Optimization formulation and monotonic solution method for the Witsenhausen problem

The Witsenhausen counterexample is examined. The problem is reduced to an optimization problem over the space of quantile functions. Calculus of variation methods are applied, and necessary conditions are generated. Aspects of the structure of the problem and the solution are discussed. A numerical method generating a sequence of solution approximations with monotonically decreasing cost is constructed, based on the necessary conditions. The limit of the method satisfies the necessary conditions.