# A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators

## TL;DR

This paper introduces a new splitting method for solving complex monotone inclusions involving Lipschitzian and cocoercive operators, extending existing algorithms and improving step size ranges for better convergence.

## Contribution

It proposes a novel reflected forward-backward splitting method for monotone inclusions with Lipschitzian operators, including cases with cocoercivity, and demonstrates its application to composite problems.

## Key findings

- Developed a simple method for zero points of sums of two monotone operators with Lipschitzian components.
- Extended the method to improve step size ranges when the Lipschitzian operator is cocoercive.
- Applied the new splitting method to composite monotone inclusions with promising results.

## Abstract

The proximal extrapolated gradient method \cite{Malitsky18a} is an extension of the projected reflected gradient method \cite{Malitsky15}. Both methods were proposed for solving the classic variational inequalities. In this paper, we investigate the projected reflected gradient method, in the general setting, for solving monotone inclusions involving Lipschitzian operators. As a result, we obtain a simple method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. We also show that one can improve the range of the stepsize of this method for the case when the Lipschitzian operator is restricted to be cocoercive. A nice combination of this method and the forward-backward splitting was proposed. As a result, we obtain a new splitting method for finding a zero point of the sum of three operators ( maximally monotone + monotone Lipschitzian + cocoercive). Application to composite monotone inclusions are demonstrated.