We propose a mathematical programming approach for the classical PSPACE - hard problem of n restless bandits in stochastic optimization. We introduce a series of n increasingly stronger linear programming relaxations, the last of which is exact and corresponds to the formulation of the problem as a Markov decision process that has exponential size, while other relaxations provide bounds and are efficiently solvable. We also propose a heuristic for solving the problem that naturally arises from the first of these relaxations and uses indices that are computed through optimal dual variables from the first relaxation. In this way we propose a policy and a suboptimality guarantee. We report computational results that suggest that the value of the proposed heuristic policy is extremely close to the optimal value. Moreover, the second order relaxation provides strong bounds for the optimal solution value.