Accelerated Optimization of the Linear-Quadratic Regulator Problem

Core Concepts

This paper introduces an accelerated optimization framework for solving the linear-quadratic regulator (LQR) problem, which is a landmark problem in optimal control. The authors present novel continuous-time and discrete-time algorithms that achieve Nesterov-optimal convergence rates for the state-feedback LQR (SLQR) problem. For the output-feedback LQR (OLQR) problem, a Hessian-free accelerated framework is proposed that can find an ϵ-stationary point with second-order guarantee in a time of O(ϵ^(-7/4) log(1/ϵ)).

Abstract

The paper introduces an accelerated optimization framework for solving the linear-quadratic regulator (LQR) problem, which is a fundamental problem in optimal control. The authors distinguish between the state-feedback LQR (SLQR) and output-feedback LQR (OLQR) problems based on whether the full state is available. For the SLQR problem: The authors prove the Lipschitz Hessian property of the LQR performance criterion, which is crucial for the application of modern optimization techniques. 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. A Nesterov-type method with a restarting rule is proposed for the discrete-time algorithm, which preserves the continuous-time convergence rate. 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. The method can find an ϵ-stationary point of the performance criterion in a time of O(ϵ^(-7/4) log(1/ϵ)), which improves upon the O(ϵ^(-2)) complexity of vanilla gradient descent. The method provides the second-order guarantee of stationary point. The key contributions of the paper include: Proving the Lipschitz Hessian property of the LQR performance criterion, which is essential for the convergence analysis. Proposing an accelerated optimization framework for the SLQR problem, with both continuous-time and discrete-time algorithms achieving Nesterov-optimal convergence rates. Developing a Hessian-free accelerated framework for the OLQR problem, which can find an ϵ-stationary point with second-order guarantee in an improved time complexity.

Stats

None.

Quotes

None.

Key Insights Distilled From

Accelerated Optimization Landscape of Linear-Quadratic Regulator

by Lechen Feng,... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2307.03590.pdf

Deeper Inquiries

How can the proposed accelerated optimization framework be extended to handle additional constraints in the LQR problem, such as decentralized control or risk constraints

The proposed accelerated optimization framework for the LQR problem can be extended to handle additional constraints by incorporating them into the objective function or the optimization algorithm. For example, for decentralized control constraints, the framework can be modified to optimize the feedback gain matrices for multiple subsystems simultaneously while ensuring coordination and stability between them. This can be achieved by introducing coupling terms or constraints that capture the interactions between the subsystems. Additionally, for risk constraints in the LQR problem, the framework can be adapted to include constraints on the performance criterion to limit the risk exposure or ensure robustness in the control system. By incorporating these constraints into the optimization process, the framework can provide solutions that meet both the control objectives and the specified constraints.

What are the potential limitations or drawbacks of the Hessian-free accelerated framework for the OLQR problem, and how can they be addressed

The Hessian-free accelerated framework for the OLQR problem may have limitations or drawbacks related to convergence speed, robustness, or scalability. One potential limitation is the sensitivity to the choice of parameters, such as the step size or the threshold for restarting the algorithm. If these parameters are not properly tuned, the algorithm may converge slowly or get stuck in local minima. To address this, a more robust parameter tuning strategy or adaptive algorithms can be implemented to automatically adjust the parameters during optimization. Another limitation could be the complexity of the negative curvature exploitation method, which may require additional computational resources or introduce computational overhead. This can be mitigated by optimizing the implementation of the negative curvature exploitation step or exploring alternative optimization techniques that are more efficient for the OLQR problem.

Can the insights and techniques developed in this paper be applied to other types of optimal control problems beyond the LQR setting

The insights and techniques developed in this paper for the LQR setting can be applied to other types of optimal control problems beyond LQR. For example, the accelerated optimization framework and the hybrid dynamic system approach can be adapted to solve nonlinear optimal control problems, stochastic control problems, or dynamic programming problems. By incorporating the principles of accelerated optimization, continuous-time optimization, and semiconvex function optimization, these techniques can be extended to a wide range of control problems with different dynamics, constraints, or objectives. The key lies in understanding the underlying principles and adapting the algorithms to suit the specific characteristics of the problem at hand.