# Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control

arXiv: https://arxiv.org/abs/2507.19895 · HTML: https://mlsanity.com/paper/2507.19895/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 26 Jul 2025.

## Builds on (most cited first, 16 of 16)

- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems](https://mlsanity.com/paper/1410.1390/) (2014)
- [Distributed Reinforcement Learning for Decentralized Linear Quadratic Control: A Derivative-Free Policy Optimization Approach](https://mlsanity.com/paper/1912.09135/) (2019)
- [Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results](https://mlsanity.com/paper/1709.05747/) (2017)
- [Global convergence of splitting methods for nonconvex composite optimization](https://mlsanity.com/paper/1407.0753/) (2014)
- [Convergence for nonconvex ADMM, with applications to CT imaging](https://mlsanity.com/paper/2006.07278/) (2020)
- [Structured Policy Iteration for Linear Quadratic Regulator](https://mlsanity.com/paper/2007.06202/) (2020)
- [Sparsity Invariance for Convex Design of Distributed Controllers](https://mlsanity.com/paper/1906.06777/) (2019)
- [Policy Optimization over Submanifolds for Linearly Constrained Feedback Synthesis](https://mlsanity.com/paper/2201.11157/) (2022)
- [Structured Sparsity Promoting Functions](https://mlsanity.com/paper/1809.06777/) (2018)
- [Structured Output Feedback Control for Linear Quadratic Regulator Using Policy Gradient Method](https://mlsanity.com/paper/2203.00289/) (2022)
- [An Accelerated Proximal Alternating Direction Method of Multipliers for Optimal Decentralized Control of Uncertain Systems](https://mlsanity.com/paper/2304.11037/) (2023)
- [Optimal Decentralized Control for Uncertain Systems by Symmetric Gauss-Seidel Semi-Proximal ALM](https://mlsanity.com/paper/2001.00306/) (2020)
- [Sparse Linear-Quadratic Feedback Design Using Affine Approximation](https://mlsanity.com/paper/1507.08592/) (2015)
- [Sparsity-Promoting Optimal Control of Cyber-Physical Systems over Shared Communication Networks](https://mlsanity.com/paper/1905.07400/) (2019)
- [Avoiding strict saddle points of nonconvex regularized problems](https://mlsanity.com/paper/2401.09274/) (2024)

## Abstract (arXiv)

In this paper, we study the distributed linear quadratic problem with fixed communication topology (DFT-LQ) and the sparse feedback linear quadratic (SF-LQ) problem through a unified optimization framework. Specifically, both problems are formulated as a nonconvex, nonsmooth optimization problem equipped with an ℓ0-penalty under affine constraints. To solve this problem, we first investigate the application of the Douglas-Rachford (DR) splitting algorithm. Under the local condition that the generated iterates remain on a fixed smooth manifold, we establish the convergence of the DR splitting to a stationary point. Furthermore, we characterize this stationary point as the global minimizer of a corresponding DFT-LQ problem. To bypass the restriction of the smooth manifold assumption, we introduce a projected subgradient descent algorithm that achieves global convergence without relying on smooth-manifold structures. This algorithm may serve as a warm-start mechanism that effectively drives the iterates toward the desired smooth manifolds, thereby establishing a favorable initialization where the convergence theory of the DR splitting algorithm becomes fully applicable. Numerical experiments shed light on the effectiveness of the proposed methods in distributed group-sparse controller design.