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 ÃÂ CA<sub>0</sub> over RCA<sub>0</sub>, i.e. relatively strong.
It also leads to a quick proof of the Caristi fixed point theorem.
Ekeland was associated with the Paris Dauphine University when he proposed this theorem.
A function valued in the extended real numbers is said to be if and it is called if it has a non-empty , which by definition is the set
and it is never equal to In other words, a map is if is valued in and not identically The map is proper and bounded below if and only if or equivalently, if and only if
A function is at a given if for every real there exists a neighborhood of such that for all A function is called if it is lower semicontinuous at every point of which happens if and only if is an open set for every or equivalently, if and only if all of its lower level sets are closed.
For example, if and are as in the theorem's statement and if happens to be a global minimum point of then the vector from the theorem's conclusion is
The principle could be thought of as follows: For any point which nearly realizes the infimum, there exists another point , which is at least as good as , it is close to and the perturbed function, , has unique minimum at . A good compromise is to take in the preceding result.