A proximal average for prox-bounded functions

TL;DR

This paper introduces a new proximal average for prox-bounded functions that generalizes classical convex cases, ensuring continuity, differentiability, and providing new insights into proximal mappings and their properties.

Contribution

It develops a novel proximal average for prox-bounded functions that extends classical convex analysis and explores its properties and implications.

Findings

The proximal average transforms continuously in epi-topology.

When one function is differentiable, the average is differentiable.

Convex combination of convexified proximal mappings yields a proximal mapping.

Abstract

In this work, we construct a proximal average for two prox-bounded functions, which recovers the classical proximal average for two convex functions. The new proximal average transforms continuously in epi-topology from one proximal hull to the other. When one of the functions is differentiable, the new proximal average is differentiable. We give characterizations for Lipschitz and single-valued proximal mappings and we show that the convex combination of convexified proximal mappings is always a proximal mapping. Subdifferentiability and behaviors of infimal values and minimizers are also studied.