MONOTONE OPTIMAL POLICIES IN DISCOUNTED MARKOV DECISION PROCESSES WITH TRANSITION PROBABILITIES INDEPENDENT OF THE CURRENT STATE: EXISTENCE AND APPROXIMATION

In this paper there are considered Markov decision processes (MDPs) that have the discounted cost as the objective function, state and decision spaces that are subsets of the real line but are not necessarily finite or denumerable. The considered MDPs have a cost function that is possibly unbounded, and dynamic independent of the current state. The considered decision sets are possibly non-compact. In the context described, conditions to obtain either an increasing or decreasing optimal stationary policy are provided; these conditions do not require assumptions of convexity. Versions of the policy iteration algorithm (PIA) to approximate increasing or decreasing optimal stationary policies are detailed. An illustrative example is presented. Finally, comments on the monotonicity conditions and the monotone versions of the PIA that are applied to discounted MDPs with rewards are given.