Nonconvex Optimization Framework for Group-Sparse Feedback Linear-Quadratic Optimal Control: Non-Penalty Approach
This work addresses a specific problem in control theory for designing sparse controllers, offering theoretical improvements over prior penalty-based approaches, but it is incremental as it builds on existing formulations with new algorithmic insights.
The paper tackles the challenge of designing group-sparse feedback gains in linear-quadratic optimal control by directly solving constrained nonconvex optimization problems, avoiding penalty methods that require parameter tuning and risk spurious solutions, and establishes convergence guarantees for algorithms like ADMM under certain assumptions.
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.