Accelerated Point-Wise Maximum Approach to Approximate Dynamic Programming

In this article, we describe an approximate dynamic programming (ADP) approach to compute lower bounds on the optimal value function for a discrete time, continuous space, and infinite horizon setting. The approach iteratively constructs a family of lower bounding approximate value functions by using the so-called Bellman inequality. The novelty of our approach is that, at each iteration, we aim to compute an approximate value function that maximizes the point-wise maximum taken with the family of approximate value functions computed thus far. This leads to a nonconvex objective, and we propose a gradient ascent algorithm to find stationary points by solving a sequence of convex optimization problems. We provide convergence guarantees for our algorithm and an interpretation for how the gradient computation relates to the state-relevance weighting parameter appearing in related ADP approaches. We demonstrate through numerical examples that, when compared to the existing approaches, the algorithm we propose computes tighter suboptimality bounds with comparable computation time.