
Journal of Convex Analysis 21 (2014), No. 4, 901923 Copyright Heldermann Verlag 2014 Parametrically ProxRegular Functions Warren L. Hare University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna, BC, Canada warren.hare@ubc.ca Chayne Planiden University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna, BC, Canada chayneplaniden@hotmail.com Proxregularity is a generalization of convexity that includes all C^{2}, lowerC^{2}, strongly amenable, and primallowernice functions. The study of proxregular functions provides insight on a broad spectrum of important functions. Parametrically proxregular (paraproxregular) functions are a further extension of this family, produced by adding a parameter. Such functions have been shown to play a key role in understanding stability of minimizers in optimization problems. This document discusses paraproxregular functions in R^{n}. We begin with some basic examples of paraproxregular functions, and move on to the more complex examples of the convex and nonconvex proximal average. We develop an alternate representation of a paraproxregular function, related to the monotonicity of an fattentive εlocalization as was done for proxregular functions by R. A. Poliquin and R. T. Rockafellar ["Proxregular functions in variational analysis", Trans. Amer. Math. Soc. 348(5) (1996) 18051838]. This extends a result of A. B. Levy ["Calm minima in parameterized finitedimensional optimization", SIAM J. Optim. 11(1) (2000) 160178 (electronic)], who used an alternate approach to show one implication of the relationship (we provide a characterization). We analyze two common forms of parametrized functions that appear in optimization: finite parametrized sum of functions, and finite parametrized max of functions. The example of strongly amenable functions by R. A. Poliquin and R. T. Rockafellar ["A calculus of proxregularity", J. Convex Analyis 17(1) (2010) 203210] is given, and a relaxation of its necessary conditions is presented. [ Fulltextpdf (204 KB)] for subscribers only. 