Generalized Hessian properties of regularized nonsmooth functions

We take up the question of second-order expansions for a class of functions of importance in optimization, namely, Moreau envelope regularizations of nonsmooth functions f. It is shown that when f is prox-regular, which includes convex functions and the extended real-valued functions representing problems of nonlinear programming, the many second-order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself. This clarifies the circumstances conducive to developing computational methods based on envelope functions, such as second-order approximations in nonsmooth optimization and variants of the proximal point algorithm. The results establish that generalized second-order expansions of Moreau envelopes, at least, can be counted on in most situations of interest in finite-dimensional optimization.