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.<ref> {{harvnb|Bonnans|Shapiro|2000|pp=496–526, 581}} {{harvnb|Goberna|López|1998}} {{harvnb|Hettich|Kortanek|1993|pp=380–429}} </ref>

==Mathematical formulation of the problem== The problem can be stated simply as: :<math> \min_{x \in X}\;\; f(x) </math>

:<math> \text{subject to: }</math>

::<math> g(x,y) \le 0, \;\; \forall y \in Y </math>

where :<math>f: R^n \to R</math> :<math>g: R^n \times R^m \to R</math> :<math>X \subseteq R^n</math> :<math>Y \subseteq R^m.</math>

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== {{Empty section|date=July 2010}}

In the meantime, see external links below for a complete tutorial.

==Examples== {{Empty section|date=July 2010}}

In the meantime, see external links below for a complete tutorial.

==References== {{reflist}} {{refbegin}} *{{cite book |first1=Edward J. |last1=Anderson |first2=Peter |last2=Nash |title=Linear Programming in Infinite-Dimensional Spaces |publisher=Wiley |date=1987 |isbn=0-471-91250-6 |oclc=15053031 }} *{{cite book |last1=Bonnans |first1=J. Frédéric |last2=Shapiro |first2=Alexander |chapter=5.4, 7.4.4 Semi-infinite programming |title=Perturbation analysis of optimization problems |series=Springer Series in Operations Research |publisher=Springer |year=2000 |pages=496–526, 581|isbn=978-0-387-98705-7|mr=1756264}} *{{cite book |first1=M.A. |last1=Goberna |first2=M.A. |last2=López |title=Linear Semi-Infinite Optimization |publisher=Wiley |date=1998 }} *{{cite book |first1=M.A. |last1=Goberna |first2=M.A. |last2=López |title=Post-Optimal Analysis in Linear Semi-Infinite Optimization |doi=10.1007/978-1-4899-8044-1 |url=https://link.springer.com/book/10.1007/978-1-4899-8044-1 |isbn=978-1-4899-8044-1 |series=SpringerBriefs in Optimization |publisher=Springer |date=2014 }} * {{cite journal|last1=Hettich|first1=R.|last2=Kortanek|first2=K.O.|title=Semi-infinite programming: Theory, methods, and applications|journal=SIAM Review|volume=35|year=1993|number=3|pages=380–429|doi=10.1137/1035089|mr=1234637 | jstor = 2132425}} *{{cite book |first=David G. |last=Luenberger |title=Optimization by Vector Space Methods |publisher=Wiley |location= |date=1997 |isbn=0-471-18117-X |oclc=52405793 }} *{{cite book |editor-first=Rembert |editor-last=Reemtsen and |editor2-first=Jan-J. |editor2-last=Rückmann |title=Semi-Infinite Programming |publisher=Springer |date=1998 |isbn=978-1-4757-2868-2 |doi=10.1007/978-1-4757-2868-2 |url=https://link.springer.com/book/10.1007/978-1-4757-2868-2 |series=Nonconvex Optimization and Its Applications |volume=25 }} *{{cite journal |first1=F. |last1=Guerra Vázquez |first2=J.-J. |last2=Rückmann |first3=O. |last3=Stein |first4=G. |last4=Still |title=Generalized semi-infinite programming: A tutorial |journal=Journal of Computational and Applied Mathematics |volume=217 |issue=2 |pages=394–419 |date=1 August 2008 |doi=10.1016/j.cam.2007.02.012 |bibcode=2008JCoAM.217..394G |url=http://www.sciencedirect.com/science/article/pii/S0377042707000982 |url-access=subscription }} {{refend}}

==External links== * [https://glossary.informs.org/ver2/mpgwiki/index.php?title=Semi-infinite_program Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science)].

Category:Optimization in vector spaces Category:Approximation theory Category:Numerical analysis