Ekeland's variational principle

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.