Ekeland's variational principle
In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland,[1][2][3] is a theorem that asserts that there exist nearly optimal solutions to some optimization problems.
Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem cannot be applied. Ekeland's principle relies on the completeness of the metric space.[4]
Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.[4][5]
Ekeland's principle has been shown to be equivalent to completeness of metric spaces.[6]
Ekeland was associated with the Paris Dauphine University when he proposed this theorem.[1]
Ekeland's variational principle[]
Preliminary definitions[]
Let be a function valued in the extended real numbers Then
- denotes the effective domain of
- is proper if (that is, if is not identically ).
- is bounded below if
- given say that is lower semicontinuous at if for every real there exists a neighborhood of such that for all in
- is lower semicontinuous if it is lower semicontinuous at every point of
- A function is lower semi-continuous if and only if is an open set for every ; alternatively, a function is lower semicontinuous if and only if all of its lower level sets are closed.
Statement of the theorem[]
Ekeland's variational principle[7] — Let be a complete metric space and a proper (that is, not identically ) lower semicontinuous function that is bounded below. Pick and such that (or equivalently, ). There exists some such that
Proof
|
---|
Define a function by
which is lower semicontinuous because it is the sum of the lower semicontinuous function and the continuous function
Given define the functions
and define the set
It may be verified that for all
Let which is a real number because was assumed to be bounded below. Pick such that Having defined and let and pick
such that
These sequences have the following properties:
It follows that for all
which proves that is a Cauchy sequence.
Because is a complete metric space, there exists some such that converges to
The fact that for all implies that for all where in particular,
The conclusion of the theorem will follow once it is shown that So let Because for all it follows as above that which implies that converges to Since the limit of is unique, it follows that Thus as desired. |
Corollaries[]
Corollary[8] — Corollary: Let be a complete metric space, and let be a lower semicontinuous functional on that is bounded below and not identically equal to Fix and a point such that
A good compromise is to take in the preceding result.[8]
References[]
- ^ a b Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47 (2): 324–353. doi:10.1016/0022-247X(74)90025-0. ISSN 0022-247X.
- ^ Ekeland, Ivar (1979). "Nonconvex minimization problems". Bulletin of the American Mathematical Society. New Series. 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. MR 0526967.
- ^ Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. Vol. 28 (Corrected reprinting of the (1976) North-Holland ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362.
- ^ a b Kirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 0-521-38289-0.
- ^ Ok, Efe (2007). "D: Continuity I". Real Analysis with Economic Applications (PDF). Princeton University Press. p. 664. ISBN 978-0-691-11768-3. Retrieved January 31, 2009.
- ^ Sullivan, Francis (October 1981). "A characterization of complete metric spaces". Proceedings of the American Mathematical Society. 83 (2): 345–346. doi:10.1090/S0002-9939-1981-0624927-9. MR 0624927.
- ^ Zalinescu 2002, p. 29.
- ^ a b Zalinescu 2002, p. 30.
Bibliography[]
- Ekeland, Ivar (1979). "Nonconvex minimization problems". Bulletin of the American Mathematical Society. New Series. 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. MR 0526967.
- Kirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 0-521-38289-0.
- Zalinescu, C (2002). Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. ISBN 981-238-067-1. OCLC 285163112.
- (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
- Convex analysis
- Theorems in functional analysis
- Variational principles