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.

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