Bregman three-operator splitting methods

TL;DR

This paper introduces Bregman three-operator splitting methods for convex optimization, extending existing primal-dual algorithms with Bregman distances, providing a unified convergence framework, and including a line search for stepsize selection.

Contribution

It develops Bregman extensions of primal-dual splitting algorithms, unifies their convergence analysis, and proposes a new line search method for improved stepsize selection.

Findings

Unified convergence analysis for Bregman extensions

Introduction of a line search procedure for stepsize selection

Convergence guarantees for the proposed Bregman PD3O method

Abstract

The paper presents primal-dual proximal splitting methods for convex optimization, in which generalized Bregman distances are used to define the primal and dual proximal update steps. The methods extend the primal and dual Condat-Vu algorithms and the primal-dual three-operator (PD3O) algorithm. The Bregman extensions of the Condat-Vu algorithms are derived from the Bregman proximal point method applied to a monotone inclusion problem. Based on this interpretation, a unified framework for the convergence analysis of the two methods is presented. We also introduce a line search procedure for stepsize selection in the Bregman dual Condat-Vu algorithm applied to equality-constrained problems. Finally, we propose a Bregman extension of PD3O and analyze its convergence.