Gradient-Bounded Dynamic Programming for Submodular and Concave Extensible Value Functions with Probabilistic Performance Guarantees

Abstract

We consider stochastic dynamic programming problems with high-dimensional, discrete state-spaces and finite, discrete-time horizons that prohibit direct computation of the value function from a given Bellman equation for all states and time steps due to the "curse of dimensionality". For the case where 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 in the realm of dual dynamic programming. We show that the proposed algorithm terminates after a finite number of iterations. Furthermore, we derive probabilistic guarantees on the value accumulated under the associated policy for a single realisation of the dynamic program and for the expectation of this value. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.