# Ekeland variational principle

There are usually three ways for getting existence results in analysis, namely compactness, Hahn–Banach-type results and completeness properties (cf. Compactness; Hahn–Banach theorem; Completeness (in topology)). The Ekeland variational principle [a10] (which provides a characterization of complete metric spaces [a14], cf. also Complete metric space) illustrates the third method in the framework of optimization. Let $f$ be a lower semi-continuous function defined on a complete metric space $( X,d )$, with values in the extended line $\mathbf R \cup \{ + \infty \}$, and bounded from below. It is well known that the lower bound of $f$ over $X$ need not be attained. Ekeland's basic principle asserts that there exists a slight perturbation of $f$ which attains its minimum on $X$. More precisely, there exists a point $a \in X$ such that $f ( a ) < f ( x ) + d ( a,x )$ for all $x \in X \setminus \{ a \}$; this says that the function $f ( \cdot ) + d ( a, \cdot )$ has a strict minimum on $X$ at $a$. It is interesting to observe that the conclusion of the basic principle is equivalent to the existence of a maximal element in the epigraph ${ \mathop{\rm epi} } f = \{ {( x,r ) \in X \times \mathbf R } : {f ( x ) \leq r } \}$ for the order defined on $X \times \mathbf R$ by $( x _ {1} ,r _ {1} ) \cle ( x _ {2} ,r _ {2} )$ if and only if $r _ {2} - r _ {1} + d ( x _ {2} ,x _ {1} ) \leq 0$[a3]. Figure: e110030a

From this basic principle one can deduce some variants which are in fact equivalent to the basic statement. The first one is as follows: given $\epsilon > 0$, $x _ {0} \in X$ such that $f ( x _ {0} ) < \inf _ {x \in X } f ( x ) + \epsilon$ and applying the basic principle to the complete metric space ${\widetilde{X} } = \{ {z \in X } : {f ( z ) + d ( x _ {0} ,z ) \leq f ( x _ {0} ) } \}$, one obtains the existence of a point $a \in X$ such that $f ( a ) + d ( a,x _ {0} ) \leq f ( x _ {0} )$ and $f ( a ) < f ( x ) + d ( a,x )$ for all $x \in X \setminus \{ a \}$. In particular, this implies that $| {f ( a ) - f ( x _ {0} ) } | \leq \epsilon$. Applying the previous result with the metric $\gamma d$, $\gamma > 0$, yields the second variant: there exists an $a \in X$ such that

$$d ( a,x _ {0} ) \leq \gamma ^ {- 1 } \epsilon,$$

$$\left | {f ( a ) - f ( x _ {0} ) } \right | \leq \epsilon,$$

$$f ( a ) < f ( x ) + \gamma d ( a,x ) \textrm{ for all } x \in X \setminus \{ a \} .$$

This variational principle has several equivalent geometric formulations. For instance, the Phelps extremization principle and the Drop theorem [a7], [a12] (see [a13] for the versions as stated here). Let $A$ be a closed subset of a Banach space $X$, let $x \in A$ and let $C$ be a closed convex bounded subset of $X$ such that $A \cap ( x + C ) = \emptyset$. Then there exist a $z \in A \cap ( x + [ 0,1 ] C )$ and a $\delta > 0$ such that $A \cap ( z + \left ] 0, \delta \right ] C ) = \emptyset$.

Among the great number of applications is the celebrated Bröndsted–Rockafellar theorem in convex analysis [a6]. Let $f$ be a closed convex function defined on a real Banach space $( X, \| \cdot \| )$ with values in $\mathbf R \cup \{ + \infty \}$( cf. also Convex function (of a real variable)). Let $x _ {0} \in { \mathop{\rm dom} } f = \{ {x \in X } : {f ( x ) < + \infty } \}$, $\epsilon > 0$, and let ${\mathcal l} _ {0} \in X ^ {*}$ be such that $f ( x ) \geq f ( x _ {0} ) + {\mathcal l} _ {0} ( x - x _ {0} ) - \epsilon$ for all $x \in X$. One can apply the third version of the theorem, with $\gamma = \sqrt \epsilon$, to the function $g = f - {\mathcal l} _ {0}$ when endowing $X$ with the equivalent norm $\| \cdot \| + | { {\mathcal l} ( \cdot ) } |$[a4]. This yields the existence of an $x _ \epsilon \in X$ and an

$${\mathcal l} _ \epsilon \in \partial f ( x _ \epsilon ) =$$

$$= \left \{ { {\mathcal l} \in X ^ {*} } : {f ( x ) \geq f ( x _ \epsilon ) + {\mathcal l} ( x - x _ \epsilon ) \textrm{ for all } x \in X } \right \}$$

such that

$$\left \| {x _ \epsilon - x _ {0} } \right \| \leq \sqrt \epsilon ,$$

$$\left | {f ( x _ \epsilon ) - f ( x _ {0} ) } \right | \leq 2 \epsilon + \sqrt \epsilon .$$

Hence the set

$${ \mathop{\rm dom} } \partial f = \left \{ {x \in { \mathop{\rm dom} } f } : {\partial f ( x ) \neq \emptyset } \right \}$$

is dense in ${ \mathop{\rm dom} } f$ for the epigraph topology, i.e. the supremum of the norm topology on $X$ and of the initial topology associated to $f$.

Another easy consequence of the Ekeland variational principle is a generalization to multi-functions of the Kirk–Caristi fixed-point theorem [a2].

Finally, it should be mentioned that analogous results hold in Banach spaces with the perturbation $d ( a, \cdot )$ replaced by some smooth one [a5].

How to Cite This Entry:
Ekeland variational principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ekeland_variational_principle&oldid=46797
This article was adapted from an original article by H. AttouchD. AzÃ© (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article