Éditeur(s) : HAL CCSD
Résumé : International audience
We consider a second-order differential system with Hessian-driven damping . An interesting property of this system is that, after introduction of an auxiliary variable y , it can be equivalently written as a first-order system in time and space. This allows us to extend the analysis to the case of a convex lower semicontinuous potential and so to introduce constraints in the model. When considering the indicator function of a closed convex set, the subdifferential operator takes account of the contact forces. In this setting, by playing with the geometrical damping parameter, we can describe nonelastic shock laws with restitution coefficient. Taking advantage of the infinite dimensional framework, we introduce a nonlinear hyperbolic PDE describing a damped oscillating system with obstacle. The system is dissipative; in the convex case each trajectory weakly converges to a minimizer of the global potential energy function. Exponential stabilization is obtained under strong convexity assumptions.
Differential Equations & Applications

hal-00803214
https://hal.archives-ouvertes.fr/hal-00803214
https://hal.archives-ouvertes.fr/hal-00803214/document
https://hal.archives-ouvertes.fr/hal-00803214/file/Attouch-Mainge-Redont-30August2011.pdf

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 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
We show that the properties of dense subdifferentiability and of trustworthiness are equivalent for any subdifferential satisfying a small set of natural axioms. The proof relies on a remarkable property of the subdifferential of the inf-convolution of two (non necessarily convex) functions. We also show the equivalence of the dense subdifferentiability property with other basic properties of subdifferentials such as a weak∗ Lipschitz Separation property, a strong Compact Separation property and a Minimal property for the analytic approximate subdifferential of Ioffe.
Serdica. Mathematical Journal

hal-00694603
https://hal.archives-ouvertes.fr/hal-00694603

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 Society for Industrial and Applied Mathematics
Résumé : International audience
We establish a general Fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a Banach space. Our basic tool is a constrained variational principle derived from the "smooth" variational principle of Borwein and Preiss. Specializing the Fermat rule to the case when the convex set is a "drop," we obtain a multidirectional Rolle-type inequality from which, in turn, we deduce a multidirectional mean value inequality, in the line of Clarke and Ledyaev. We follow the abstract approach of our previous paper [Trans. Amer. Math. Soc., 347 (1995), pp. 4147-4161], thus covering all standard situations met in applications, while stressing the links between the results and the few key properties that are needed.
ISSN: 1052-6234

hal-00699204
https://hal.archives-ouvertes.fr/hal-00699204

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Given a pair of Banach spaces X and Y such that one is the dual of the other, we study the relationships between generic Fréchet differentiability of convex continuous functions on Y (Asplund property), generic existence of linear perturbations for lower semicontinuous functions on X to have a strong minimum (Stegall variational principle), and dentability of bounded subsets of X (Radon-Nikodým Property).
ISSN: 0927-6947

hal-00694604
https://hal.archives-ouvertes.fr/hal-00694604

Éditeur(s) : HAL CCSD
Résumé : The convex feasibility problem (CFP) is to find a member of the intersection of finitely many closed convex sets in Euclidean space. When the intersection is empty, one can minimize a proximity function to obtain an approximate solution to the problem. The split feasibility problem (SFP) and the split equality problem (SEP) are generalizations of the CFP. The approximate SFP (ASFP) and approximate SEP (ASEP) involve finding only approximate solutions to the SFP and SEP, respectively. We present here the SSEA, a simultaneous iterative algorithm for solving the ASEP. When this algorithm is applied to the ASFP it resembles closely, but is not equivalent to, the CQ algorithm. The SSEA involves orthogonal projection onto the given closed convex sets. The relaxed SSEA (RSSEA) is an easily implementable variant of the SSEA that uses orthogonal projection onto half-spaces at each step to solve the SEP. The perturbed version of the SSEA (PSSEA) is similar to the RSSEA, but uses orthogonal projection onto a sequence of epi-convergent closed convex sets.
https://hal.univ-antilles.fr/hal-00776640

Droits : info:eu-repo/semantics/OpenAccess
hal-00776640
https://hal.univ-antilles.fr/hal-00776640
https://hal.univ-antilles.fr/hal-00776640/document
https://hal.univ-antilles.fr/hal-00776640/file/CBAM10.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
It is an admitted fact that mainstream boosting algorithms like AdaBoost do not perform well to estimate class conditional probabilities. In this paper, we analyze, in the light of this problem, a recent algorithm, unn, which leverages nearest neighbors while minimizing a convex loss. Our contribution is threefold. First, we show that there exists a subclass of surrogate losses, elsewhere called balanced, whose minimization brings simple and statistically efficient estimators for Bayes posteriors. Second, we show explicit convergence rates towards these estimators for unn, for any such surrogate loss, under a Weak Learning Assumption which parallels that of classical boosting results. Third and last, we provide experiments and comparisons on synthetic and real datasets, including the challenging SUN computer vision database. Results clearly display that boosting nearest neighbors may provide highly accurate estimators, sometimes more than a hundred times more accurate than those of other contenders like support vector machines.
European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML PKDD)
Bristol, United Kingdom

hal-00702771
https://hal.archives-ouvertes.fr/hal-00702771
https://hal.archives-ouvertes.fr/hal-00702771/document
https://hal.archives-ouvertes.fr/hal-00702771/file/ecml12-dnbnb-sub.pdf

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 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 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 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