Point-wise maximum approach to approximate dynamic programming

Abstract

We describe an approximate dynamic programming approach to compute lower bounds on the optimal value function for a discrete time, continuous space, 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 non-convex 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 approximate dynamic programming approaches. We demonstrate through numerical examples that, when compared to existing approaches, the algorithm we propose computes tighter suboptimality bounds with less computation time. Paul N. Beuchat received the B.Eng. degree in mechanical engineering and B.Sc. in physics from the University of Melbourne, Australia, in 2008, and the M.Sc. degree in robotics, systems and control from ETH Zürich, Switzerland, in 2014, where he is currently working towards the Ph.D degree at the Automatic Control Laboratory. From 2009-2012 he was as a subsurface engineer for ExxonMobil. His research interests are control and optimization of large scale systems, with a focus towards developing approximate dynamic programming techniques for applications in the areas of building control, and coordinated flight.

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