A deterministic annealing approach to Witsenhausen´s counterexample

This paper proposes an optimization method, based on information theoretic ideas, to a class of distributed control problems. As a particular test case, the well-known and numerically “over-mined” problem of decentralized control and implicit communication, commonly referred to as Witsenhausen´s counterexample, is considered. The key idea is to randomize the zero-delay mappings. which become “soft”, probabilistic mappings to be optimized in a deterministic annealing process, by incorporating a Shannon entropy constraint in the problem formulation. The entropy of the mapping is controlled and gradually lowered to zero to obtain deterministic mappings, while avoiding poor local minima. For the particular test case, our approach obtains new mappings that shed light on the structure of the optimal solution, as well as achieving a small improvement in total cost over the state of the art in numerical approaches to this problem. Proposed method is general and applicable to any problem of similar nature.