A linear programming relaxation for stochastic control problems with non-classical information patterns

It was shown recently [1] that stochastic control problems with non-classical information patterns can be cast equivalently as nonconvex optimization problems and that lower bounds could be obtained by using the information-theoretic data processing inequality for creating convex relaxations. We present a linear programming relaxation to these problems. This provides a new avenue for obtaining lower bounds and yields the largest known class of inverse optimal cost functions. The linear programming relaxation improves on information theory based relaxations since it retains in its set of extreme points, all the extreme points of the original nonconvex problem. Lower bounds on the optimal cost of the original nonconvex problem can be found by constructing feasible points for the dual of the relaxation. We show that our relaxation is tight for the problem of minimizing the expected distortion of a binary source over binary memoryless symmetric channel for block length of 1 and 2. This gives a new, direct and non-communication-theoretic proof of the optimal distortion. Incorporating the data processing inequality in the linear programming relaxation, we obtain the largest known class of inverse optimal cost functions.