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 ∈ 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

  • 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 978-0-387-98705-7. 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): 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 978-0-387-98705-7. 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): 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


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