Douglas--Rachford for multioperator comonotone inclusions with applications to multiblock optimization

TL;DR

This paper analyzes the convergence of an adaptive Douglas--Rachford algorithm for multioperator problems involving maximally comonotone operators, extending classical methods to nonconvex and multiblock optimization scenarios.

Contribution

It introduces a convergence analysis for aDR in nonconvex, multiblock settings using duality, and derives a multiblock ADMM-type algorithm applicable to structured convex and nonconvex problems.

Findings

Convergence of aDR established under comonotonicity assumptions.

Extension of Douglas--Rachford and ADMM duality to multiblock nonconvex problems.

Guarantees provided for convex and strongly convex-weakly convex regimes.

Abstract

We study the convergence of the adaptive Douglas--Rachford (aDR) algorithm for solving a multioperator inclusion problem involving the sum of maximally comonotone operators. To address such problems, we adopt a product space reformulation that accommodates nonconvex-valued operators, which is essential when dealing with comonotone mappings. We establish convergence of the aDR method under comonotonicity assumptions, subject to suitable conditions on the algorithm parameters and comonotonicity moduli of the operators. Our analysis leverages the Attouch--Th\'{e}ra duality framework, which allows us to study the convergence of the aDR algorithm via its application to the dual inclusion problem. As an application, we derive a multiblock ADMM-type algorithm for structured convex and nonconvex optimization problems by applying the aDR algorithm to the operator inclusion formulation of the KKT system. The resulting method extends to multiblock and nonconvex settings the classical duality between the Douglas--Rachford algorithm and the alternating direction method of multipliers in the convex two-block case. Moreover, we establish convergence guarantees for both the fully convex and strongly convex-weakly convex regimes.