A Calculus of Prox-Reguarity

We show that the operations of composition and addition, under appropriate conditions, preserve prox-regularity. The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-C2 functions, strongly amenable functions (i.e. convexly composite functions), and pln functions, hence a large core of functions of interest in variational analysis and optimization. These functions, despite being in general nonconvex, possess many of the properties that one would expect only to find in convex or near convex (lower-C2) functions e.g. the Moreau-envelopes are C1+, a localization of the subgradient mapping is hypomonotone, etc... In this paper, we add to this list of convex-like properties by showing, under suitable conditions, that locally the subdifferential of the sum of prox-regular functions is equal to the sum of subdifferentials.