In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, 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. 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.

History

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

Ekeland's variational principle

Preliminary definitions

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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,, has unique minimum at v. A good compromise is to take in the preceding result.