Résumé. Il
Abstract. It is shown by Mentagui [ESAIM : COCV 9 (2003) 297-315] that, in the case of generalBanach spaces, the Attouch-Wets convergence is stable by a class of classical operations of convex analysis, when the limits satisfy some natural qualification conditions. This fails with the slice convergence. We establish here uniform qualification conditions ensuring the stability of the slice convergence under the same operations which play a basic role in convex optimization. We obtain as consequences, some key stability results of epi-convergence established by Mc Linden and Bergstrom [Trans. Amer. Math. Soc. 286 (1981) 127-142] in finite dimension. As an application, we give a model of convergence and stability for a wide class of problems in convex optimization and duality theory. The key ingredients in our methodology are, the horizon analysis, the notions of quasi-continuity and inf-local compactness of convex functions, and the bicontinuity of the Legendre-Fenchel transform relatively to the slice convergence.Classification Mathématique. 90C25, 90C31, 49K40, 46N10.Reçu le 22 janvier 2003.
Résumé. Soit X un espace de Banach de dual topologique X . C (X) (resp. C (X )) désigne l'ensemble des parties non vides convexes fermées de X (resp. w * -fermées de X ) muni de la topologie de la convergence uniforme sur les bornés des fonctions distances. Cette topologie se réduità celle de la métrique de Hausdorff sur les convexes fermés bornés [16] et admet en général une représentation en terme de cette dernière [11]. De plus, la métrique qui lui est associée s'est révélée très adéquate pour l'étude quantitative de la stabilité et l'approximation des solutions d'une large classe de problèmes en optimisation convexe [6][7][8][9]. Dans cet article, nous montrons que, sous des conditions de qualification naturelles, la stabilité de la convergence associéeà la topologie définie sur C (X) (resp. C (X )) est conservée par une classe de transformations linéaires. En identifiant ensuite toute fonction convexeà sonépigraphe et en se basant sur la version ensembliste de la stabilité, nous montrons que la convergence précitée est stable par certaines opérations de l'analyse convexe dont le rôle est fondamental en optimisation et en théorie de la dualité. L'hypothèse clé dans les conditions de qualification assurant la stabilité au niveau fonctionnel, est la notion d'inf-locale compacité d'une fonction convexe, introduite dans [28] et qui se traduit dans l'espace X par la quasi-continuité de sa conjuguée. Abstract. Let X be a Banach space and X its continuous dual. C (X) (resp. C (X )) denotes the set of nonempty convex closed subsets of X (resp. w * -closed subsets of X ) endowed with the topology of uniform convergence of distance functions on bounded sets. This topology reduces to the Hausdorff metric topology on the closed and bounded convex sets [16] and in general has a Hausdorff-like presentation [11]. Moreover, this topology is well suited for estimations and constructive approximations [6][7][8][9]. We prove here, that under natural qualification conditions, the stability of the convergence associated to the topology defined on C (X) (resp. C (X )) is preserved by a class of linear transformations. Building on these results, and by identifing each convex function with its epigraph, the stability at the functional level is acquired towards some operations of convex analysis which play a basic role in convex optimization and duality theory. The key hypothesis in the qualification conditions ensuring the functional stability is the notion of inf-local compactness of a convex function introduced in [28] and expressed in the space X by the quasi-continuity of its conjugate. Mots Clés. Fonction convexe, opérateur linéaire, convergence au sens d'Attouch-Wets, Mosco/épi-convergence, convergence uniforme sur les bornés, inf-(locale) compacité, quasi-continuité, cône (fonction) horizon, dualité, stabilité, approximation et optimisation.
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