A stochastic preconditioned Douglas-Rachford splitting method for saddle-point problems

TL;DR

This paper introduces a stochastic, preconditioned Douglas-Rachford splitting method for saddle-point problems, proving convergence and demonstrating high efficiency through numerical experiments.

Contribution

It presents a novel stochastic and relaxed preconditioned Douglas-Rachford method with proven convergence for nonsmooth saddle-point problems.

Findings

Almost sure convergence of the method.

Sublinear ergodic convergence rate.

Numerical experiments confirm high efficiency.

Abstract

In this article, we propose and study a stochastic and relaxed preconditioned Douglas--Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas--Rachford splitting methods.