Ekeland's 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 cannot 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.
Ekeland's principle has been shown to be equivalent to completeness of metric spaces.
Ekeland was associated with the Paris Dauphine University when he proposed this theorem.

Ekeland's variational principle

Preliminaries

Let be a function.
Then,
Theorem :
Let be a complete metric space and a proper lower semicontinuous function that is bounded below.
Pick and such that .
There exists some such that
and for all,

Proof of theorem

Define a function by
and note that G is lower semicontinuous.
Given, define the functions and and define the set
It is straightforward to show that for all,
  1. is closed ;
  2. if then ;
  3. if then ; in particular, ;
  4. if then.
Let, which is a real number since f was assumed to be bounded below.
Pick such that.
Having defined and, define and pick such that.
Observe the following:
It follows that for all,, thus showing that is a Cauchy sequence.
Since X is a complete metric space, there exists some such that converges to v.
Since for all, we have for all, where in particular,.
We will show that from which the conclusion of the theorem will follow.
Let and note that since for all, we have as above that and note that this implies that converges to x.
Since the limit of is unique, we must have.
Thus, as desired. Q.E.D.

Corollaries

Corollary:
Let be a complete metric space, and let f: X → R ∪ be a lower semicontinuous functional on X that is bounded below and not identically equal to +∞. Fix ε >; 0 and a point ∈ X such that
Then, for every λ > 0, there exists a point v ∈ X such that
and, for all x ≠ v,
Note that a good compromise is to take in the preceding result.