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 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. The principle relies on the completeness of the metric space.[4]

The principle has been shown to be equivalent to completeness of metric spaces.[5] In proof theory, it is equivalent to Π1
1
CA0 over RCA0
, i.e. relatively strong.

It also leads to a quick proof of the Caristi fixed point theorem.[4][6]

History

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Ekeland was associated with the Paris Dauphine University when he proposed this theorem.[1]

Ekeland's variational principle

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Preliminary definitions

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A function   valued in the extended real numbers   is said to be bounded below if   and it is called proper if it has a non-empty effective domain, which by definition is the set   and it is never equal to   In other words, a map is proper if is valued in   and not identically   The map   is proper and bounded below if and only if   or equivalently, if and only if  

A function   is lower semicontinuous at a given   if for every real   there exists a neighborhood   of   such that   for all   A function is called lower semicontinuous if it is lower semicontinuous at every point of   which happens if and only if   is an open set for every   or equivalently, if and only if all of its lower level sets   are closed.

Statement of the theorem

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Ekeland's variational principle[7] — Let   be a complete metric space and let   be a proper lower semicontinuous function that is bounded below (so  ). Pick   such that   (or equivalently,  ) and fix any real   There exists some   such that   and for every   other than   (that is,  ),  

Proof

Define a function   by   which is lower semicontinuous because it is the sum of the lower semicontinuous function   and the continuous function   Given   denote the functions with one coordinate fixed at   by     and define the set   which is not empty since   An element   satisfies the conclusion of this theorem if and only if   It remains to find such an element.

It may be verified that for every  

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

Let   which is a real number because   was assumed to be bounded below. Pick   such that   Having defined   and   let   and pick   such that   For any     guarantees that   and   which in turn implies   and thus also   So if   then   and   which guarantee  

It follows that for all positive integers     which proves that   is a Cauchy sequence. Because   is a complete metric space, there exists some   such that   converges to   For any   since   is a closed set that contain the sequence   it must also contain this sequence's limit, which is   thus   and in particular,  

The theorem will follow once it is shown that   So let   and it remains to show   Because   for all   it follows as above that   which implies that   converges to   Because   also converges to   and limits in metric spaces are unique,     Q.E.D.

For example, if   and   are as in the theorem's statement and if   happens to be a global minimum point of   then the vector   from the theorem's conclusion is  

Corollaries

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Corollary[8] — 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   Then, for every   there exists a point   such that     and, for all    

The principle could be thought of as follows: For any point   which nearly realizes the infimum, there exists another point  , which is at least as good as  , it is close to   and the perturbed function,  , has unique minimum at  . A good compromise is to take   in the preceding result.[8]

See also

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References

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  1. ^ 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.
  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.
  3. ^ 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.
  4. ^ a b Kirk, William A.; Goebel, Kazimierz (1990). Topics in Metric Fixed Point Theory. Cambridge University Press. ISBN 0-521-38289-0.
  5. ^ 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.
  6. ^ 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.
  7. ^ Zalinescu 2002, p. 29.
  8. ^ a b Zalinescu 2002, p. 30.

Bibliography

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