New insights into Witsenhausen's counterexample

Optimization Letters

Volumn 6, Issue 7, 2012, Pages 1425-1446

  Gnecco, Giorgio ^a   Sanguineti, Marcello ^a  

^a UNIVERSITY OF GENOA   (Italy)

Author keywords

Decision in the presence of uncertainty; Distributed (or decentralized) optimization; Information structure; Suboptimal solutions; Team optimization

Indexed keywords

DECISION IN THE PRESENCE OF UNCERTAINTY; INFORMATION STRUCTURES; NUMERICAL RESULTS; OPTIMIZATION PROBLEMS; SUBOPTIMAL SOLUTION; TEAM OPTIMIZATION;

CONTROL; OPTIMIZATION;

STRUCTURAL OPTIMIZATION;

EID: 84867224336     PISSN: 18624472     EISSN: 18624480     Source Type: Journal    
DOI: 10.1007/s11590-011-0339-6     Document Type: Article

References (38)

1. Alessandri A., Sanguineti M.: Optimization of approximating networks for optimal fault diagnosis. Optim. Methods Softw. 20, 235-260 (2005).
2. Baglietto M., Parisini T., Zoppoli R.: Numerical solutions to the Witsenhausen counterexample by approximating networks. IEEE Trans. Autom. Control 46, 1471-1477 (2001).
3. Barron, A. R.: Neural net approximation. In: Narendra, K. (ed.) Proceedings of 7th Yale Workshops on Adaptive and Learning Systems, pp. 69-72. Yale University Press (1992).
4. Barron A. R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39, 930-945 (1993).
5. Bennet K. P., Mangasarian O. L.: Neural network training via linear programming. In: Pardalos, P. M. (eds) Advances in Optimization and Parallel Computing, pp. 56-67. North-Holland, Amsterdam (1992).
6. Chu K. C.: Team decision theory and information structures in optimal control, part II. IEEE Trans. Autom. Control 17, 22-28 (1972).
7. Daniel J. W.: The Approximate Minimization of Functionals. Prentice-Hall, Englewood Cliffs (1971).
8. Darken, C., Donahue, M., Gurvits, L., Sontag, E.: Rate of approximation results motivated by robust neural network learning. In: Proceedings of 6th Annual ACM Conference on Computational Learning Theory, pp. 303-309. The Association for Computing Machinery, New York (1993).
9. Deng M., Ho Y. C.: An ordinal optimization approach to optimal control problems. Automatica 35, 331-338 (1999).
10. Giulini S., Sanguineti M.: Approximation schemes for functional optimization problems. J. Optim. Theory Appl. 140, 33-54 (2009).
11. Gnecco, G., Sanguineti, M.: Suboptimal solutions to network team optimization problems. In: CD-Proc. Int. Network Optimization Conference (INOC) (2009).
12. Gnecco, G., Sanguineti, M.: Smooth optimal decision strategies for static team optimization problems and their approximations. In: Lecture Notes in Computer Science, vol. 5901, pp. 440-451. Springer, Berlin (2010).
13. Gnecco G., Sanguineti M.: Team optimization problems with Lipschitz continuous strategies. Optim. Lett. 5, 333-346 (2011).
14. Grover P., Sahai A.: Witsenhausen's counterexample as assisted interference suppression. Int. J. Syst. Control Commun. 2, 197-237 (2010).
15. Grover, P., Sahai, A.: Is Witsenhausen's counterexample a relevant toy? In: Proceedings of 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA, pp. 585-590, December 15-17, 2010.
16. Grover, P., Sahai, A., Park, S. Y.: The finite-dimensional Witsenhausen counterexample. In: Proc. 7th Int. Symp. on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), June 23-27, Seoul, Korea (2009).
17. Haykin S.: Neural Networks: A Comprehensive Foundation, 2nd edn. Prentice-Hall, Englewood Cliffs (1998).
18. Ho, Y. C.: Review of the Witsenhausen problem. In: Proceedings of 47th IEEE Conference on Decision and Control (CDC), pp. 1611-1613 (2008).
19. Ho Y. C., Chu K. C.: Team decision theory and information structures in optimal control problems, part I. IEEE Trans. Autom. Control 17, 15-22 (1972).
20. Jones L. K.: A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. Ann. Stat. 20, 608-613 (1992).
21. Juditsky A., Hjalmarsson H., Benveniste A., Delyon B., Ljung L., Sjöberg J., Zhang Q.: Nonlinear black-box models in system identification: mathematical foundations. Automatica 31, 1725-1750 (1995).
22. Kůrková V., Kainen P. C., Kreinovich V.: Estimates of the number of hidden units and variation with respect to half-spaces. Neural Netw. 10, 1061-1068 (1997).
23. Kůrková V., Sanguineti M.: Error estimates for approximate optimization by the extended Ritz method. SIAM J. Optim. 15, 261-287 (2005).
24. Kůrková V., Sanguineti M.: Learning with generalization capability by kernel methods of bounded complexity. J. Complex. 21, 350-367 (2005).
25. Kůrková V., Sanguineti M.: Approximate minimization of the regularized expected error over kernel models. Math. Oper. Res. 33, 747-756 (2008).
26. Kůrková V., Sanguineti M.: Geometric upper bounds on rates of variable-basis approximation. IEEE Trans. Inf. Theory 54, 5681-5688 (2008).
27. Kainen P. C., Kůrková V., Sanguineti M.: Complexity of Gaussian radial-basis networks approximating smooth functions. J. Complex. 25, 63-74 (2009).
28. Kim K. H., Roush F. W.: Team Theory. Ellis Horwood Limited, Chichester (1988).
29. Kolmogorov A. N., Fomin S. V.: Introductory Real Analysis. Dover, New York (1970).
30. Lee J. T., Lau E., Ho Y. C.: The Witsenhausen counterexample: a hierarchical search approach for nonconvex optimization problems. IEEE Trans. Autom. Control 46, 382-397 (2001).
31. Li, N., Marden, J. R., Shamma, J. S.: Learning approaches to the Witsenhausen counterexample from a view of potential games. In: Proceedings of 48th IEEE Conference on Decision and Control (CDC), pp. 157-162 (2009).
32. Murphey, R., Pardalos, P. M. (eds): Cooperative Control and Optimization. Kluwer, Dordrecht (2002).
33. Papadimitriou C. H., Tsitsiklis J. N.: Intractable problems in control theory. SIAM J. Control Optim. 24, 639-654 (1986).
34. Pisier, G.: Remarques sur un résultat non publié de B. Maurey. In: Séminaire d'Analyse Fonctionnelle 1980-81, vol. I, no. 12. École Polytechnique, Centre de Mathématiques, Palaiseau, France (1981).
35. Sanguineti M.: Universal approximation by ridge computational models: a survey. Open Appl. Math. J. 2, 31-58 (2008).
36. Sjöberg J., Zhang Q., Ljung L., Benveniste A., Glorennec P.-Y., Delyon B., Hjalmarsson H., Juditsky A.: Nonlinear black-box modeling in system identification: a unified overview. Automatica 31, 1691-1724 (1995).
37. Witsenhausen H. S.: A counterexample in stochastic optimum control. SIAM J. Control Optim. 6, 131-147 (1968).
38. Witsenhausen H. S.: Equivalent stochastic control problems. Math. Control Signals Syst. 1, 3-11 (1988).