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 Π11CA0 over RCA0, i.e. relatively strong.
It also leads to a quick proof of the Caristi fixed point theorem.
[4][6] Ekeland was associated with the Paris Dauphine University when he proposed this theorem.
valued in the extended real numbers
and it is called proper if it has a non-empty effective domain, which by definition is the set
In other words, a map is proper if is valued in
A function is called lower semicontinuous if it is lower semicontinuous at every point of
or equivalently, if and only if all of its lower level sets
Ekeland's variational principle[7] — Let
be a complete metric space and let
be a proper lower semicontinuous function that is bounded below (so
which is lower semicontinuous because it is the sum of the lower semicontinuous function
denote the functions with one coordinate fixed at
satisfies the conclusion of this theorem if and only if
It remains to find such an element.
It follows that for all positive integers
is a complete metric space, there exists some
is a closed set that contain the sequence
The theorem will follow once it is shown that
and limits in metric spaces are unique,
happens to be a global minimum point of
be a complete metric space, and let
be a lower semicontinuous functional on
that is bounded below and not identically equal to
The principle could be thought of as follows: For any point
which nearly realizes the infimum, there exists another point