# Shadow Douglas--Rachford Splitting for Monotone Inclusions

## TL;DR

This paper introduces a novel algorithm derived from a non-standard discretization of a continuous dynamical system for solving monotone inclusions involving a Lipschitz continuous operator, requiring one forward and one backward evaluation per iteration.

## Contribution

It presents a new explicit discretization-based algorithm for monotone inclusions, expanding the Douglas--Rachford splitting framework with a different discretization approach.

## Key findings

- Convergence of the proposed algorithm is established.
- The method efficiently handles problems with a Lipschitz continuous operator.
- It offers an alternative to traditional implicit discretization methods.

## Abstract

In this work, we propose a new algorithm for finding a zero in the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas--Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator.