Numerical solutions to the Witsenhausen counterexample by approximating networks

Approximate solutions to the Witsenhausen counterexample are derived by con straining the unknown control functions to take on fixed structures contain ing "free" parameters to be optimized. Such structures are given by "nonlin ear approximating networks," i.e., linear combinations of parametrized basi s functions that benefit by density properties in normed linear spaces. Thi s reduces the original functional problem to a nonlinear programming one wh ich is solved via stochastic approximation. The method yields lower values of the costs than the ones achieved so far in the literature, and, most of all, provides rather a complete overview of the shapes of the optimal contr ol functions when the two parameters that characterize the Witsenhausen cou nterexample vary. One-hidden-layer neural networks are chosen as approximat ing networks.