Semi-infinite programming

In mathematics, 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 [1]. In the former case the constraints are typically parameterized.

Mathematical formulation of the problem

The problem can be stated simply as:

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

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

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

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 (multilevel programming) in which the lower-level variables do not participate in the objective function.

See also

• Optimization
• Generalized semi-infinite programming (GSIP)

References

• 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. MR1756264. {{cite book}}: Cite has empty unknown parameter: |1= (help)
• M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
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• 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 07923505451998

External links

• Mathematical Programming Glossary