Solving the Infinite-horizon Constrained LQR Problem using Accelerated Dual Proximal Methods

TL;DR

This paper introduces an accelerated proximal gradient algorithm for solving infinite-horizon constrained LQR problems, achieving fast convergence and practical applicability to large systems without requiring terminal invariant sets.

Contribution

It develops a novel accelerated Forward-Backward Splitting method with proven convergence for infinite-dimensional LQR problems, suitable for large-scale systems and robust to disturbances.

Findings

Achieves optimal convergence rates of O(1/k^2) for function values.

Requires only finite memory per iteration, making it computationally efficient.

Demonstrates effectiveness through numerical examples.

Abstract

This work presents an algorithmic scheme for solving the infinite-time constrained linear quadratic regulation problem. We employ an accelerated version of a popular proximal gradient scheme, commonly known as the Forward-Backward Splitting (FBS), and prove its convergence to the optimal solution in our infinite-dimensional setting. Each iteration of the algorithm requires only finite memory, is computationally cheap, and makes no use of terminal invariant sets; hence, the algorithm can be applied to systems of very large dimensions. The acceleration brings in optimal convergence rates O(1/k^2) for function values and O(1/k) for primal iterates and renders the proposed method a practical alternative to model predictive control schemes for setpoint tracking. In addition, for the case when the true system is subject to disturbances or modelling errors, we propose an efficient warm-starting procedure, which significantly reduces the number of iterations when the algorithm is applied in closed-loop. Numerical examples demonstrate the approach.