Extending Douglas-Rachford Splitting for Convex Optimization

TL;DR

This paper extends the understanding of Douglas-Rachford splitting by characterizing all unconditionally convergent variants in convex optimization, leading to new ADMM and Chambolle-Pock algorithms.

Contribution

It provides a complete characterization of all frugal, no-lifting resolvent-splitting methods that are unconditionally convergent for convex optimization.

Findings

Characterization of all such splitting methods in convex optimization.

Identification of parameter regions ensuring unconditional convergence.

Derivation of new ADMM and Chambolle-Pock variants from the characterization.

Abstract

The Douglas-Rachford splitting method is a classical and widely used algorithm for solving monotone inclusions involving the sum of two maximally monotone operators. It was recently shown to be the unique frugal, no-lifting resolvent-splitting method that is unconditionally convergent in the general two-operator setting. In this work, we show that this uniqueness does not hold in the convex optimization case: when the operators are subdifferentials of proper, closed, convex functions, a strictly larger class of frugal, no-lifting resolvent-splitting methods is unconditionally convergent. We provide a complete characterization of all such methods in the convex optimization setting and prove that this characterization is sharp: unconditional convergence holds exactly on the identified parameter regions. These results immediately yield new families of convergent ADMM-type and Chambolle-Pock-type methods obtained through their Douglas-Rachford reformulations.