Exact worst-case convergence rates for Douglas--Rachford and Davis--Yin splitting methods

TL;DR

This paper precisely determines the worst-case convergence rates, including constants, for Douglas--Rachford and Davis--Yin splitting methods in convex optimization, providing tight bounds and examples to verify their optimality.

Contribution

It establishes the first exact worst-case convergence rates with constants for DRS and DYS, and compares different variants to highlight discrepancies.

Findings

Exact worst-case convergence rates including constants are derived.

Worst-case examples verify the tightness of the rates.

Discrepancies between DYS variants are demonstrated.

Abstract

In this work, we aim to establish the exact worst-case convergence rates of Douglas--Rachford splitting (DRS) and Davis--Yin splitting (DYS) when applied to convex optimization problems. Both DRS and DYS have two variants as swapping the roles of the two nonsmooth convex functions in both algorithms yields different sequences of iterates. For both variants of DRS and one variant of DYS, we establish the exact worst-case convergence rates, including the constant factor, using the primal--dual gap function as the performance metric. We provide worst-case examples to verify the tightness of these rates. To the best of our knowledge, this is the first result that establishes the exact worst-case convergence rates for DRS and DYS that include the constant factor. For the other variant of DYS, we establish the best-known convergence rate and provide a concrete example indicating a discrepancy between the convergence rates of the two DYS variants.