Ekeland's variational principle

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 Π$1 1$CA0 over RCA0, i.e. relatively strong.

It also leads to a quick proof of the Caristi fixed point theorem.

History
Ekeland was associated with the Paris Dauphine University when he proposed this theorem.

Preliminary definitions
A function $$f : X \to \R \cup \{-\infty, +\infty\}$$ valued in the extended real numbers $$\R \cup \{-\infty, +\infty\} = [-\infty, +\infty]$$ is said to be ' if $$\inf_{} f(X) = \inf_{x \in X} f(x) > -\infty$$ and it is called ' if it has a non-empty , which by definition is the set $$\operatorname{dom} f ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \{x \in X : f(x) \neq +\infty\},$$ and it is never equal to $$-\infty.$$ In other words, a map is if is valued in $$\R \cup \{+\infty\}$$ and not identically $$+\infty.$$ The map $$f$$ is proper and bounded below if and only if $$-\infty < \inf_{} f(X) \neq +\infty,$$ or equivalently, if and only if $$\inf_{} f(X) \in \R.$$

A function $$f :X \to [-\infty, +\infty]$$ is  at a given $$x_0 \in X$$ if for every real $$y < f\left(x_0\right)$$ there exists a neighborhood $$U$$ of $$x_0$$ such that $$f(u) > y$$ for all $$u \in U.$$ A function is called if it is lower semicontinuous at every point of $$X,$$ which happens if and only if $$\{x \in X : ~f(x) > y\}$$ is an open set for every $$y \in \R,$$ or equivalently, if and only if all of its lower level sets $$\{x \in X : ~f(x) \leq y\}$$ are closed.

Statement of the theorem
$$

For example, if $$f$$ and $$(X, d)$$ are as in the theorem's statement and if $$x_0 \in X$$ happens to be a global minimum point of $$f,$$ then the vector $$v$$ from the theorem's conclusion is $$v := x_0.$$

Corollaries
The principle could be thought of as follows: For any point $$x_0$$ which nearly realizes the infimum, there exists another point $$v$$, which is at least as good as $$x_0$$, it is close to $$x_0$$ and the perturbed function, $$f(x)+\frac{\varepsilon}{\lambda} d(v, x)$$, has unique minimum at $$v$$. A good compromise is to take $$\lambda := \sqrt{\varepsilon}$$ in the preceding result.