
Journal of Convex Analysis 17 (2010), No. 1, 203210 Copyright Heldermann Verlag 2010 A Calculus of ProxReguarity René A. Poliquin Dept. of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada rene.poliquin@ualberta.ca Ralph Tyrell Rockafellar Dept. of Mathematics, University of Washington, Seattle, WA 981954350, U.S.A. rtr@math.washington.edu We show that the operations of composition and addition, under appropriate conditions, preserve proxregularity. The class of proxregular functions covers all l.s.c., proper, convex functions, lowerC^{2} 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 (lowerC^{2}) functions e.g. the Moreauenvelopes are C^{1+}, a localization of the subgradient mapping is hypomonotone, etc... In this paper, we add to this list of convexlike properties by showing, under suitable conditions, that locally the subdifferential of the sum of proxregular functions is equal to the sum of subdifferentials. Keywords: Proxregular, proximal mapping, Moreau envelope, convex, convexly composite, pln, strongly amenable, subgradients, amenable functions. MSC: 49A52, 58C06, 58C20; 90C30 [ Fulltextpdf (116 KB)] for subscribers only. 