Robust Dynamic Programming for Discounted Infinite-Horizon Markov Decision Processes with Uncertain Stationary Transition Matrice

In this paper, finite-state, finite-action, discounted infinite-horizon-cost Markov decision processes (MDPs) with uncertain stationary transition matrices are discussed in the deterministic policy space. Uncertain stationary parametric transition matrices are clearly classified into independent and correlated cases. It is pointed out in this paper that the optimality criterion of uniform minimization of the maximum expected total discounted cost functions for all initial states, or robust uniform optimality criterion, is not appropriate for solving MDPs with correlated transition matrices. A new optimality criterion of minimizing the maximum quadratic total value function is proposed which includes the previous criterion as a special case. Based on the new optimality criterion, robust policy iteration is developed to compute an optimal policy in the deterministic stationary policy space. Under some assumptions, the solution is guaranteed to be optimal or near-optimal in the deterministic policy space