Semi-infinite programming: Difference between revisions

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In [[optimization (mathematics)|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}}

* {{cite book|last1=Bonnans|first1=J.&nbsp;Frédéric|last2=Shapiro|first2=Alexander|chapter=5.4 and&nbsp;7.4.4 Semi-infinite programming|title=Perturbation analysis of optimization problems|series=Springer Series in Operations Research|publisher=Springer-Verlag|___location=New York|year=2000|pages=496–526 and&nbsp;581|isbn=0-387-98705-3|mr=1756264}}

{{harvnb|Goberna|López|1998}}

* M. A. Goberna and M. A. López, ''Linear Semi-Infinite Optimization'', Wiley, 1998.

{{harvnb|Hettich|Kortanek|1993|pp=380–429}}

* {{cite article|last1=Hettich|first1=R.|last2=Kortanek|first2=K.&nbsp;O.|title=Semi-infinite programming: Theory, methods, and applications|jstor=2132425|journal=SIAM Review|volume=35|year=1993|number=3|pages=380–429|doi=10.1137/1035089|mr=1234637 | jstor = 2132425}}

</ref>

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:<math> \min_{x \in X}\;\; f(x) </math>

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

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

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:<math>Y \subseteq R^m.</math>

SIP can be seen as a special case of bilevel programs ([[multilevelbilevel programmingprogram]])s in which the lower-level variables do not participate in the objective function.

==Methods for solving the problem==

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==References==

{{reflist}}

<references/>

{{refbegin}}

*{{cite book |first1=Edward J. |last1=Anderson and |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.&nbsp;Frédéric |last2=Shapiro |first2=Alexander |chapter=5.4, and&nbsp;7.4.4 Semi-infinite programming |title=Perturbation analysis of optimization problems |series=Springer Series in Operations Research |publisher=Springer-Verlag|___location=New York|year=2000 |pages=496–526, and&nbsp;581|isbn=978-0-387-98705-37|mr=1756264}}

*{{cite book |first1=M. A. |last1=Goberna and |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 article|last1=Hettich|first1=R.|last2=Kortanek|first2=K.&nbsp;O.|title=Semi-infinite programming: Theory, methods, and applications|jstor=2132425|journal=SIAM Review|volume=35|year=1993|number=3|pages=380–429|doi=10.1137/1035089|mr=1234637 | jstor = 2132425}}

* {{cite articlejournal|last1=Hettich|first1=R.|last2=Kortanek|first2=K.&nbsp;O.|title=Semi-infinite programming: Theory, methods, and applications|jstor=2132425|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 (1997). ''|title=Optimization by Vector Space Methods.'' John |publisher=Wiley &|___location= Sons. ISBN|date=1997 |isbn=0-471-18117-X. |oclc=52405793 }}

* Rembert Reemtsen and Jan-J. Rückmann (Editors), ''Semi-Infinite Programming (Nonconvex Optimization and Its Applications)''. Springer, 1998, ISBN 0-7923-5054-5, 1998

*{{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==

* [httphttps://glossary.computing.society.informs.org/secondver2/mpgwiki/index.php?pagetitle=S.html#Semi-infinite_program Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science)].

* [http://www.sciencedirect.com/science/article/pii/S0377042707000982# 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]

[[Category:Optimization in vector spaces]]

[[Category:Approximation theory]]

[[Category:Numerical analysis]]

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