Split-Douglas-Rachford algorithm for composite monotone inclusions and Split-ADMM

TL;DR

This paper introduces a generalized splitting algorithm for monotone inclusions involving linear operators, extending classical methods, and derives a new Split-ADMM for convex optimization problems, with demonstrated numerical efficiency.

Contribution

It generalizes Douglas-Rachford and primal-dual algorithms to include non-standard metrics and linear operators, and introduces a novel Split-ADMM method for composite problems.

Findings

Guaranteed weak convergence to Kuhn-Tucker points.

The new Split-ADMM activates linear operators implicitly and explicitly.

Numerical results show improved efficiency in image restoration and sparse minimization.

Abstract

In this paper we provide a generalization of the Douglas-Rachford splitting (DRS) and the primal-dual algorithm (Vu 2013, Condat 2013) for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual non-standard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn-Tucker point is guaranteed, generalizing the main result in Svaiter (2011). Inspired by Gabay (1983), we also derive a new Split-ADMM (SADMM) by applying our method to the dual of a convex optimization problem involving a linear operator which can be expressed as the composition of two linear operators. The proposed SADMM activates one linear operator implicitly and the other one explicitly, and we recover ADMM when the latter is set as the identity. Connections and comparisons of our theoretical results with respect to the literature are provided for the main algorithm and SADMM. The flexibility and efficiency of both methods is illustrated via a numerical simulations in total variation image restoration and a sparse minimization problem.