Parametrically prox-regular functions

TL;DR

This paper explores parametrically prox-regular functions, extending the concept of prox-regularity with parameters, and analyzes their properties, representations, and roles in optimization stability.

Contribution

It introduces the concept of para-prox-regular functions, provides new representations, and analyzes their properties and applications in optimization.

Findings

Extended the characterization of prox-regular functions to parametric versions.

Provided new representations related to monotonicity of f-attentive epsilon-localization.

Analyzed parametrized sums and maxima of functions in the context of para-prox-regularity.

Abstract

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-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 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 epsilon-localization as has been 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.