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 be a lower semi-continuous function defined on a complete metric space
, with values in the extended line
, and bounded from below. It is well known that the lower bound of
over
need not be attained. Ekeland's basic principle asserts that there exists a slight perturbation of
which attains its minimum on
. More precisely, there exists a point
such that
for all
; this says that the function
has a strict minimum on
at
. It is interesting to observe that the conclusion of the basic principle is equivalent to the existence of a maximal element in the epigraph
for the order defined on
by
if and only if
[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 ,
such that
and applying the basic principle to the complete metric space
, one obtains the existence of a point
such that
and
for all
. In particular, this implies that
. Applying the previous result with the metric
,
, yields the second variant: there exists an
such that
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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 be a closed subset of a Banach space
, let
and let
be a closed convex bounded subset of
such that
. Then there exist a
and a
such that
.
Among the great number of applications is the celebrated Bröndsted–Rockafellar theorem in convex analysis [a6]. Let be a closed convex function defined on a real Banach space
with values in
(cf. also Convex function (of a real variable)). Let
,
, and let
be such that
for all
. One can apply the third version of the theorem, with
, to the function
when endowing
with the equivalent norm
[a4]. This yields the existence of an
and an
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such that
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Hence the set
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is dense in for the epigraph topology, i.e. the supremum of the norm topology on
and of the initial topology associated to
.
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 replaced by some smooth one [a5].
References
[a1] | H. Attouch, H. Riahi, "Stability results for the Ekeland's variational principle and cone extremal solutions" Math. Oper. Res. , 18 (1993) pp. 173–201 |
[a2] | J.-P. Aubin, I. Ekeland, "Applied nonlinear analysis" , Wiley (1984) |
[a3] | E. Bishop, R.R. Phelps, "The support functional of a convex set" P. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 27–35 |
[a4] | J.M. Borwein, "A note on ![]() |
[a5] | J.M. Borwein, R. Preiss, "Smooth variational principle" Trans. Amer. Math. Soc. , 303 (1987) pp. 517–527 |
[a6] | A. Bröndsted, R.T. Rockafellar, "On the subdifferentiability of convex functions" Proc. Amer. Math. Soc. , 16 (1965) pp. 605–611 |
[a7] | F.H. Clarke, "Optimization and nonsmooth analysis" , Wiley (1983) |
[a8] | J. Daneš, "A geometric theorem useful in nonlinear functional analysis" Boll. Un. Mat. Ital. , 4 (1972) pp. 369–375 |
[a9] | D.G. de Figueiredo, "The Ekeland variational principle, tours and detours" , Lecture Notes Tata Inst. , Springer (1989) |
[a10] | I. Ekeland, "On the variational principle" J. Math. Anal. Appl. , 47 (1974) pp. 324–353 |
[a11] | I. Ekeland, "Nonconvex minimization problems" Bull. Amer. Math. Soc. (N.S.) , 1 (1979) pp. 443–474 |
[a12] | J.-P. Penot, "The drop theorem, the petal theorem and Ekeland's variational principle" Nonlinear Anal.: Theory, Methods, Appl. , 10 (1986) pp. 813–822 |
[a13] | J.S. Treiman, "Characterization of Clarke's tangent and normal cones in finite and infinite dimensions" Nonlinear Anal.: Theory, Methods, Appl. , 7 (1983) pp. 771–783 |
[a14] | J.D. Weston, "A characterization of metric completeness" Proc. Amer. Math. Soc. , 64 (1977) pp. 186–188 |
Ekeland variational principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ekeland_variational_principle&oldid=41306