Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
A set-valued upper semi-continuous map is called an Rδ -map if each of its values is an Rδ -set (we recall that an Rδ -set is a space that can be represented as the intersection of a decreasing sequence of compact AR-spaces). We prove that a compact set-valued map of an AR-space into itself has a fixed point provided it can be factorized by an arbitrary finite number of Rδ -maps through ANR-spaces. This fact is a consequence of a more general result which is the main goal of this note. The proof relies on a refinement of the approximation technique and does not make use of homological tools.
ISSN: 0166-8641

hal-00699222
https://hal.archives-ouvertes.fr/hal-00699222

Éditeur(s) : HAL CCSD American Mathematical Society
Résumé : International audience
We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the ''smooth'' variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, cone-monotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.
ISSN: 0002-9947

hal-00699220
https://hal.archives-ouvertes.fr/hal-00699220

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex.
ISSN: 0925-5001

hal-00761658
https://hal.archives-ouvertes.fr/hal-00761658
ARXIV : 1212.0532
DOI : 10.1007/s10898-013-0078-6

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Journal of Mathematical Analysis and applications

hal-00699228
https://hal.archives-ouvertes.fr/hal-00699228

Éditeur(s) : HAL CCSD
Résumé : International audience
We show that the properties of dense subdifferentiability and of trustworthiness are equivalent for any subdifferential satisfying a small set of natural axioms. The proof relies on a remarkable property of the subdifferential of the inf-convolution of two (non necessarily convex) functions. We also show the equivalence of the dense subdifferentiability property with other basic properties of subdifferentials such as a weak∗ Lipschitz Separation property, a strong Compact Separation property and a Minimal property for the analytic approximate subdifferential of Ioffe.
Serdica. Mathematical Journal

hal-00694603
https://hal.archives-ouvertes.fr/hal-00694603

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
ISSN: 0362-546X

hal-00699030
https://hal.archives-ouvertes.fr/hal-00699030

Éditeur(s) : HAL CCSD
Résumé : International audience
J. Convex Anal.

hal-00695362
https://hal.archives-ouvertes.fr/hal-00695362

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Given a pair of Banach spaces X and Y such that one is the dual of the other, we study the relationships between generic Fréchet differentiability of convex continuous functions on Y (Asplund property), generic existence of linear perturbations for lower semicontinuous functions on X to have a strong minimum (Stegall variational principle), and dentability of bounded subsets of X (Radon-Nikodým Property).
ISSN: 0927-6947

hal-00694604
https://hal.archives-ouvertes.fr/hal-00694604

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate epsilon-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate epsilon-subdifferential.
ISSN: 0233-1934

hal-00694601
https://hal.archives-ouvertes.fr/hal-00694601
DOI : 10.1080/02331934.2011.645034

Éditeur(s) : HAL CCSD Heldermann
Résumé : International audience
We study extended notions for sums of monotone operators and for precompositions of monotone operators with continuous linear mappings. First, we establish some new properties related to the notion of extended sum recently proposed by Revalski and Théra, among them the monotonicity of this sum provided the operators involved are maximal monotone. Then, we show that the equivalence which exists between usual sums and compositions remains valid also for the extended operations. This allows us to obtain some known and new results for extended compositions via the corresponding properties of extended sums.
Journal of Convex Analysis

hal-00698807
https://hal.archives-ouvertes.fr/hal-00698807

Résumé : Cette thèse est consacrée à la résolution d'un problème fondamental de l'analyse variationnelle qu'est la recherchede zéros d'opérateurs maximaux monotones dans un espace de Hilbert. Nous nous sommes tout d'abord intéressés au cas de l'opérateur somme étendue de deux opérateurs maximaux monotones; la recherche d'un zéro de cet opérateur est un problème dont la bibliographie est peu fournie: nous proposons une version modifiée de l'algorithme d'éclatement forward-backward utilisant à chaque itération, l'epsilon-élargissement d'un opérateur maximal monotone,afin de construire une solution. Nous avons ensuite étudié la convergence d'un nouvel algorithme de faisceaux pour construire ID zéro d'un opérateur maximal monotone quelconque en dimension finie. Cet algorithme fait intervenir une double approximation polyédrale de l'epsilon-élargissement de l'opérateur considéré
This thesis is devoted to solving a basic problem of variational analysis which is the search of zeros of maximal monotone operators in a Hilbert space. First of aIl, we concentrate on the case of the extended som of two maximal monotone operators; the search of a zero of this operator is a problem for which the bibliography is not abondant: we purpose a modified version of the forward-backward splitting algorithm using at each iteration, the epsilon-enlargement of a maximal monotone operator, in order to construet a solution. Secondly, we study the convergence of a new bondie algorithm to construet a zero of an arbitrary maximal monotone operator in a finite dimensional space. In this algorithm, intervenes a double polyhedral approximation of the epsilon-enlargement of the considered operator

http://www.theses.fr/2012AGUY0548/document

Éditeur(s) : HAL CCSD
Résumé : International audience
Nous démontrons un résultat du type suivant : si toutes les applications continues qui satisfont une certaine condition au bord ont la propriété du point fixe, alors il en va de même des multi-applications de Kakutani qui satisfont une condition au bord du même type. Ce travail reprend et affine les idées contenues dans un article récent de l'auteur.
Pitman Res. Notes Math. Ser.
Marseille, France

Droits : info:eu-repo/semantics/OpenAccess
hal-00699226
https://hal.archives-ouvertes.fr/hal-00699226
https://hal.archives-ouvertes.fr/hal-00699226/document
https://hal.archives-ouvertes.fr/hal-00699226/file/D_Pitman91.pdf

Éditeur(s) : HAL CCSD Heldermann
Résumé : International audience
Let f : X → R ∪ {+∞} be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdifferential ∂f is quasimonotone. As an immediate consequence, we get that f is convex if and only if ∂f is monotone.
Journal of Convex Analysis

hal-00699221
https://hal.archives-ouvertes.fr/hal-00699221

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
ISSN: 0362-546X

hal-00699203
https://hal.archives-ouvertes.fr/hal-00699203

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.
ISSN: 0927-6947

hal-00698802
https://hal.archives-ouvertes.fr/hal-00698802

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
ISSN: 0927-6947

hal-00507830
https://hal.archives-ouvertes.fr/hal-00507830

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We establish a general Fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a Banach space. Our basic tool is a constrained variational principle derived from the "smooth" variational principle of Borwein and Preiss. Specializing the Fermat rule to the case when the convex set is a "drop," we obtain a multidirectional Rolle-type inequality from which, in turn, we deduce a multidirectional mean value inequality, in the line of Clarke and Ledyaev. We follow the abstract approach of our previous paper [Trans. Amer. Math. Soc., 347 (1995), pp. 4147-4161], thus covering all standard situations met in applications, while stressing the links between the results and the few key properties that are needed.
ISSN: 1052-6234

hal-00699204
https://hal.archives-ouvertes.fr/hal-00699204

Éditeur(s) : HAL CCSD Springer
Résumé : International audience
It is known that a locally Lipschitz function f is approximately convex if, and only if, its Clarke subdifferential ∂C f is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of functions and we discuss the case of arbitrary subdifferentials
Mathematical Programming B

hal-00694605
https://hal.archives-ouvertes.fr/hal-00694605

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We present generalizations of the KKM Principle and of the Kakutani-Fan-Glicksberg fixed point theorem adapted to the study of Nash equilibria in games without ordered preferences.
Journal of Mathematical Analysis and applications

hal-00699223
https://hal.archives-ouvertes.fr/hal-00699223

Éditeur(s) : HAL CCSD Institute of Electrical and Electronics Engineers
Résumé : International audience
We show that the slope introduced by DeGiorgi-Marino-Tosques is stable with respect to the variational convergence introduced by Wijsman. Applications to the stability of subdifferentials at critical points and to subdifferential sum rules are derived.
ISSN: 0018-9219
Las Vegas, United States

Droits : info:eu-repo/semantics/OpenAccess
hal-00699033
https://hal.archives-ouvertes.fr/hal-00699033
https://hal.archives-ouvertes.fr/hal-00699033/document
https://hal.archives-ouvertes.fr/hal-00699033/file/S_IEEE02.pdf