Two-Time Scale Controlled Markov Chains: A Decomposition and Parallel Processing Approach

This correspondence deals with a class of ergodic control problems for systems described by Markov chains with strong and weak interactions. These systems are composed of a set of $m$ subchains that are weakly coupled. Using results already available in the literature one formulates a limit control problem the solution of which can be obtained via an associated nondifferentiable convex programming (NDCP) problem. The technique used to solve the NDCP problem is the Analytic Center Cutting Plane Method (ACCPM) which implements a dialogue between, on one hand, a master program computing the analytical center of a localization set containing the solution and, on the other hand, an oracle proposing cutting planes that reduce the size of the localization set at each main iteration. The interesting aspect of this implementation comes from two characteristics: (i) the oracle proposes cutting planes by solving reduced sized Markov Decision Problems (MDP) via a linear program (LP) or a policy iteration method; (ii) several cutting planes can be proposed simultaneously through a parallel implementation on $m$ processors. The correspondence concentrates on these two aspects and shows, on a large scale MDP obtained from the numerical approximation “À la Kushner-Dupuis” of a singularly perturbed hybrid stochastic control problem, the important computational speed-up obtained.