Four-operator splitting algorithms for solving monotone inclusions

TL;DR

This paper introduces three novel splitting algorithms for solving complex monotone inclusion problems involving four operators, extending existing methods and demonstrating their effectiveness through numerical experiments.

Contribution

The paper proposes three new splitting algorithms specifically designed for four-operator monotone inclusions, expanding the scope of existing splitting methods.

Findings

Algorithms successfully solve four-operator monotone inclusions.

Numerical results show improved efficiency over existing methods.

Effective application to projection problems on Minkowski sums.

Abstract

Monotone inclusions involving the sum of three maximally monotone operators or more have received much attention in recent years. In this paper, we propose three splitting algorithms for finding a zero of the sum of four monotone operators, which are two maximally monotone operators, one monotone Lipschitz operator, and one cocoercive operator. These three splitting algorithms are based on the forward-reflected-Douglas-Rachford splitting algorithm, backward-forward-reflected-backward splitting algorithm, and backward-reflected-forward-backward splitting algorithm, respectively. As applications, we apply the proposed algorithms to solve the monotone inclusions problem involving a finite sum of maximally monotone operators. Numerical results on the Projection on Minkowski sums of convex sets demonstrate the effectiveness of the proposed algorithms.