Simplex Algorithm for Countable-State Discounted Markov Decision Processes

We consider discounted Markov Decision Processes (MDPs) with countably-infinite statespaces, finite action spaces, and unbounded rewards. Typical examples of such MDPs areinventory management and queueing control problems in which there is no specific limit on thesize of inventory or queue. Existing solution methods obtain a sequence of policies that convergesto optimality in value but may not improve monotonically, i.e., a policy in the sequence maybe worse than preceding policies. Our proposed approach considers countably-infinite linearprogramming (CILP) formulations of the MDPs (a CILP is defined as a linear program (LP)with countably-infinite numbers of variables and constraints). Under standard assumptions foranalyzing MDPs with countably-infinite state spaces and unbounded rewards, we extend themajor theoretical extreme point and duality results to the resulting CILPs. Under an additionaltechnical assumption which is satisfied by several applications of interest, we present a simplex-type algorithm that is implementable in the sense that each of its iterations requires only afinite amount of data and computation. We show that the algorithm finds a sequence of policieswhich improves monotonically and converges to optimality in value. Unlike existing simplex-typealgorithms for CILPs, our proposed algorithm solves a class of CILPs in which each constraintmay contain an infinite number of variables and each variable may appear in an infinite numberof constraints. A numerical illustration for inventory management problems is also presented.