Generalized Bregman envelopes and proximity operators

TL;DR

This paper introduces generalized Bregman envelopes and proximity operators based on a family of distances induced by maximally monotone operators, extending classical Bregman concepts with new asymptotic and example-based insights.

Contribution

It defines generalized Bregman envelopes and proximity operators, extending classical Bregman theory to a broader family of distances with new asymptotic analysis and illustrative examples.

Findings

Established conditions for convexity, coercivity, and supercoercivity of generalized Bregman distances.

Extended classical Bregman envelopes and proximity operators to a generalized setting.

Provided examples demonstrating the importance of distance choice in the generalized framework.

Abstract

Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Mart\'inez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural "extreme" cases that highlight the importance of which generalized Bregman distance is chosen.