Restless bandits, linear programming relaxations, and a primal-dual index heuristic

We develop a mathematical programming approach for the classical PSPACE-hard restless bandit problem in stochastic optimization. We introduce a hierarchy of N (where N is the number of bandits) increasingly stronger linear programming relaxations, the last of which is exact and corresponds to the (exponential size) formulation of the problem as a Markov decision chain, while the other relaxations provide bounds and are efficiently computed. We also propose a priority-index heuristic scheduling policy from the solution to the first-order relaxation, where the indices are defined in terms of optimal dual variables. In this way we propose a policy and a suboptimality guarantee. We report results of computational experiments that suggest that the proposed heuristic policy is nearly optimal, Moreover, the second-order relaxation is found to provide strong bounds on the optimal value.