Ekeland's variational principle
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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]