An optimization approach to the Witsenhausen counterexample

We examine the structure of the Witsenhausen counterexample/ problem and its solution. In particular, we find it useful to work with the associated quantile function, rather than the controller itself or its distribution. With this transformation, the problem is reduced to minimization of a certain criterion over a particular function space. The optimization criterion is the sum of two functionals. The first, representing the control cost, is a simple quadratic. The second, representing the expected squared estimation error, has a more complex structure over this space. Nonetheless, it has a unique minimum (i.e., no other local minima). The problem of determining the parameter region over which the total cost criterion has a unique minimum remains open, although numerical experimentation suggests that this may “typically” be the case. Numerical results also indicate the form of the solution.