  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 sub-optimality bounds with less
computation time.
