Parametrically Prox-Regular Functions

Prox-regularity is a generalization of convexity that includes all C^(2), lower-C^(2), 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-prox-regular) 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 R^(n). 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 ε-localization as was done for prox-regular functions by R. A. Poliquin and R. T. Rockafellar ["Prox-regular functions in variational analysis", Trans. Amer. Math. Soc. 348(5) (1996) 1805--1838]. This extends a result of A. B. Levy ["Calm minima in parameterized finite-dimensional optimization", SIAM J. Optim. 11(1) (2000) 160--178 (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 prox-regularity", J. Convex Analyis 17(1) (2010) 203--210] is given, and a relaxation of its necessary conditions is presented