Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions

TL;DR

This paper introduces two weakly convergent algorithms for solving complex monotone inclusion problems by leveraging problem structure, involving Douglas-Rachford and partial inverse methods, with applications to optimization and multiple operators.

Contribution

The paper proposes novel algorithms that explicitly utilize problem structure, combining Douglas-Rachford and partial inverse techniques for monotone inclusions.

Findings

Algorithms are weakly convergent.

Methods effectively handle sums of monotone operators and cocoercive operators.

Applications demonstrated in optimization problems.

Abstract

We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas-Rachford iteration involving the maximally monotone operator and the normal cone. In the second method it is a proximal step involving the partial inverse of the maximally monotone operator with respect to the vector subspace. Connections between the proposed methods and other methods in the literature are provided. Applications to monotone inclusions with finitely many maximally monotone operators and optimization problems are examined.