A Smooth Variational Principle With Applications to Subdifferentiability and to Differentiability of Convex Functions

Borwein, Jonathan M.; Preiss, D.

doi:10.2307/2000681

Cited by 73 publications

(108 citation statements)

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“…In a normed linear space X with uniformly Gateaux differentiable norm, Zajicek [12, p.300] has shown that the distance function generated by a non-empty closed set [3] A distance function property . 61…”

Section: (X+ty)-4>(x)mentioning

confidence: 99%

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A distance function property implying differentiability

Giles

1989

Bull. Austral. Math. Soc.

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In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gateaux (uniformly Frechet) differentiable norm, then d is Gateaux (Frechet) differentiable at x e X \ K if there exists an t f l , || "i* || = 1 such thatand is Gateaux (Frechet) differentiable on X \ K if there exists a set P+(K) dense in X\K where such a limit is approached uniformly for all x £ P^.(K). When X is complete this last property implies that K is convex.Given a real normed linear space X a non-empty closed subset K generates a distance function d on X where d{x) = inf{||x -T/ II : y 6 K}. A distance function d always satisfies the Lipschitz condition \d{x) -d{y)\ ^ \\x -y\\f o r a l l x,y € X and is a convex function if and only if if is a convex set.There is a natural duality between the properties of the set K and its distance function d. We see this duality exploited in abstract approximation theory and in techniques of non-smooth optimisation.A longstanding abstract approximation problem concerns Chebyshev sets. A subset K is said to be a Chebyshev set if for each x G X \ K there exists a unique p(x) € K such that <*(*) = H X -P O O H .The problem is to determine necessary and sufficient conditions for a Chebyshev set to be convex. The best result so far was given by Vlasov [10,11] who showed that in

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Section: (X+ty)-4>(x)mentioning

confidence: 99%

“…Recently, Borwein and Preiss [3] have established a significant extension of Ekeland's Variational Principle. An important application of their result is as follows.…”

Section: That Is the Mapping X -* F-g{y) (X -* F-g) Is Uniformly Comentioning

confidence: 99%

A distance function property implying differentiability

Giles

1989

Bull. Austral. Math. Soc.

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“…Ekeland's variational principle [12], [13] and its smooth generalizations: Borwein-Preiss' [3] and Deville-GodefroyZizler's variational principles [9], [10], [11], are now one of the main tools in nonlinear and non-smooth analysis. Various applications of the Borwein-Preiss smooth variational principle are presented, for instance, in [4], [5], [6], [17].…”

Section: Introductionmentioning

confidence: 99%

Parametric Borwein-Preiss variational principle and applications

Georgiev

2005

Proc. Amer. Math. Soc.

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Abstract. A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.The tool for proving this parametric smooth variational principle is a useful lemma about continuous ε-minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

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“…With this aim, we first show that the classical Ekeland's variational principle can be generalized to partial metric spaces. Ekeland's variational principle (in short, EVP) is one of the most applicable results of nonlinear analysis: it is used for problems from fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamic systems, etc; see, for example, ([1]- [2], [12], [13], [17]- [20], [23], [27], [33], [35], [47] …”

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confidence: 99%