Edward J. Anderson and Peter Nash, Linear Programming in Infinite-Dimensional Spaces, Wiley, 1987.
• Book: Bonnans. J. Frédéric. Shapiro. Alexander. 5.4 and 7.4.4 Semi-infinite programming. Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer-Verlag. New York. 2000. 496–526 and 581. 978-0-387-98705-7. 1756264.
• M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
• Hettich. R.. Kortanek. K. O.. Semi-infinite programming: Theory, methods, and applications. SIAM Review. 35. 1993. 3. 380–429. 10.1137/1035089. 1234637 . 2132425.
• David Luenberger (1997). Optimization by Vector Space Methods. John Wiley & Sons. .
• Rembert Reemtsen and Jan-J. Rückmann (Editors), Semi-Infinite Programming (Nonconvex Optimization and Its Applications). Springer, 1998,, 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