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 Taylor & Francis
Résumé : International audience
We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate epsilon-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate epsilon-subdifferential.
ISSN: 0233-1934

hal-00694601
https://hal.archives-ouvertes.fr/hal-00694601
DOI : 10.1080/02331934.2011.645034

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 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 Springer Verlag
Résumé : 4 pages
International audience
We show that a point is solution of the Minty variational inequality of subdifferential type for a given function if and only if the function is increasing along rays starting from that point. This provides a characterization of the monotone polar of subdifferentials of lower semicontinuous functions: it is a common subset of their graphs which depends only on the function.
ISSN: 1862-4472

hal-00842670
https://hal.archives-ouvertes.fr/hal-00842670
ARXIV : 1307.1826
DOI : 10.1007/s11590-013-0693-7

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 Springer
Résumé : International audience
It is known that a locally Lipschitz function f is approximately convex if, and only if, its Clarke subdifferential ∂C f is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of functions and we discuss the case of arbitrary subdifferentials
Mathematical Programming B

hal-00694605
https://hal.archives-ouvertes.fr/hal-00694605

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 Institute of Electrical and Electronics Engineers
Résumé : International audience
We show that the slope introduced by DeGiorgi-Marino-Tosques is stable with respect to the variational convergence introduced by Wijsman. Applications to the stability of subdifferentials at critical points and to subdifferential sum rules are derived.
ISSN: 0018-9219
Las Vegas, United States

Droits : info:eu-repo/semantics/OpenAccess
hal-00699033
https://hal.archives-ouvertes.fr/hal-00699033
https://hal.archives-ouvertes.fr/hal-00699033/document
https://hal.archives-ouvertes.fr/hal-00699033/file/S_IEEE02.pdf

Éditeur(s) : HAL CCSD
Résumé : 6 pages
We provide a new proof that the subdifferential of a proper lower semicontinuous convex function on a Banach space is maximal monotone by adapting the pattern commonly used in the Hilbert setting. We then extend the arguments to show more precisely that subdifferentials of proper lower semicontinuous prox-bounded functions possess the Brondsted-Rockafellar property.
https://hal.archives-ouvertes.fr/hal-00838205

hal-00838205
https://hal.archives-ouvertes.fr/hal-00838205
ARXIV : 1306.5466

Éditeur(s) : HAL CCSD Heldermann
Résumé : International audience
Let f : X → R ∪ {+∞} be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdifferential ∂f is quasimonotone. As an immediate consequence, we get that f is convex if and only if ∂f is monotone.
Journal of Convex Analysis

hal-00699221
https://hal.archives-ouvertes.fr/hal-00699221

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

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

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 Taylor & Francis
Résumé : International audience
Recently Auslender & Crouzeix introduced a well-behaved asymptotical notion for convex functions. The aim of this work is to extend this notion to the saddle case
ISSN: 0163-0563

hal-00773195
https://hal.univ-antilles.fr/hal-00773195
DOI : 10.1080/01630569708816806

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