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

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 $\ell_0$-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.