The Ekeland variational principle

In mathematical analysis, Ekeland’s variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems. Ekeland’s variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland’s principle relies on the completeness of the metric space. Ekeland’s principle leads to a quick proof of the Caristi fixed point theorem.

Theorem (Ekeland’s variational principle). Let be a complete metric space, and let be a lower semicontinuous functional on that is bounded below and not identically equal to . Fix and a point such that

Then there exists a point such that

1. ,
2. ,
3. and for all , .

Source: Wiki.