Proximal Limited-Memory Quasi-Newton Methods for Scenario-based Stochastic Optimal Control

Stochastic optimal control problems are typically of rather large scale involving millions of decision variables, but possess a certain structure which can be exploited by first-order methods such as forward-backward splitting and the alternating direction method of multipliers (ADMM). In this paper, we use the forward-backward envelope, a real-valued continuously differentiable penalty function, to recast the dual of the original nonsmooth problem as an unconstrained problem which we solve via the limited-memory BFGS algorithm. We show that the proposed method leads to a significant improvement of the convergence rate without increasing much the computational cost per iteration.