  We consider dynamic programming problems with finite, discrete-time horizons
and prohibitively high-dimensional, discrete state-spaces for direct
computation of the value function from the Bellman equation. For the case that
the value function of the dynamic program is concave extensible and submodular
in its state-space, we present a new algorithm that computes deterministic
upper and stochastic lower bounds of the value function similar to dual dynamic
programming. We then show that the proposed algorithm terminates after a finite
number of iterations. Finally, we demonstrate the efficacy of our approach on a
high-dimensional numerical example from delivery slot pricing in attended home
delivery.
