Proximal mappings and Moreau envelopes of single-variable convex piecewise cubic functions and multivariable gauge functions

TL;DR

This paper explores properties of the Moreau envelope for convex functions, focusing on gauge and piecewise cubic functions, providing characterizations related to convexity, Lipschitz continuity, and other properties.

Contribution

It introduces new characterizations of the Moreau envelope for gauge and piecewise cubic convex functions, expanding understanding of their properties.

Findings

Characterizations of convex Moreau envelopes established

Properties related to strict and strong convexity analyzed

Lipschitz continuity of envelopes discussed

Abstract

This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their Moreau envelopes categorized. Characterizations of convex Moreau envelopes are established; topics include strict convexity, strong convexity and Lipschitz continuity.