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 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 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
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 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 North Atlantic Science Publishing Company
Résumé : International audience
By means of the Yosida approximate of an accretive operator, we extended two recent results by Chidume and Chidume and Zegeye (2003) to set-valued operators, and we made the connection with two recent convergence results obtained by Benavides etal for a relaxed version of the so-called proximal point algorithm.
ISSN: 1048-9533

hal-00780243
https://hal.univ-antilles.fr/hal-00780243
https://hal.univ-antilles.fr/hal-00780243/document
https://hal.univ-antilles.fr/hal-00780243/file/056704.pdf
DOI : 10.1155/JAMSA/2006/56704

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 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 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

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