Bregman Douglas-Rachford Splitting Method
This work addresses optimization problems in computational mathematics, particularly for researchers in monotone operator theory and optimal transport, but it appears incremental as it builds on existing splitting methods with Bregman extensions.
The paper tackles the maximal monotone inclusion problem by proposing the Bregman Douglas-Rachford splitting (BDRS) method and its variant, showing equivalence to Bregman ADMM and applying it to the discrete optimal transport problem, with convergence proven under certain assumptions.
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.