# Regularizing with Bregman-Moreau envelopes

## TL;DR

This paper analyzes Bregman-Moreau envelopes, extending previous work by exploring both left and right variants and providing new asymptotic results, with applications in convex and nonconvex optimization.

## Contribution

It offers a comprehensive analysis of both left and right Bregman-Moreau envelopes, including new asymptotic properties, expanding the theoretical understanding of these regularization tools.

## Key findings

- Extended analysis of Bregman-Moreau envelopes for convex and nonconvex functions.
- Derived new asymptotic properties of the envelopes.
- Provided multiple illustrative examples.

## Abstract

Moreau's seminal paper, introducing what is now called the Moreau envelope and the proximity operator (also known as the proximal mapping), appeared in 1965. The Moreau envelope of a given convex function provides a regularized version which has additional desirable properties such as differentiability and full domain. Fifty years ago, Attouch proposed using the Moreau envelope for regularization. Since then, this branch of convex analysis has developed in many fruitful directions. In 1967, Bregman introduced what is nowadays known as the Bregman distance as a measure of discrepancy between two points generalizing the square of the Euclidean distance. Proximity operators based on the Bregman distance have become a topic of significant research as they are useful in the algorithmic solution of optimization problems. More recently, in 2012, Kan and Song studied regularization aspects of the left Bregman-Moreau envelope even for nonconvex functions. In this paper, we complement previous works by analyzing the left and right Bregman-Moreau envelopes and by providing additional asymptotic results. Several examples are provided.