# A Note on the Forward-Douglas--Rachford Splitting for Monotone Inclusion and Convex Optimization

arXiv: https://arxiv.org/abs/1704.06948 · HTML: https://mlsanity.com/paper/1704.06948/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2017.

## Built on by (most cited first, 2 of 2)

- [Is there anything left? Measuring semantic residuals of objects removed from 3D Gaussian Splatting](https://mlsanity.com/paper/2503.17574/) (2025)
- [Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers](https://mlsanity.com/paper/2503.19190/) (2025)

## Abstract (arXiv)

We shed light on the structure of the "three-operator" version of the forward-Douglas--Rachford splitting algorithm for finding a zero of a sum of maximally monotone operators A + B + C, where B is cocoercive, involving only the computation of B and of the resolvent of A and of C, separately. We show that it is a straightforward extension of a fixed-point algorithm proposed by us as a generalization of the forward-backward splitting algorithm, initially designed for finding a zero of a sum of an arbitrary number of maximally monotone operators sumi=1n Ai + B, where B is cocoercive, involving only the computation of B and of the resolvent of each Ai separately. We argue that, the former is the "true" forward-Douglas--Rachford splitting algorithm, in contrast to the initial use of this designation in the literature. Then, we highlight the extension to an arbitrary number of maximally monotone operators in the splitting, sumi=1n Ai + B + C, in a formulation admitting preconditioning operators. We finally demonstate experimentally its interest in the context of nonsmooth convex optimization.