A convergent relaxation of the Douglas-Rachford algorithm

TL;DR

This paper introduces a unified algorithm that generalizes and extends the Douglas-Rachford method, providing convergence analysis and demonstrating improved numerical performance in structured optimization and feasibility problems.

Contribution

It proposes a convergent relaxation of the Douglas-Rachford algorithm that encompasses existing methods and offers new convergence criteria, especially for nonconvex and inconsistent cases.

Findings

Characterized fixed points of the algorithm in various cases.

Established convergence criteria for general fixed point operators.

Demonstrated improved numerical performance over RAAR in sparse feasibility problems.

Abstract

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward-backward and the Douglas-Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the algorithm is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point operators are established. When applying to nonconvex feasibility including the inconsistent case, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets which need not intersect. In this special case, we refine known linear convergence criteria for the Douglas-Rachford algorithm (DR). As a consequence, for feasibility with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.