An outer reflected forward-backward splitting algorithm for solving monotone inclusions

TL;DR

This paper introduces a new outer reflected forward-backward splitting algorithm for solving complex monotone inclusions, with proven convergence and applications to composite problems in signal processing and machine learning.

Contribution

It proposes a novel splitting algorithm that handles multiple operators explicitly and converges under mild conditions, expanding the toolkit for monotone inclusion problems.

Findings

Proves convergence of the proposed algorithm.

Demonstrates the algorithm's applicability to composite monotone inclusions.

Highlights the algorithm's ability to process operators explicitly.

Abstract

Monotone inclusions have wide applications in solving various convex optimization problems arising in signal and image processing, machine learning, and medical image reconstruction. In this paper, we propose a new splitting algorithm for finding a zero of the sum of a maximally monotone operator, a monotone Lipschitzian operator, and a cocoercive operator, which is called outer reflected forward-backward splitting algorithm. Under mild conditions on the iterative parameters, we prove the convergence of the proposed algorithm. As applications, we employ the proposed algorithm to solve composite monotone inclusions involving monotone Lipschitzian operator, cocoercive operator, and the parallel sum of operators. The advantage of the obtained algorithm is that it is a completely splitting algorithm, in which the Lipschitzian operator and the cocoercive operator are processed via explicit steps and the maximally monotone operators are processed via their resolvents.