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

This work is a companion paper of [8], where the distributed linear-quadratic problem with fixed communication topology (DFT-LQ) and the sparse feedback LQ problem (SF-LQ) are formulated into a nonsmooth and nonconvex optimization problem with affine constraints. Moreover, a penalty approach is considered in \cite{feng-part1}, and the PALM (proximal alternating linearized minimization) algorithm is studied with convergence and complexity analysis. In this paper, we aim to address the inherent drawbacks of the penalty approach, such as the challenge of tuning the penalty parameter and the risk of introducing spurious stationary points. Specifically, we first reformulate the SF-LQ problem and the DFT-LQ problem from an epi-composition function perspective, aiming to solve the constrained problem directly. Then, from a theoretical viewpoint, we revisit the alternating direction method of multipliers (ADMM) and establish its convergence to the set of cluster points under certain assumptions. When these assumptions do not hold, we can effectively utilize alternative approaches combining subgradient descent with Difference-of-Convex relaxation methods. In summary, our results enable the direct design of group-sparse feedback gains with theoretical guarantees, without resorting to convex surrogates, restrictive structural assumptions, or penalty formulations that incorporate constraints into the cost function.