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
1CA0 over RCA0, i.e. relatively strong.

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

  1. ^ 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.