Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The aim of this paper is to investigate the asymptotic behavior of an inertial alternating algorithm based on the composition of resolvents of monotone operators. The proposed algorithm is a generalization of those proposed in Attouch et al. (2007) and Bauschke et al. (2005). As a special case, we also recover the classical alternating minimization algorithm (Acker, 1980), which itself is a natural extension of the alternating projection algorithm of von Neumann (1950) [4]. An application to equilibrium problems is also proposed.
ISSN: 0893-9659

hal-00776658
https://hal.univ-antilles.fr/hal-00776658
DOI : 10.1016/j.aml.2010.01.023

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
A perturbed method for solving a new class of variational inclusions, is presented and a convergence result which includes, as a special case, some known results in this field, is given.
Journal of Mathematical Analysis and applications

hal-00816721
https://hal.univ-antilles.fr/hal-00816721
DOI : 10.1006/jmaa.1994.1277

Éditeur(s) : HAL CCSD
Résumé : International audience
The paper deals with iterative schemes for finding a zero of a maximal monotone operator. A combination of the auxiliary principle with the Tikhonov method is considered. Under some assumptions the sequence generated by such an iterative scheme is shown to converge to a solution of the problem that is required to satisfy an associated variational inequality
Lecture notes in economics and mathematical systems

hal-00773198
https://hal.univ-antilles.fr/hal-00773198

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
An approximation method which combines Tikhonov method with the proximal point algorithm, is presented. Conditions which guarantee the convergence to a particular element of the solution set, are provided. A particular attention is given to the convex and convex-concave optimization cases.
ISSN: 0233-1934

hal-00815513
https://hal.univ-antilles.fr/hal-00815513
DOI : 10.1080/02331939608844217

Éditeur(s) : HAL CCSD
Résumé : International audience
In this note we study the convergence of a descent-proximal method for finding critical points of a DC function. From a practical point of view, we propose an algorithmic pattern obtained by coupling the latter with cutting plane approximations.
Journal of Applied Functional Analysis

hal-00815499
https://hal.univ-antilles.fr/hal-00815499

Éditeur(s) : HAL CCSD Yokohama
Résumé : International audience
We introduce a convergent method for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a monotone equilibrium problem. Using the main convergence theorem we obtain results which improve, develop and unify several results in fixed point problems, variational inequalities and equilibrium problems.
Journal of Nonlinear and Convex Analysis

hal-00779989
https://hal.univ-antilles.fr/hal-00779989

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
ISSN: 0233-1934

hal-00816722
https://hal.univ-antilles.fr/hal-00816722
DOI : 10.1080/02331939408843968

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We present iterative methods for finding the critical points and/or the minima of extended real valued functions of the form $phi = psi+ g-h$, where $psi$ is a differentiable function and g and h are convex, proper, and lower semicontinuous. The underlying idea relies upon the discretization of a first order dissipative dynamical system which allows us to preserve the local feature and to obtain some convergence results. The main theorems not only recover known convergence results in this field but also provide a theoretical basis for the development of new iterative methods.
ISSN: 1052-6234

hal-00779987
https://hal.univ-antilles.fr/hal-00779987
DOI : 10.1137/060655183

Éditeur(s) : HAL CCSD
Résumé : International audience
Abstract In this paper we prove the convergence of the approximate proximal method for DC functions proposed by Sun et al [6]. Our analysis also permits to treat the exact method. We then propose an interesting result in the case where the second component of the DC function is differentiable and provide some computational experiences which proved the efficiency of our method.
Journal of Computational Mathematics,

hal-00780885
https://hal.univ-antilles.fr/hal-00780885

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Based on the very recent work by Censor-Gibali-Reich (http://arxiv.org/abs/1009.3780), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel'skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.
ISSN: 0022-3239

hal-00776651
https://hal.univ-antilles.fr/hal-00776651
DOI : 10.1007/s10957-011-9814-6

Éditeur(s) : HAL CCSD SpringerOpen
Résumé : International audience
We introduce methods which seem to be a new and promising tool in hierarchical fixed-point problems. The goal of this note is to analyze the convergence properties of these new types of approximating methods for fixed-point problems. The limit attained by these curves is the solution of the general variational inequality,, where denotes the normal cone to the set of fixed point of the original nonexpansive mapping and a suitable nonexpansive mapping criterion. The link with other approximation schemes in this field is also made.
ISSN: 1687-1820

hal-00780237
https://hal.univ-antilles.fr/hal-00780237
https://hal.univ-antilles.fr/hal-00780237/document
https://hal.univ-antilles.fr/hal-00780237/file/1687-1812-2006-95453.pdf
DOI : 10.1155/FPTA/2006/95453

Éditeur(s) : HAL CCSD
Résumé : International audience
The main purpose of this note is to propose viscosity approximation methods which amount to selecting a particular fixed-point of a given nonexpansive self mapping in a general Hilbert space. The connection with the selection principles of Attouch, in the context of convex minimization and monotone inclusion problems, is made and an application to a semi-coercive elliptic problem is then stated. Key words: nonexpansive mappings, fixed-points, monotone operators, iterative methods, convex optimization, selection, penalty ...
Lecture notes in economics and mathematical systems

hal-00776623
https://hal.univ-antilles.fr/hal-00776623

Éditeur(s) : HAL CCSD IOP Publishing
Résumé : International audience
Based on the very recent work by Censor and Segal (2009 J. Convex Anal. 16 587-600) and inspired by Xu (2006 Inverse Problems 22 2021-34) and Yang (2004 Inverse Problems 20 1261-6), we investigate an algorithm for solving the split common fixed-point problem for the class of demicontractive operators in a Hilbert space. Our results improve and/or develop previously discussed feasibility problems and related algorithms. It is worth mentioning that the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems, for instance, in sensor networks, in radiation therapy treatment planning, in computerized tomography and data compression, see Censor et al (2006 Phys. Med. Biol. 51 2353-65) and Combettes (1996 Adv. Imaging Electron. Phys. 95 155-270) and references therein.
ISSN: 0266-5611

hal-00778174
https://hal.univ-antilles.fr/hal-00778174
DOI : 10.1088/0266-5611/26/5/055007

Éditeur(s) : HAL CCSD
Résumé : International audience
Panamerican Mathematical Journal

hal-00814853
https://hal.univ-antilles.fr/hal-00814853

Éditeur(s) : HAL CCSD Yokohama
Résumé : International audience
We make use of the viscosity technique to develop a convergent method for approximating a common element of the set of fixed points of a demicontractive operator and the set of solutions of a monotone equilibrium problem. The proposed algorithm is obtained by coupling a modified hybrid steepest descent method with the extragradient algorithm. Under mild conditions, the strong convergence of the sequences generated by the algorithm is obtained. Using this result we obtain two corollaries which improve or develop several corresponding results in this field.
Journal of Nonlinear and Convex Analysis

hal-00779988
https://hal.univ-antilles.fr/hal-00779988

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We consider a variable Krasnosel'skii-Mann algorithm for approximating critical points of a prox-regular function or equivalently for finding fixed-points of its proximal mapping proxλf. The novelty of our approach is that the latter is not non-expansive any longer. We prove that the sequence generated by such algorithm (via the formula xk+1=(1−αk)xk+αkproxλkfxk, where (αk) is a sequence in (0,1)), is an approximate fixed-point of the proximal mapping and converges provided that the function under consideration satisfies a local metric regularity condition.
ISSN: 0362-546X

hal-00778175
https://hal.univ-antilles.fr/hal-00778175
DOI : 10.1016/j.na.2009.07.011

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

hal-00780893
https://hal.univ-antilles.fr/hal-00780893

Éditeur(s) : HAL CCSD School of Communications and Informatics
Résumé : International audience
An approximate procedure for solving equilibrium problems is proposed and its convergence is established under natural conditions. The result obtained in this paper includes, as a special case, some known results in convex minimization and monotone inclusion fields.
ISSN: 1443-5756

hal-00780881
https://hal.univ-antilles.fr/hal-00780881

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 School of Communications and Informatics
Résumé : International audience
The hybrid proximal point algorithm introduced by Solodov and Svaiter allowing significant relaxation of the tolerance requirements imposed on the solution of proximal subproblems will be combined with the inertial method introduced by Alvarez and Attouch which incorporates second order information to achieve faster convergence. The weak convergence of the resulting method will be investigated for finding zeroes of a maximal monotone operator in a Hilbert space.
ISSN: 1443-5756

hal-00780745
https://hal.univ-antilles.fr/hal-00780745
https://hal.univ-antilles.fr/hal-00780745/document
https://hal.univ-antilles.fr/hal-00780745/file/170_03_www.pdf