PARTIAL EXTENSIONS OF ATTOUCHS THEOREM WITH APPLICATIONS TO PROTO-DERIVATIVES OF SUBGRADIENT MAPPINGS

Attouch's Theorem, which gives on a reflexive Banach space the equival ence between the Mosco epi-convergence of a sequence of convex functio ns and the graph convergence of the associated sequence of subgradient s, has many important applications in convex optimization. In particul ar, generalized derivatives have been defined in terms of the epi-conv ergence or graph convergence of certain difference quotient mappings, and Attouch's Theorem has been used to relate these various generalize d derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of opt imization problems. We prove in a Hilbert space several ''partial exte nsions'' of Attouch's Theorem to functions more general than convex; t hese functions are called primal-lower-nice. Furthermore, we use our e xtensions to derive a relationship between the second-order epi-deriva tives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.