Nonconvex Optimization Framework for Group-Sparse Feedback Linear-Quadratic Optimal Control: Penalty Approach

TL;DR

This paper introduces a nonconvex optimization framework for designing group-sparse feedback controllers in infinite-horizon LQ problems, enabling direct, structure-aware control with theoretical convergence guarantees.

Contribution

It formulates the controller design as a nonconvex problem with group sparsity, unifies distributed and sparse LQ problems, and proposes a convergent PALM algorithm.

Findings

Unified framework for distributed and sparse LQ control.

Proposed PALM algorithm guarantees convergence to critical points.

Enables direct design of group-sparse controllers without convex relaxations.

Abstract

This paper develops a unified nonconvex optimization framework for the design of group-sparse feedback controllers in infinite-horizon linear-quadratic (LQ) problems. We address two prominent extensions of the classical LQ problem: the distributed LQ problem with fixed communication topology (DFT-LQ) and the sparse feedback LQ problem (SF-LQ), both of which are motivated by the need for scalable and structure-aware control in large-scale systems. Unlike existing approaches that rely on convex relaxations or are limited to block-diagonal structures, we directly formulate the controller synthesis as a finite-dimensional nonconvex optimization problem with group $\ell_0$-norm regularization, capturing general sparsity patterns. We establish a connection between DFT-LQ and SF-LQ problems, showing that both can be addressed within our unified framework. Furthermore, we propose a penalty-based proximal alternating linearized minimization (PALM) algorithm and provide a rigorous convergence analysis under mild assumptions, overcoming the lack of coercivity in the objective function. The proposed method admits efficient solvers for all subproblems and guarantees global convergence to critical points. Our results fill a key gap in the literature by enabling the direct design of group-sparse feedback gains with theoretical guarantees, without resorting to convex surrogates or restrictive structural assumptions.