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Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings

Authors: A. B. Levy, R. Poliquin and L. Thibault
Journal: Trans. Amer. Math. Soc. 347 (1995), 1269-1294
MSC: Primary 49J52; Secondary 58C20
MathSciNet review: 1290725
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Abstract: Attouch's Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch's Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch's Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.

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Keywords: Attouch's Theorem, epi-derivatives, Mosco epi-convergence, primal lower-nice functions, Attouch-Wets convergence, graph convergence, nonsmooth analysis, sensitivity analysis, optimization, proto-derivatives, Painlevé-Kuratowski convergence, set-valued analysis
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