An Adaptive Restart Heavy-Ball Projected Primal-Dual Method for Solving Constrained Linear Quadratic Optimal Control Problems

This paper presents a fast and flexible projected primal-dual method for solving linear quadratic optimal control problems with box constraints. Using a specific preconditioning, the algorithm achieves dead-beat convergence for unconstrained problems and has fast convergence for constrained problems. Accelerated convergence is obtained by applying a heavy-ball method to accelerate the projected primal-dual algorithm. In order to avoid missing critical points due to high momentum, an adaptive restarting procedure is used to slow the algorithm down if the solution diverges. Furthermore, convergence is proven by representing the algorithm as a Lur'e-type dynamic system and applying LaSalle's invariance principle to show the fixed point is asymptotically stable. The resulting algorithm is simple, while also achieving competitive computational times.