Douglas--Rachford algorithm for nonmonotone multioperator inclusion problems

TL;DR

This paper extends the Douglas--Rachford algorithm to solve multioperator inclusion problems involving multiple weakly and strongly monotone operators, providing convergence guarantees and applicability to nonconvex optimization.

Contribution

The work generalizes the Douglas--Rachford algorithm to handle multioperator problems with convergence analysis and applications to nonconvex minimization.

Findings

Algorithm converges to fixed points under appropriate parameters.

Applicable to sum-of-functions minimization with weakly and strongly convex functions.

Provides global subsequential convergence for certain nonconvex problems.

Abstract

The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we extend the Douglas--Rachford algorithm to address multioperator inclusion problems involving $m$ ($m\geq 2$) weakly and strongly monotone operators, reformulated as a two-operator inclusion in a product space. By selecting appropriate parameters, we establish the convergence of the algorithm to a fixed point, from which solutions can be extracted. Furthermore, we illustrate its applicability to sum-of-$m$-functions minimization problems characterized by weakly convex and strongly convex functions. For general nonconvex problems in finite-dimensional spaces, comprising Lipschitz continuously differentiable functions and a proper closed function, we provide global subsequential convergence guarantees.