Parametrically prox-regular functions
Prox-regularity is a generalization of convexity that includes all C2, lower-C2, strongly amenable, and primal-lower-nice functions. The study of prox-regular functions provides insight on a broad spectrum of important functions. Parametrically prox-regular (para-proxregular) 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 para-prox-regular functions in Rn. We begin with some basic examples of para-prox-regular functions, and move on to the more complex examples of the convex and nonconvex proximal average. We develop an alternate representation of a para-prox-regular function, related to the monotonicity of an f-attentive-localization as was done for prox-regular functions. This extends a result of Levy, 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 Poliquin and Rockafellar is given, and a relaxation of its necessary conditions is presented.