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In [[mathematical analysis]], '''Ekeland's variational principle''', discovered by [[Ivar Ekeland]],<ref name="Eke74">{{cite journal
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|    doi = 10.1016/0022-247X(74)90025-0
|    last = Ekeland
|    first = Ivar
|    title = On the variational principle
|  journal = J. Math. Anal. Appl.
|  volume = 47
|    year = 1974
|    pages = 324&ndash;353
|    issn = 0022-247X
}}</ref><ref>{{cite journal|last=Ekeland|first=Ivar|<!-- authorlink=Ivar Ekeland -->|title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|ref=harv}}</ref><ref>{{cite book|last1=Ekeland|first1=Ivar|last2=Temam|first2=Roger|authorlink=Roger Temam|title=Convex analysis and variational problems|edition=Corrected reprinting of the (1976) North-Holland|series=Classics in applied mathematics|volume=28 |publisher=Society for Industrial and Applied Mathematics&nbsp;(SIAM)|location=Philadelphia,&nbsp;PA|year=1999|pages=357–373|isbn=0-89871-450-8|mr=1727362|ref=harv}}</ref> is a theorem that asserts that there exists nearly optimal solutions to some [[optimization problem]]s.
 
Ekeland's variational principle can be used when the lower [[level set]] of a minimization problems is not [[compactness (topology)|compact]], so that the [[Bolzano&ndash;Weierstrass theorem]] can not be applied. Ekeland's principle relies on the [[completeness]] of the [[metric space]].<ref name="KG90"/>
 
Ekeland's principle leads to a quick proof of the [[Caristi fixed point theorem]].<ref name="KG90">{{cite book
| author    = Kirk, William A.; Goebel, Kazimierz
| title      = Topics in Metric Fixed Point Theory
| year      = 1990
| publisher  = Cambridge University Press
| isbn        = 0-521-38289-0
}}</ref><ref name="realanalysisok">{{cite book|last=Ok|first=Efe|title=Real Analysis with Economic Applications|publisher=Princeton University Press|year=2007|pages=664|chapter=D: Continuity I|isbn=978-0-691-11768-3|url=http://homepages.nyu.edu/~eo1/Book-PDF/Ekeland.pdf|accessdate=January 31, 2009}}</ref>
 
Ekeland was associated with the [[Paris Dauphine University]] when he proposed this theorem.<ref name="Eke74"/>
 
==Statement of the theorem==
 
Let (''X'',&nbsp;''d'') be a [[complete metric space]], and let ''F'':&nbsp;''X''&nbsp;→&nbsp;'''R'''&nbsp;∪&nbsp;{+∞} be a [[lower semicontinuous]] functional on ''X'' that is bounded below and not identically equal to +∞. Fix ''&epsilon;''&nbsp;&gt;&nbsp;0 and a point ''u''&nbsp;∈&nbsp;''X'' such that
:<math>F(u) \leq \varepsilon + \inf_{x \in X} F(x).</math>
Then there exists a point ''v''&nbsp;∈&nbsp;''X'' such that
:<math>F(v) \leq F(u),</math>
:<math>d(u, v) \leq 1,</math>
and, for all ''w''&nbsp;≠&nbsp;''v'',
:<math>F(w) > F(v) - \varepsilon d(v, w).</math>
 
== References ==
<references />
 
==Further reading==
* {{cite journal|last=Ekeland|first=Ivar|authorlink=Ivar Ekeland|title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|ref=harv}}
*{{cite book
| author    = Kirk, William A.; Goebel, Kazimierz
| title      = Topics in Metric Fixed Point Theory
| year      = 1990
| publisher  = Cambridge University Press
| isbn        = 0-521-38289-0
}}
 
[[Category:Variational analysis]]
[[Category:Convex analysis]]
[[Category:Calculus of variations]]
[[Category:Functional analysis]]
[[Category:Mathematical principles]]
[[Category:Variational principles]]

Latest revision as of 14:22, 8 September 2014

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