# Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results

arXiv: https://arxiv.org/abs/1709.05747 · HTML: https://mlsanity.com/paper/1709.05747/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2017.

## Built on by (most cited first, 21 of 21)

- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [FedADMM: A Federated Primal-Dual Algorithm Allowing Partial Participation](https://mlsanity.com/paper/2203.15104/) (2022)
- [FedDR -- Randomized Douglas-Rachford Splitting Algorithms for Nonconvex Federated Composite Optimization](https://mlsanity.com/paper/2103.03452/) (2021)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser](https://mlsanity.com/paper/2301.13731/) (2023)
- [Training Recurrent Neural Networks by Sequential Least Squares and the Alternating Direction Method of Multipliers](https://mlsanity.com/paper/2112.15348/) (2021)
- [Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter](https://mlsanity.com/paper/2311.01216/) (2023)
- [Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penalties](https://mlsanity.com/paper/2309.03094/) (2023)
- [Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems](https://mlsanity.com/paper/2505.08909/) (2025)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [An Operator Splitting View of Federated Learning](https://mlsanity.com/paper/2108.05974/) (2021)
- [Decentralized Smoothing ADMM for Quantile Regression with Non-Convex Sparse Penalties](https://mlsanity.com/paper/2408.01307/) (2024)
- [A Convergent ADMM Framework for Efficient Neural Network Training](https://mlsanity.com/paper/2112.11619/) (2021)
- [Efficient Alternating Minimization Solvers for Wyner Multi-View Unsupervised Learning](https://mlsanity.com/paper/2303.15866/) (2023)
- [Moreau Envelope ADMM for Decentralized Weakly Convex Optimization](https://mlsanity.com/paper/2308.16752/) (2023)
- [Anderson Acceleration in Nonsmooth Problems: Local Convergence via Active Manifold Identification](https://mlsanity.com/paper/2410.09420/) (2024)
- [Smoothing ADMM for Non-convex and Non-smooth Hierarchical Federated Learning](https://mlsanity.com/paper/2503.08869/) (2025)
- [Parameter-free structure-texture image decomposition by unrolling](https://mlsanity.com/paper/2503.13354/) (2025)
- [Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control](https://mlsanity.com/paper/2507.19895/) (2025)
- [An Asynchronous Decentralised Optimisation Algorithm for Nonconvex Problems](https://mlsanity.com/paper/2507.22311/) (2025)
- [Signs Beat Floats: Low-Rank Double-Binary Adaptation for On-Device Fine-Tuning](https://mlsanity.com/paper/2605.24058/) (2026)

## Abstract (arXiv)

Although originally designed and analyzed for convex problems, the alternating direction method of multipliers (ADMM) and its close relatives, Douglas-Rachford splitting (DRS) and Peaceman-Rachford splitting (PRS), have been observed to perform remarkably well when applied to certain classes of structured nonconvex optimization problems. However, partial global convergence results in the nonconvex setting have only recently emerged. In this paper we show how the Douglas-Rachford envelope (DRE), introduced in 2014, can be employed to unify and considerably simplify the theory for devising global convergence guarantees for ADMM, DRS and PRS applied to nonconvex problems under less restrictive conditions, larger prox-stepsizes and over-relaxation parameters than previously known. In fact, our bounds are tight whenever the over-relaxation parameter ranges in (0,2]. The analysis of ADMM uses a universal primal equivalence with DRS that generalizes the known duality of the algorithms.