# Uniqueness of DRS as the 2 Operator Resolvent-Splitting and Impossibility of 3 Operator Resolvent-Splitting

## TL;DR

This paper proves the uniqueness of Douglas--Rachford splitting (DRS) as the only 2-operator resolvent-splitting with certain properties and shows that direct generalization to 3 operators is impossible without lifting, providing a minimal lifting solution.

## Contribution

It establishes the uniqueness of DRS among 2-operator resolvent-splittings and introduces a minimal lifting method for 3-operator generalization.

## Key findings

- DRS is the unique 2-operator resolvent-splitting with favorable properties.
- Direct 3-operator generalization of DRS is impossible without lifting.
- A novel minimal lifting 3-operator resolvent-splitting is proposed.

## Abstract

Given the success of Douglas--Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.