# Semi-infinite programming

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Semi-infinite_programming
> Markdown URL: https://mediated.wiki/source/Semi-infinite_programming.md
> Source: https://en.wikipedia.org/wiki/Semi-infinite_programming
> Source revision: 1307765661
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [optimization theory](/source/optimization_(mathematics)), '''semi-infinite programming''' ('''SIP''') is an [optimization problem](/source/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 program](/source/bilevel_program)s 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.

==See also==
* [Optimization](/source/optimization_(mathematics))
* [Generalized semi-infinite programming (GSIP)](/source/Generalized_semi-infinite_programming)

==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.&nbsp;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

{{Mathapplied-stub}}

---
Adapted from the Wikipedia article [Semi-infinite programming](https://en.wikipedia.org/wiki/Semi-infinite_programming) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Semi-infinite_programming?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.