Ekeland's variational principle explained

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.

The principle has been shown to be equivalent to completeness of metric spaces.^[4] In proof theory, it is equivalent to ΠCA₀ over RCA₀, i.e. relatively strong.

It also leads to a quick proof of the Caristi fixed point theorem.^{[5] [6]}

History

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

Ekeland's variational principle

Preliminary definitions

A function

f : X \to \R \cup \{ -inf ty, +inf ty\} valued in the extended real numbers \R \cup \{ -inf ty, +inf ty\} = [ -inf ty, +inf ty] is said to be if inf f(X) = inf _{x} f(x) > -inf ty and it is called if it has a non-empty , which by definition is the set $$\backslash operatorname\; f\; ~\backslash stackrel~\; \backslash ,$$and it is never equal to -inf ty. In other words, a map is if is valued in \R \cup \{ +inf ty\} and not identically +inf ty. The map f is proper and bounded below if and only if -inf ty < inf f(X) ≠ +inf ty, or equivalently, if and only if inf f(X) \in \R.

A function

f :X \to [ -inf ty, +inf ty] is at a given x _{0} \in X if for every real y < f\left(x _{0\right)} there exists a neighborhood U of x _{0} such that f(u) > y for all u \in U. A function is called if it is lower semicontinuous at every point of X, which happens if and only if \{x \in X : ~f(x) > y\} is an open set for every y \in \R, or equivalently, if and only if all of its lower level sets \{x \in X : ~f(x) \leq y\} are closed.

Statement of the theorem

For example, if

f and (X, d) are as in the theorem's statement and if x _{0} \in X happens to be a global minimum point of f, then the vector v from the theorem's conclusion is v : = x _{0.}

Corollaries

The principle could be thought of as follows: For any point

x _{0} which nearly realizes the infimum, there exists another point v , which is at least as good as x _{0} , it is close to x _{0} and the perturbed function, f(x) + (\varepsilon)/(λ) d(v, x) , has unique minimum at v . A good compromise is to take λ : = \sqrt{\varepsilon} in the preceding result.

Bibliography

• Ekeland. Ivar. Ivar Ekeland. Nonconvex minimization problems. Bulletin of the American Mathematical Society. New Series. 1. 1979. 3. 443–474. 10.1090/S0273-0979-1979-14595-6. 526967. free.
• Book: Kirk, William A.. Goebel, Kazimierz. Topics in Metric Fixed Point Theory. 1990. Cambridge University Press. 0-521-38289-0.
• Book: Zalinescu, C. Convex analysis in general vector spaces. World Scientific. River Edge, N.J. London. 2002. 981-238-067-1. 285163112.

Notes and References

1. 10.1016/0022-247X(74)90025-0. Ekeland. Ivar. Ivar Ekeland. On the variational principle. J. Math. Anal. Appl.. 47. 1974. 2. 324–353. 0022-247X. free.
2. Ekeland. Ivar. Nonconvex minimization problems. Bulletin of the American Mathematical Society. New Series. 1. 1979. 3. 443–474. 10.1090/S0273-0979-1979-14595-6. 526967. free.
3. Book: Ekeland. Ivar. Temam. Roger. Roger Temam. Convex analysis and variational problems. Corrected reprinting of the (1976) North-Holland. Classics in applied mathematics. 28 . Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA. 1999. 357–373. 0-89871-450-8. 1727362.
4. Sullivan. Francis. A characterization of complete metric spaces. Proceedings of the American Mathematical Society. 83. 2. October 1981. 345–346. 10.1090/S0002-9939-1981-0624927-9. 624927. free.
5. Book: Kirk. William A.. Kazimierz. Goebel. Topics in Metric Fixed Point Theory. 1990. Cambridge University Press. 0-521-38289-0.
6. Book: Ok, Efe. Real Analysis with Economic Applications. Princeton University Press. 2007. 664. D: Continuity I. 978-0-691-11768-3. January 31, 2009.