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
Relying on prox-regularity we suggest and analyze a class of iterative schemes for solving nonconvex variational inequalities, our result can be considered as an improvement and a significant extension of previously known results in this field.
ISSN: 0096-3003

hal-00780630
https://hal.univ-antilles.fr/hal-00780630
DOI : 10.1016/j.amc.2003.06.003

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The aim of this paper is twofold. First, it is to extend the sensitivity analysis framework, developed recently for variational inequalities, to mixed equilibrium problems. The second is to propose iterative methods for solving this kind of problems. In the process, we establish an equivalence between an extended version of Wiener-Hopf equations and the given problems relying on a generalization of the Yosida approximation notion. Our results generalize results obtained for optimization, variational inequalities, complementarity problems, and problems of Nash equilibria.
ISSN: 0898-1221

hal-00779234
https://hal.univ-antilles.fr/hal-00779234
DOI : 10.1016/S0898-1221(02)00218-3

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a simple proximal method. Under mild conditions, the weak convergence of the sequences generated by the algorithm is obtained. Using this result we obtain corollaries which improve several corresponding results in this field.
ISSN: 0925-5001

hal-00776657
https://hal.univ-antilles.fr/hal-00776657
DOI : 10.1007/s10898-009-9476-1

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

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

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
This paper is concerned with the simultaneous null controllability with constraint on the control, for a system of coupled linear heat equations. First, we show that, by means of a change of variable, a system of two linear heat equations with a same control can be rewritten in the form of a system of linear heat equations with a control function acting only in one equation. Then, we establish an observability inequality, and we state an appropriate Carleman estimate adapted to the constraint that we use to prove the main result.
ISSN: 0096-3003

hal-01679125
https://hal.archives-ouvertes.fr/hal-01679125
DOI : 10.1016/j.amc.2012.12.022

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
Relying on the prox-regularity notion, we introduce and establish the convergence of two-step projection methods for a system of nonconvex variational inequality problems. We obtain as a particular case some known results in this field.
ISSN: 0362-546X

hal-00779983
https://hal.univ-antilles.fr/hal-00779983
DOI : 10.1016/j.na.2008.10.119

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Two splitting procedures for solving equilibrium problems involving the sum of two bifunctions are proposed and their convergence is established under mild assumptions.
Journal of Mathematical Analysis and applications

hal-00779982
https://hal.univ-antilles.fr/hal-00779982
DOI : 10.1016/j.jmaa.2009.06.005

Éditeur(s) : HAL CCSD Yokohama Publishers
Résumé : International audience
ISSN: 1348-9151

hal-00780002
https://hal.univ-antilles.fr/hal-00780002

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 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 Taylor & Francis
Résumé : International audience
We introduce the notions of conditioning and well-posedness for equilibrium problems. Using these concepts, we obtain finite and strong convergence results for the proximal method that improve, develop, and unify several theorems in optimization and nonlinear analysis.
ISSN: 0163-0563

hal-00779994
https://hal.univ-antilles.fr/hal-00779994
DOI : 10.1080/01630560701766684

Éditeur(s) : HAL CCSD BioMed Central
Résumé : International audience
INTRODUCTION: In order to study social health inequalities, contextual (or ecologic) data may constitute an appropriate alternative to individual socioeconomic characteristics. Indices can be used to summarize the multiple dimensions of the neighborhood socioeconomic status. This work proposes a statistical procedure to create a neighborhood socioeconomic index. METHODS: The study setting is composed of three French urban areas. Socioeconomic data at the census block scale come from the 1999 census. Successive principal components analyses are used to select variables and create the index. Both metropolitan area-specific and global indices are tested and compared. Socioeconomic categories are drawn with hierarchical clustering as a reference to determine "optimal" thresholds able to create categories along a one-dimensional index. RESULTS: Among the twenty variables finally selected in the index, 15 are common to the three metropolitan areas. The index explains at least 57% of the variance of these variables in each metropolitan area, with a contribution of more than 80% of the 15 common variables. CONCLUSIONS: The proposed procedure is statistically justified and robust. It can be applied to multiple geographical areas or socioeconomic variables and provides meaningful information to public health bodies. We highlight the importance of the classification method. We propose an R package in order to use this procedure.
ISSN: 1475-9276

inserm-00809310
http://www.hal.inserm.fr/inserm-00809310
http://www.hal.inserm.fr/inserm-00809310/document
http://www.hal.inserm.fr/inserm-00809310/file/1475-9276-12-21.pdf
DOI : 10.1186/1475-9276-12-21

Éditeur(s) : HAL CCSD BioMed Central
Résumé : International audience
Introduction In order to study social health inequalities, contextual (or ecologic) data may constitute an appropriate alternative to individual socioeconomic characteristics. Indices can be used to summarize the multiple dimensions of the neighborhood socioeconomic status. This work proposes a statistical procedure to create a neighborhood socioeconomic index. Methods The study setting is composed of three French urban areas. Socioeconomic data at the census block scale come from the 1999 census. Successive principal components analyses are used to select variables and create the index. Both metropolitan area-specific and global indices are tested and compared. Socioeconomic categories are drawn with hierarchical clustering as a reference to determine "optimal" thresholds able to create categories along a one-dimensional index. Results Among the twenty variables finally selected in the index, 15 are common to the three metropolitan areas. The index explains at least 57% of the variance of these variables in each metropolitan area, with a contribution of more than 80% of the 15 common variables. Conclusions The proposed procedure is statistically justified and robust. It can be applied to multiple geographical areas or socioeconomic variables and provides meaningful information to public health bodies. We highlight the importance of the classification method. We propose an R package in order to use this procedure.
International Journal for Equity in Health

hal-00854028
https://hal.archives-ouvertes.fr/hal-00854028
https://hal.archives-ouvertes.fr/hal-00854028/document
DOI : 10.1186/1475-9276-12-21