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 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 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 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 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

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 EDP Sciences
Résumé : International audience
In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order damped evolution equations, which are governed by the sum of the gradient operator of a convex differentiable potential function , and a maximal monotone operator which is assumed to be cocoercive. Under a sharp condition involving the viscous and the cocoercive parameters, it is proved that each trajectory weakly converges to an equilibrium. Passing from weak to strong convergence is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.
ISSN: 1292-8119

hal-00803774
https://hal.archives-ouvertes.fr/hal-00803774
https://hal.archives-ouvertes.fr/hal-00803774/document
https://hal.archives-ouvertes.fr/hal-00803774/file/ATTOUCH-MAINGE-09-04-2010-COCV.pdf
DOI : 10.1051/cocv/2010024

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 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
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 Bulgarian Academy of Sciences
Résumé : International audience
Using monotone bifunctions, we introduce a recession concept for general equilibrium problems relying on a variational convergence notion. The interesting purpose is to extend some results of P. L. Lions on variational problems. In the process we generalize some results by H. Brezis and H. Attouch relative to the convergence of the resolvents associated with maximal monotone operators.
Serdica Mathematical Journal

Droits : info:eu-repo/semantics/OpenAccess
hal-00776626
https://hal.univ-antilles.fr/hal-00776626
https://hal.univ-antilles.fr/hal-00776626/document
https://hal.univ-antilles.fr/hal-00776626/file/sjm-vol26-num3-2000-207p-220p.pdf

Éditeur(s) : HAL CCSD
Résumé : In this note we show that the splitting scheme of Passty [7] as well as the barycentric-proximal method of Lehdili & Lemaire [4] can be used to approximate a zero of the extended sum of maximal monotone operators. When the extended sum is maximal monotone, we extend the convergence result obtained by Lehdili & Lemaire for convex functions to the case of maximal monotone operators. Moreover, we recover the main convergence results by Passty and Lehdili & Lemaire when the pointwise sum of the involved operators is maximal monotone.
https://hal.univ-antilles.fr/hal-00783905

hal-00783905
https://hal.univ-antilles.fr/hal-00783905
https://hal.univ-antilles.fr/hal-00783905/document
https://hal.univ-antilles.fr/hal-00783905/file/10.1.1.165.7044.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
The author proposes a generalized proximal method for solving an equilibrium problem which consists in finding x 2 K such that F(x, y) 0, for all y 2 K, where K is a nonempty, convex and closed set of a real Hilbert space X, and F:K ×K !Ris a given bifunction with F(x, x) = 0 for all x 2 K. The weak convergence of the sequence generated by the method is proved under the assumptions of monotonicity and convexity, with respect to the second argument y (for every fixed x 2 K), of the bifunction F. Replacing the assumption of monotonicity with the one of strong monotonicity on F, a strong convergence result is obtained. A second strong convergence theorem is proved under the hypotheses of monotonicity and a new assumption of "co-Lipschitz continuity at 0" of the operator F. Applications to convex optimization, to the problem of finding a zero of a maximal monotone operator and to Nash equilibria problems are provided.
Journal of Natural Geometry

hal-00773194
https://hal.univ-antilles.fr/hal-00773194

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

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 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 Hindawi Publishing Corporation
Résumé : International audience
This paper is concerned with the study of a penalization-gradient algorithmfor solving variational inequalities, namely, find x ∈ C such that Ax, y − x ≥ 0 for all y ∈ C, where A : H → H is a single-valued operator, C is a closed convex set of a real Hilbert space H. Given Ψ : H → ∪ { ∞} which acts as a penalization function with respect to the constraint x ∈ C, and a penalization parameter βk, we consider an algorithm which alternates a proximal step with respect to ∂Ψ and a gradient step with respect to A and reads as xk I λkβk∂Ψ −1 xk−1 − λkAxk−1 . Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing themultivalued operator by its Yosida approximatewhich is always Lipschitz continuous.
ISSN: 0161-1712

hal-00776654
https://hal.univ-antilles.fr/hal-00776654
https://hal.univ-antilles.fr/hal-00776654/document
https://hal.univ-antilles.fr/hal-00776654/file/305856.pdf
DOI : 10.1155/2011/305856

Éditeur(s) : HAL CCSD
Résumé : International audience
In the paper the following mixed equilibrium problem in a Hilbert space is considered: find x 2 C such that F(x, x)+hG(x), x−xi 0, 8x 2 C, where C is a nonempty convex closed subset of a real Hilbert space. For its solution a version of the proximal approach is studied. The general proximal method always includes a quadratic term of the kind |x−xn|2, which provides a process of transition from a point xn to a point xn+1. The authors replace this quadratic term by its linearization. Thus, they obtain a method of the kind F(xn+1, x)+hG(xn)+ −1(h 0(xn+1)−h 0(xk)), x−xn+1i 0, 8x 2 C, where h0 is the derivative of a given function on C. It is proved that if the function F(x, y) is monotone and convex in y for any x, F(x, x) = 0, and the operator G(x) is co-coercive, then the method weakly converges in a real Hilbert space to an equilibrium solution. Note that this result will also be true if the co-coercive condition is replaced by a monotonicity property. The Tikhonov regularization method (in the absence of disturbances and calculating auxiliary solutions approximately) is justified and its weak convergence in a Hilbert space is proved. It is known that the proximal method can always be interpreted as an implicit discret approximation of some differential equation or inclusion. Therefore the authors consider a differential inclusion for the extreme mapping. Permitting "-approximation for the function F(x, y) by means of a parametric family F(x, y, ") and assuming the strong monotonicity of the operator @F(x, y, ") (it is a very strong condition), the authors prove the convergence of the trajectory of the differential inclusion to an equilibrium solution.
Ill-posed variational problems and regularization techniques

hal-00773204
https://hal.univ-antilles.fr/hal-00773204

Éditeur(s) : HAL CCSD Academic Publishing Ltd
Résumé : International audience
An approximate procedure for solving the problem of nding a zero of a maximal monotone operator is proposed and its convergence is established under various conditions. More precisely, it is shown that this method weakly converges under natural assumptions and strongly converges provided that either the inverse of the involved operator is Lipschitz continuous around zero or the interior of the solution set is nonempty. A particular attention is given to the convex minimization case. AMS Subject Classication : Primary, 90C ; Secondary, 49M45, 65C.
ISSN: 1311-8080

hal-00779255
https://hal.univ-antilles.fr/hal-00779255
https://hal.univ-antilles.fr/hal-00779255/document
https://hal.univ-antilles.fr/hal-00779255/file/10.1.1.13.4261_3_.pdf