mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Ekeland's variational principle, discovered by

Ivar Ekeland Ivar I. Ekeland (born 2 July 1944, Paris) is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well a ...

, is a theorem that asserts that there exist nearly optimal solutions to some

optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables ...

s. Ekeland's principle can be used when the lower

level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...

of a minimization problems is not

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, so that the Bolzano–Weierstrass theorem cannot be applied. The principle relies on the completeness of the

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

. The principle has been shown to be equivalent to completeness of metric spaces. In proof theory, it is equivalent to ΠCA₀ over RCA₀, i.e. relatively strong. It also leads to a quick proof of the

Caristi fixed point theorem In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ''ε''- va ...

History

Ekeland was associated with the

Paris Dauphine University Paris Dauphine University - PSL (french: Université Paris-Dauphine, also known as Paris Dauphine - PSL or Dauphine - PSL) is a public research university based in Paris, France. It is one of the 13 universities formed by the division of the ancie ...

when he proposed this theorem.

Ekeland's variational principle

Preliminary definitions

A function

f : X \to \R \cup \

valued in the

extended real numbers In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...

\R \cup \ = \infty, +\infty /math> is said to be  if \inf_ f(X) = \inf_ f(x) > -\infty and it is called  if it has a non-empty , which by definition is the set \operatorname f ~\stackrel~ \, and it is never equal to -\infty. In other words, a map is  if is valued in \R \cup \ 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 /math> 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 \ is an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

for every

y \in \R,

or equivalently, if and only if all of its lower

\

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

A good compromise is to take

\lambda := \sqrt

in the preceding result.

References

Bibliography

* * * * {{Functional analysis Convex analysis Theorems in functional analysis Variational analysis Variational principles