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

TL;DR

This paper introduces a non-penalty, direct approach for designing group-sparse feedback gains in linear-quadratic control problems, avoiding common issues of penalty methods and providing theoretical convergence guarantees.

Contribution

It reformulates the sparse feedback LQ problem from an epi-composition perspective and establishes convergence of ADMM without penalty parameters, offering a novel direct solution method.

Findings

Reformulation of SF-LQ and DFT-LQ as constrained problems.

Convergence proof of ADMM under certain assumptions.

Effective alternative methods when assumptions do not hold.

Abstract

In [1], 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 [1], 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 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 show that alternative approaches combining subgradient descent with Difference-of-Convex relaxation methods can be effectively utilized. 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.