Yuan-Hua Ni

Lechen Feng, Yuan-Hua Ni

Linear-quadratic regulator (LQR) is a landmark problem in the field of optimal control, which is the concern of this paper. Generally, LQR is classified into state-feedback LQR (SLQR) and output-feedback LQR (OLQR) based on whether the full state is obtained. It has been suggested in existing literature that both SLQR and OLQR could be viewed as \textit{constrained nonconvex matrix optimization} problems in which the only variable to be optimized is the feedback gain matrix. In this paper, we introduce a first-order accelerated optimization framework of handling the LQR problem, and give its convergence analysis for the cases of SLQR and OLQR, respectively. Specifically, a Lipschiz Hessian property of LQR performance criterion is presented, which turns out to be a crucial property for the application of modern optimization techniques. For the SLQR problem, a continuous-time hybrid dynamic system is introduced, whose solution trajectory is shown to converge exponentially to the optimal feedback gain with Nesterov-optimal order $1-\frac{1}{\sqrtκ}$ ($κ$ the condition number). Then, the symplectic Euler scheme is utilized to discretize the hybrid dynamic system, and a Nesterov-type method with a restarting rule is proposed that preserves the continuous-time convergence rate, i.e., the discretized algorithm admits the Nesterov-optimal convergence order. For the OLQR problem, a Hessian-free accelerated framework is proposed, which is a two-procedure method consisting of semiconvex function optimization and negative curvature exploitation. In a time $\mathcal{O}(ε^{-7/4}\log(1/ε))$, the method can find an $ε$-stationary point of the performance criterion; this entails that the method improves upon the $\mathcal{O}(ε^{-2})$ complexity of vanilla gradient descent. Moreover, our method provides the second-order guarantee of stationary point.

Jun Xie, Yuan-Hua Ni, Yiqin Yang et al.

This paper proposes efficient policy iteration and value iteration algorithms for the continuous-time linear quadratic regulator problem with unmeasurable states and unknown system dynamics, from the perspective of direct data-driven control. Specifically, by re-examining the data characteristics of input-output filtered vectors and introducing QR decomposition, an improved substitute state construction method is presented that further eliminates redundant information, ensures a full row rank data matrix, and enables a complete parameterized representation of the feedback controller. Furthermore, the original problem is transformed into an equivalent linear quadratic regulator problem defined on the substitute state with a known input matrix, verifying the stabilizability and detectability of the transformed system. Consequently, model-free policy iteration and value iteration algorithms are designed that fully exploit the full row rank substitute state data matrix. The proposed algorithms offer distinct advantages: they avoid the need for prior knowledge of the system order or the calculation of signal derivatives and integrals; the iterative equations can be solved directly without relying on the traditional least-squares paradigm, guaranteeing feasibility in both single-output and multi-output settings; and they demonstrate superior numerical stability, reduced data demand, and higher computational efficiency. Moreover, the heuristic results regarding trajectory generation for continuous-time systems are discussed, circumventing potential failure modes associated with existing approaches.

Yiqin Yang, Xu Yang, Yuhua Jiang et al.

In the realm of multi-agent systems, the challenge of \emph{partial observability} is a critical barrier to effective coordination and decision-making. Existing approaches, such as belief state estimation and inter-agent communication, often fall short. Belief-based methods are limited by their focus on past experiences without fully leveraging global information, while communication methods often lack a robust model to effectively utilize the auxiliary information they provide. To solve this issue, we propose Global State Diffusion Algorithm~(GlobeDiff) to infer the global state based on the local observations. By formulating the state inference process as a multi-modal diffusion process, GlobeDiff overcomes ambiguities in state estimation while simultaneously inferring the global state with high fidelity. We prove that the estimation error of GlobeDiff under both unimodal and multi-modal distributions can be bounded. Extensive experimental results demonstrate that GlobeDiff achieves superior performance and is capable of accurately inferring the global state.

Lechen Feng, Xun Li, Yuan-Hua Ni

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.

Lechen Feng, Xun Li, Yuan-Hua Ni

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.

Lechen Feng, Yuan-Hua Ni, Xuebo Zhang

A $\mathcal{H}_2$-guaranteed sparse-feedback linear-quadratic (LQ) optimal control with convex parameterization and convex-bounded uncertainty is studied in this paper, where $\ell_0$-penalty is added into the $\mathcal{H}_2$ cost to penalize the number of communication links among distributed controllers. Then, the sparse-feedback gain is investigated to minimize the modified $\mathcal{H}_2$ cost together with the stability guarantee, and the corresponding main results are of three parts. First, the $\ell_1$ relaxation sparse-feedback LQ problem is of concern, and a two-timescale algorithm is developed based on proximal coordinate descent and primal-dual splitting approach. Second, piecewise quadratic relaxation sparse-feedback LQ control is investigated, which exhibits an accelerated convergence rate. Third, sparse-feedback LQ problem with $\ell_0$-penalty is directly studied through BSUM (Block Successive Upper-bound Minimization) framework, and precise approximation method and variational properties are introduced.