Éditeur(s) : HAL CCSD Springer Verlag
Résumé : 4 pages
International audience
We show that a point is solution of the Minty variational inequality of subdifferential type for a given function if and only if the function is increasing along rays starting from that point. This provides a characterization of the monotone polar of subdifferentials of lower semicontinuous functions: it is a common subset of their graphs which depends only on the function.
ISSN: 1862-4472

hal-00842670
https://hal.archives-ouvertes.fr/hal-00842670
ARXIV : 1307.1826
DOI : 10.1007/s11590-013-0693-7

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
A multifunction Γ is called a Kakutani multifunction if there exist two nonempty convex sets X and Y , each in a Hausdorff topological vector space, such that Γ : X → Y is upper semi-continuous with nonempty compact convex values. We prove the following extension of the Kakutani fixed point theorem : Let Γ : X → X be a multi-function from a simplex X into itself ; if Γ can be factorized by an arbitrary finite number of Kakutani multifunctions, then Γ has a fixed point. The proof relies on a simplicial approximation technique and the Brouwer fixed point theorem. Extensions to infinite-dimensional spaces and applications to game theory are given.
Journal of Mathematical Analysis and applications

hal-00699225
https://hal.archives-ouvertes.fr/hal-00699225

Éditeur(s) : HAL CCSD
Résumé : 6 pages
We provide a new proof that the subdifferential of a proper lower semicontinuous convex function on a Banach space is maximal monotone by adapting the pattern commonly used in the Hilbert setting. We then extend the arguments to show more precisely that subdifferentials of proper lower semicontinuous prox-bounded functions possess the Brondsted-Rockafellar property.
https://hal.archives-ouvertes.fr/hal-00838205

hal-00838205
https://hal.archives-ouvertes.fr/hal-00838205
ARXIV : 1306.5466

Éditeur(s) : HAL CCSD
Résumé : International audience
CMS Conf. Proc.
Limoges, France

hal-00508320
https://hal.archives-ouvertes.fr/hal-00508320

Éditeur(s) : HAL CCSD American Mathematical Society
Résumé : International audience
We propose to study fragmentability of set-valued mappings not only for a given single mapping, but also for a sequence of mappings associated with the initial one. It turns out that this property underlies several variational principles, such as for example the Deville-Godefroy-Zizler variational principle and the Stegall variational principle, by providing a common path for proof.
ISSN: 0002-9939

hal-00698981
https://hal.archives-ouvertes.fr/hal-00698981

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We propose an extended forward-backward algorithm for approximating a zero of a maximal monotone operator which can be split as the extended sum of two maximal monotone operators. We establish the weak convergence in average of the sequence generated by the algorithm under assumptions similar to those used in classical forward-backward algorithms. This provides as a special case an algorithm for solving convex constrained minimization problems without qualification condition.
Journal of Mathematical Analysis and applications

hal-00761657
https://hal.archives-ouvertes.fr/hal-00761657
ARXIV : 1212.0523
DOI : 10.1016/j.jmaa.2013.02.022

Éditeur(s) : HAL CCSD INSTYTUT MATEMATYCZNY * POLSKA AKADEMIA NAUK
Résumé : International audience
Studia Mathematica

hal-00699224
https://hal.archives-ouvertes.fr/hal-00699224

Éditeur(s) : HAL CCSD
Résumé : International audience
Topological Methods in Nonlinear Analysis

hal-00699218
https://hal.archives-ouvertes.fr/hal-00699218

Éditeur(s) : HAL CCSD
Résumé : International audience
Topological Methods in Nonlinear Analysis

hal-00699219
https://hal.archives-ouvertes.fr/hal-00699219

Éditeur(s) : HAL CCSD American Mathematical Society
Résumé : International audience
We show that if n sets in a topological space are given so that all the sets are closed or all are open, and for each k ≤ n every k of the sets have a (k − 2)-connected union, then the n sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every n + 1 or fewer members of a finite family of closed sets in Rn have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.
ISSN: 0002-9939

hal-00699217
https://hal.archives-ouvertes.fr/hal-00699217

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Nous montrons d'abord que les versions ''fermée'' et ''ouverte'' du Principe KKM se déduisent l'une de l'autre. Nous montrons ensuite que, comme la version ''fermée'' (dont les applications sont bien connues), la version ''ouverte'' possède quelques conséquences remarquables : par exemple, elle conduit directement au théorème de ''matching'' de Fan et permet une démonstration particulièrement simple du théorème de point fixe de Kakutani-Fan-Glicksberg.
ISSN: 0764-4442

hal-00699227
https://hal.archives-ouvertes.fr/hal-00699227

Éditeur(s) : HAL CCSD
Résumé : International audience
We start from a basic version of the Hahn-Banach theorem, of which we provide a proof based on Tychonoff's theorem on the product of compact intervals. Then, in the first section, we establish conditions ensuring the existence of affine functions lying between a convex function and a concave one in the setting of vector spaces -- this directly leads to the theorems of Hahn-Banach, Mazur-Orlicz and Fenchel. In the second section, we caracterize those topological vector spaces for which certain convex functions are continuous -- this is connected to the uniform boundedness theorem of Banach-Steinhaus and to the closed graph and open mapping theorems of Banach. Combining both types of results readily yields topological versions of the theorems of the first section.
Nonconvex Optim. Appl.
Erice, Italy

Droits : info:eu-repo/semantics/OpenAccess
hal-00699215
https://hal.archives-ouvertes.fr/hal-00699215
https://hal.archives-ouvertes.fr/hal-00699215/document
https://hal.archives-ouvertes.fr/hal-00699215/file/M_NonconvexOptimAppl98.pdf