The Douglas-Rachford Algorithm for Weakly Convex Penalties

TL;DR

This paper extends the Douglas-Rachford algorithm to handle sums of convex and weakly convex functions, providing convergence conditions and a modified algorithm that relaxes smoothness requirements, supported by numerical experiments.

Contribution

It introduces convergence conditions for the Douglas-Rachford algorithm with weakly convex penalties and proposes a modified version that does not require smoothness.

Findings

Convergence is guaranteed under specific conditions for weakly convex functions.

The modified algorithm relaxes smoothness constraints on the convex function.

Numerical experiments validate the theoretical convergence and performance.

Abstract

The Douglas-Rachford algorithm is widely used in sparse signal processing for minimizing a sum of two convex functions. In this paper, we consider the case where one of the functions is weakly convex but the other is strongly convex so that the sum is convex. We provide a condition that ensures the convergence of the same Douglas-Rachford iterations, provided that the strongly convex function is smooth. We also present a modified Douglas-Rachford algorithm that does not impose a smoothness condition for the convex function. We then provide a discussion on the convergence speed of the two types of algorithms and demonstrate the discussion with numerical experiments.