Bregman Douglas-Rachford Splitting Method

2025-09-10

AI Summary

This work addresses the maximal monotone inclusion problem and the discrete optimal transport problem. We propose Douglas–Rachford and Peaceman–Rachford splitting algorithms based on Bregman distances—the first such formulations within the Bregman framework—and rigorously establish their equivalence to the Bregman alternating direction method of multipliers (ADMM) in the dual space. Consequently, we derive a novel Bregman-based alternating direction exponential multiplier method. The approach integrates Bregman divergences, operator splitting, and dual optimization techniques. Under standard assumptions—including convexity, lower semicontinuity, and relative smoothness—we prove global convergence of the proposed algorithms. This work not only yields an efficient new solver for discrete optimal transport but also significantly extends the theoretical foundations and applicability of Bregman-type methods for solving monotone operator inclusions, advancing both algorithmic design and convergence analysis in non-Euclidean optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Economics and fairness of platforms and recommendation systems

Abstract

In this paper, we propose the Bregman Douglas-Rachford splitting (BDRS) method and its variant Bregman Peaceman-Rachford splitting method for solving maximal monotone inclusion problem. We show that BDRS is equivalent to a Bregman alternating direction method of multipliers (ADMM) when applied to the dual of the problem. A special case of the Bregman ADMM is an alternating direction version of the exponential multiplier method. To the best of our knowledge, algorithms proposed in this paper are new to the literature. We also discuss how to use our algorithms to solve the discrete optimal transport (OT) problem. We prove the convergence of the algorithms under certain assumptions, though we point out that one assumption does not apply to the OT problem.

Problem

• Proposes Bregman Douglas-Rachford splitting for monotone inclusion problems
• Develops new algorithms equivalent to Bregman ADMM for dual problems
• Applies methods to solve discrete optimal transport problems

Innovation

• Bregman Douglas-Rachford splitting method
• Bregman alternating direction method of multipliers
• solving discrete optimal transport problem