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 ∈ 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

  • Anderson, Edward J.; Nash, Peter (1987). Linear Programming in Infinite-Dimensional Spaces. Wiley. ISBN 0-471-91250-6. OCLC .
  • Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer. pp. 496–526, 581. ISBN 978-0-387-98705-7. MR .
  • Goberna, M.A.; López, M.A. (1998). Linear Semi-Infinite Optimization. Wiley.
  • Goberna, M.A.; López, M.A. (2014). . SpringerBriefs in Optimization. Springer. doi:. ISBN 978-1-4899-8044-1.
  • Hettich, R.; Kortanek, K.O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3): 380–429. doi:. JSTOR . MR .
  • Luenberger, David G. (1997). Optimization by Vector Space Methods. Wiley. ISBN 0-471-18117-X. OCLC .
  • Reemtsen and, Rembert; Rückmann, Jan-J., eds. (1998). . Nonconvex Optimization and Its Applications. Vol. 25. Springer. doi:. ISBN 978-1-4757-2868-2.
  • Guerra Vázquez, F.; Rückmann, J.-J.; Stein, O.; Still, G. (1 August 2008). . Journal of Computational and Applied Mathematics. 217 (2): 394–419. Bibcode:. doi:.

External links

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