Bregman three-operator splitting methods

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-Vũ algorithms and the primal-dual threeoperator (PD3O) algorithm. The Bregman extensions of the Condat-Vũ 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-Vũ algorithm applied to equality-constrained problems. Finally, we propose a Bregman extension of PD3O and analyze its convergence.

Understanding the convergence of the preconditioned PDHG method: a view of indefinite proximal ADMM

“…Note that a similar condition of (1.6) is extended for infinite dimensional Hilbert space in [3]. Very recently, under condition (1.6), Jiang and Vandenberghe [31] showed convergence of iterates for Bregman PDHG, of which PrePDHG is a special case.…”