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 exists 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

Preliminaries

Let be a function. Then,

  • .
  • f is proper if (i.e. if f is not identically ).
  • f is bounded below if .
  • given , say that f is lower semicontinuous at if for every there exists a neighborhood of such that for all in .
  • f is lower semicontinuous if it is lower semicontinuous at every point of X.
    • A function is lower semi-continuous if and only if is an open set for every ; alternatively, a function is lower semi-continuous if and only if all of its lower level sets are closed.

Statement of the theorem

Theorem (Ekeland):[7] Let be a complete metric space and a proper (i.e. not identically ) lower semicontinuous function that is bounded below. Pick and such that (or equivalently, ). There exists some such that

and for all ,

.

Proof of theorem

Define a function by

and note that G is lower semicontinuous (being the sum of the lower semicontinuous function f and the continuous function ). Given , define the functions and and define the set

.

It is straightforward to show that for all ,

  1. is closed (because is lower semicontinuous);
  2. if then ;
  3. if then ; in particular, ;
  4. if then .

Let , which is a real number since f was assumed to be bounded below. Pick such that . Having defined and , define and pick such that .

Observe the following:

  • for all , (because , where this now implies that ;
  • for all , because

It follows that for all , , thus showing that is a Cauchy sequence. Since X is a complete metric space, there exists some such that converges to v. Since for all , we have for all , where in particular, .

We will show that from which the conclusion of the theorem will follow. Let and note that since for all , we have as above that and note that this implies that converges to x. Since the limit of is unique, we must have . Thus , as desired. Q.E.D.

Corollaries

Corollary:[8] Let (X, d) be a complete metric space, and let f: X  R  {+∞} be a lower semicontinuous functional on X that is bounded below and not identically equal to +∞. Fix ε > 0 and a point   X such that

Then, for every λ > 0, there exists a point v  X such that

and, for all x  v,

Note that a good compromise is to take in the preceding result.[8]

References

  1. Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47: 324–353. doi:10.1016/0022-247X(74)90025-0. ISSN 0022-247X.
  2. 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.CS1 maint: ref=harv (link)
  3. Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. 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.CS1 maint: ref=harv (link)
  4. Kirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 0-521-38289-0.
  5. 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.
  6. 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.
  7. Zalinescu 2002, p. 29.
  8. Zalinescu 2002, p. 30.

Further reading

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