# Accelerated forward-backward and Douglas-Rachford splitting dynamics

arXiv: https://arxiv.org/abs/2407.20620 · HTML: https://mlsanity.com/paper/2407.20620/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 27 references that are in the map.
- First posted on arXiv 30 Jul 2024.

## Builds on (most cited first, 14 of 14)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [A Dynamical Systems Perspective on Nesterov Acceleration](https://mlsanity.com/paper/1905.07436/) (2019)
- [The proximal augmented Lagrangian method for nonsmooth composite optimization](https://mlsanity.com/paper/1610.04514/) (2016)
- [Gradient flows and proximal splitting methods: A unified view on accelerated and stochastic optimization](https://mlsanity.com/paper/1908.00865/) (2019)
- [Control Barrier Function Based Design of Gradient Flows for Constrained Nonlinear Programming](https://mlsanity.com/paper/2204.01930/) (2022)
- [Douglas-Rachford Splitting: Complexity Estimates and Accelerated Variants](https://mlsanity.com/paper/1407.6723/) (2014)
- [Forward-backward truncated Newton methods for convex composite optimization](https://mlsanity.com/paper/1402.6655/) (2014)
- [Further properties of the forward-backward envelope with applications to difference-of-convex programming](https://mlsanity.com/paper/1605.00201/) (2016)
- [Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics](https://mlsanity.com/paper/2103.12675/) (2021)
- [A Contraction Theory Approach to Optimization Algorithms from Acceleration Flows](https://mlsanity.com/paper/2105.08832/) (2021)
- ["Second-Order Primal'' + "First-Order Dual'' Dynamical Systems with Time Scaling for Linear Equality Constrained Convex Optimization Problems](https://mlsanity.com/paper/2103.12931/) (2021)
- [Fast inertial dynamics and FISTA algorithms in convex optimization. Perturbation aspects](https://mlsanity.com/paper/1507.01367/) (2015)
- [Envelope Functions: Unifications and Further Properties](https://mlsanity.com/paper/1606.01327/) (2016)

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

- [Finding Structure in Continual Learning](https://mlsanity.com/paper/2602.04555/) (2026)
- [Task Switching Without Forgetting via Proximal Decoupling](https://mlsanity.com/paper/2604.18857/) (2026)

## Abstract (arXiv)

We examine convergence properties of continuous-time variants of accelerated Forward-Backward (FB) and Douglas-Rachford (DR) splitting algorithms for nonsmooth composite optimization problems. When the objective function is given by the sum of a quadratic and a nonsmooth term, we establish accelerated sublinear and exponential convergence rates for convex and strongly convex problems, respectively. Moreover, for FB splitting dynamics, we demonstrate that accelerated exponential convergence rate carries over to general strongly convex problems. In our Lyapunov-based analysis we exploit the variable-metric gradient interpretations of FB and DR splittings to obtain smooth Lyapunov functions that allow us to establish accelerated convergence rates. We provide computational experiments to demonstrate the merits and the effectiveness of our analysis.