The Witsenhausen counterexample: a hierarchical search approach for nonconvex optimization problems

The Witsenhausens counterexample (1968) is a difficult nonconvex functional optimization problem which has been outstanding for more than 30 years. Considerable amount of literature has been accumulated, but optimal solutions remain elusive. In this paper, we develop a framework that allows one to gain additional new insights to the properties of a better solution for a benchmark instance. Through our approach, we are able to zero in on a solution that is 13% better than the previously known best solution, and more than 54% better than previous results obtained by other authors. More importantly, we demonstrate that our approach, called hierarchical search, can be useful in general optimization problems.

[1]  T. Başar,et al.  Stochastic Teams with Nonclassical Information Revisited: When is an Affine Law Optimal? , 1986, 1986 American Control Conference.

[2]  S. Mitter,et al.  Information and control: Witsenhausen revisited , 1999 .

[3]  D. Wolpert,et al.  No Free Lunch Theorems for Search , 1995 .

[4]  Y. Ho,et al.  Team decision theory and information structures in optimal control problems--Part II , 1972 .

[5]  J. Meditch,et al.  Applied optimal control , 1972, IEEE Transactions on Automatic Control.

[6]  Yu-Chi Ho,et al.  Massively parallel simulation of a class of discrete event systems , 1992, [Proceedings 1992] The Fourth Symposium on the Frontiers of Massively Parallel Computation.

[7]  Chun-Hung Chen,et al.  Rates of Convergence of Ordinal Comparison for Dependent Discrete Event Dynamic Systems , 1997 .

[8]  Nikos Theodore Patsis.  Pricing American-style exotic options using ordinal optimization , 1998 .

[9]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[10]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[11]  Mike Shang-Yu Yang.  Ordinal optimization and its application to complex deterministic problems , 1998 .

[12]  H. Witsenhausen.  A Counterexample in Stochastic Optimum Control , 1968 .

[13]  Y. Ho,et al.  Another look at the nonclassical information structure problem , 1980 .

[14]  Christos Papadimitriou,et al.  Intractable problems in control theory , 1985, 1985 24th IEEE Conference on Decision and Control.

[15]  Yu-Chi Ho,et al.  Ordinal optimization approach to rare event probability problems , 1995, Discret. Event Dyn. Syst..

[16]  Loo Hay Lee,et al.  Explanation of goal softening in ordinal optimization , 1999, IEEE Trans. Autom. Control..

[17]  Thomas Parisini,et al.  Numerical solutions to the Witsenhausen counterexample by approximating networks , 2001, IEEE Trans. Autom. Control..

[18]  M. L. Cohen,et al.  The Fisher information and convexity (Corresp.) , 1968, IEEE Trans. Inf. Theory.

[19]  Thomas Parisini,et al.  Nonlinear approximations for the solution of team optimal control problems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[20]  Yu-Chi Ho,et al.  Ordinal optimization of DEDS , 1992, Discret. Event Dyn. Syst..

[21]  Y. Ho.  Heuristics, rules of thumb, and the 80/20 proposition , 1994, IEEE Trans. Autom. Control..

[22]  Yu-Chi Ho,et al.  An ordinal optimization approach to optimal control problems , 1999, Autom..

[23]  T. W. E. Lau,et al.  Universal Alignment Probabilities and Subset Selection for Ordinal Optimization , 1997 .

[24]  J. Tsitsiklis,et al.  Intractable problems in control theory , 1986 .