Anderson Accelerated Douglas-Rachford Splitting

TL;DR

This paper introduces an Anderson accelerated Douglas-Rachford splitting algorithm for non-smooth convex optimization with linear constraints, demonstrating its convergence, scalability, and practical performance across various applications.

Contribution

The paper proposes a novel accelerated algorithm that improves convergence and scalability for convex optimization problems with linear constraints, applicable in multiple fields.

Findings

Algorithm converges globally or provides infeasibility certificates

Partial decoupling allows parallel computation

Demonstrated efficiency on diverse examples

Abstract

We consider the problem of non-smooth convex optimization with linear equality constraints, where the objective function is only accessible through its proximal operator. This problem arises in many different fields such as statistical learning, computational imaging, telecommunications, and optimal control. To solve it, we propose an Anderson accelerated Douglas-Rachford splitting (A2DR) algorithm, which we show either globally converges or provides a certificate of infeasibility/unboundedness under very mild conditions. Applied to a block separable objective, A2DR partially decouples so that its steps may be carried out in parallel, yielding an algorithm that is fast and scalable to multiple processors. We describe an open-source implementation and demonstrate its performance on a wide range of examples.