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

hal-00816761
https://hal.univ-antilles.fr/hal-00816761
DOI : 10.1016/0362-546X(93)90065-Z

Éditeur(s) : HAL CCSD
Résumé : In this work, we develop a variant of a bundle method in order to find a zero of a maximal monotone operator. This algorithm relies on two polyhedral approximations of the epsilon-enlargement of the considered operator, via a systematic use of the transportation formula. Moreover, the use of a double polyhedral approximation in our algorithm could inspire other bundle methods for the case where the given operator can be split as the sum of two other maximal monotone operators.
https://hal.archives-ouvertes.fr/hal-00828930

hal-00828930
https://hal.archives-ouvertes.fr/hal-00828930
ARXIV : 1305.5810

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
The aim of this note consists in studying the solvability of the following problem find xA such that yT(x) T is a maximal monotone operator and A a subspace of a real Hilbert space H.
ISSN: 0163-0563

hal-00779243
https://hal.univ-antilles.fr/hal-00779243
DOI : 10.1080/01630569508816642

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Relying on the Yosida approximate, we investigate the problem of finding zeroes for a difference of two maximal monotone operators in Hilbert spaces. These zeroes are compared to those of the corresponding regularized and dual problems. The behavior of the regularized operator is also studied and a splitting algorithm involving the resolvents of both operators is suggested via a fixed-point formulation of the regularized problem. A particular attention is given to the DC programming case.
ISSN: 0096-3003

hal-00779986
https://hal.univ-antilles.fr/hal-00779986
DOI : 10.1016/j.amc.2008.01.024

Éditeur(s) : HAL CCSD
Résumé : International audience
An iterative method for solving set-valued variational inequalities is considered and its convergence properties are studied under strong monotonicity and coercivity conditions. The results obtained in this paper include, as a special case, some known results in this field.
Mathematical Inequalities & Applications

hal-00776616
https://hal.univ-antilles.fr/hal-00776616

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a significant extension of previously known results in this field.
ISSN: 1385-1292

hal-00780762
https://hal.univ-antilles.fr/hal-00780762
https://hal.univ-antilles.fr/hal-00780762/document
https://hal.univ-antilles.fr/hal-00780762/file/10.1.1.13.5012.pdf
DOI : 10.1023/B:POST.0000023199.34934.9f

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 Elsevier
Résumé : International audience
The aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed-point of a given nonexpansive self-mapping.
Journal of Mathematical Analysis and applications

hal-00776600
https://hal.univ-antilles.fr/hal-00776600
DOI : 10.1006/jmaa.1999.6615

Éditeur(s) : HAL CCSD
Résumé : International audience
Relying on an idea of T. Pennanen, we use a localized version of maximal hypomonotonicity, and prove that it is enough to guarantee the local convergence of the Tikhonov regularization. In the process, we revisit this method which together with the help of the analysis developed by T. Pennanen allows us to obtain the desired result.
Mathematical Inequalities and Apllications

hal-00779258
https://hal.univ-antilles.fr/hal-00779258

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
A forward-backward inertial procedure for solving the problem of finding a zero of the sum of two maximal monotone operators is proposed and its convergence is established under a cocoercivity condition with respect to the solution set.
ISSN: 0377-0427

hal-00780778
https://hal.univ-antilles.fr/hal-00780778
https://hal.univ-antilles.fr/hal-00780778/document
https://hal.univ-antilles.fr/hal-00780778/file/10.1.1.12.4189.pdf
DOI : 10.1016/S0377-0427(02)00906-8

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a significant extension of previously known results in this field.
ISSN: 1385-1292

hal-00815511
https://hal.univ-antilles.fr/hal-00815511
DOI : 10.1023/B:POST.0000023199.34934.9f

Éditeur(s) : HAL CCSD Heldermann
Résumé : International audience
We present an iterative method for finding zeroes of maximal monotone operators in a real Hilbert space. The underlying idea relies upon the discretization of a first order dissipative dynamical system which allows us to preserve the local feature, as well as to obtain convergence results. The main theorems do not only recover known convergence results of standard and inertial proximal methods, but also provide a theoretical basis for the application of new iterative methods.
Journal of Convex Analysis

hal-00780210
https://hal.univ-antilles.fr/hal-00780210

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Based on the very recent work by Censor and Segal (2009), and inspired by Xu (2006), Zhao and Yang (2005) , and Bauschke and Combettes (2001), we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.
ISSN: 0362-546X

hal-00776645
https://hal.univ-antilles.fr/hal-00776645
DOI : 10.1016/j.na.2011.03.041

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
In this paper, using the penalty method in conjunction with graph-convergence, we study the existence of solutions for a class of generalized variational inequalities with variational problem constraints. Our results extend, improve and develop some known results in this field. Our method of proofs is very simple and does not use the hemicontinuity nor the pseudo-continuity properties of monotone operators.
ISSN: 1862-4472

hal-00776643
https://hal.univ-antilles.fr/hal-00776643
DOI : 10.1007/s11590-010-0271-1

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Identifying operators with their graphs, we study the continuity of the parallel sum of two monotone operators with respect to set convergence and we examine its stability from a quantitative point of view.
Journal of Mathematical Analysis and applications

hal-00816097
https://hal.univ-antilles.fr/hal-00816097
DOI : 10.1006/jmaa.1996.0155

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 Taylor & Francis
Résumé : International audience
This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems
ISSN: 0163-0563

hal-00776641
https://hal.univ-antilles.fr/hal-00776641
DOI : 10.1080/01630563.2011.619676

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

hal-00776631
https://hal.univ-antilles.fr/hal-00776631
DOI : 10.1016/S0362-546X(99)00136-4

Éditeur(s) : HAL CCSD
Résumé : International audience
In this note we show that the splitting scheme of Passty [13] as well as the barycentric-proximal method of Lehdili & Lemaire [8] can be used to approximate a zero of the extended sum of maximal monotone operators. When the extended sum is maximal monotone, we generalize a convergence result obtained by Lehdili & Lemaire for convex functions to the case of maximal monotone operators. Moreover, we recover the main convergence results of Passty and Lehdili & Lemaire when the pointwise sum of the involved operators in maximal monotone.
Studies in Computational Mathematics

hal-00778164
https://hal.univ-antilles.fr/hal-00778164
DOI : 10.1016/S1570-579X(01)80022-7