We present an optimization-based approach to stochastic control problems with nonclassical information structures. We cast these problems equivalently as optimization prob- lems on joint distributions. The resulting problems are necessarily nonconvex. Our approach to solving them is through convex relaxation. We solve the instance solved by Bansal and Basar with a particular application of this approach that uses the data processing inequality for constructing the convex relaxation. Using certain f-divergences, we obtain a new, larger set of inverse optimal cost functions for such problems. Insights are obtained on the relation between the structure of cost functions and of convex relaxations for inverse optimal control.