Abstract
The approximation of the optimal policy functions is investigated for dynamic optimization problems with an objective that is additive over a finite number of stages. The distance between optimal and suboptimal values of the objective functional is estimated, in terms of the errors in approximating the optimal policy functions at the various stages. Smoothness properties are derived for such functions and exploited to choose the approximating families. The approximation error is measured in the supremum norm, in such a way to control the error propagation from stage to stage. Nonlinear approximators corresponding to Gaussian radial-basis-function networks with adjustable centers and widths are considered. Conditions are defined, guaranteeing that the number of Gaussians (hence, the number of parameters to be adjusted) does not grow “too fast” with the dimension of the state vector. The results help to mitigate the curse of dimensionality in dynamic optimization. An example of application is given and the use of the estimates is illustrated via a numerical simulation.
Similar content being viewed by others
References
Montrucchio, L.: Lipschitz continuous policy functions for strongly concave optimization problems. J. Math. Econ. 16, 259–273 (1987)
Bertsekas, D.P., Tsitsiklis, J.: Neuro-Dynamic Programming. Athena Scientific, Belmont (1996)
Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 2. Athena Scientific, Belmont (2007)
Powell, W.B.: Approximate Dynamic Programming—Solving the Curses of Dimensionality. John Wiley & Sons, Hoboken (2007)
Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)
Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont (2005)
Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, Berlin (1970)
Ekeland, I., Turnbull, T.: Infinite-Dimensional Optimization and Convexity. The University of Chicago Press, Chicago (1983)
Daniel, J.W.: The Approximate Minimization of Functionals. Prentice Hall, Englewood Cliffs (1971)
Alt, W.: On the approximation of infinite optimization problems with an application to optimal control problems. Appl. Math. Optim. 12, 15–27 (1984)
Zoppoli, R., Sanguineti, M., Parisini, T.: Approximating networks and extended Ritz method for the solution of functional optimization problems. J. Optim. Theory Appl. 112, 403–439 (2002)
Kůrková, V., Sanguineti, M.: Error estimates for approximate optimization by the extended Ritz method. SIAM J. Optim. 18, 461–487 (2005)
Giulini, S., Sanguineti, M.: Approximation schemes for functional optimization problems. J. Optim. Theory Appl. 140, 33–54 (2009)
Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice Hall, Englewood Cliffs (1963)
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)
Narendra, K.S., Mukhopadhyay, S.: Adaptive control using neural networks and approximate models. IEEE Trans. Neural Netw. 8, 475–485 (1997)
Barron, A.R.: Neural net approximation. In: Narendra, K. (ed.) Proc. 7th Yale Workshop on Adaptive and Learning Systems, pp. 69–72. Yale University Press, New Haven (1992)
Girosi, F., Anzellotti, G.: Rates of convergence for radial basis functions and neural networks. In: Mammone, R.J. (ed.) Artificial Neural Networks for Speech and Vision, pp. 97–114. Chapman and Hall, London (1993)
Hornik, K., Stinchcombe, M., White, H., Auer, P.: Degree of approximation results for feedforward networks approximating unknown mappings and their derivatives. Neural Comput. 6, 1262–1275 (1994)
Girosi, F.: Approximation error bounds that use VC-bounds. In: Proc. Int. Conf. on Artificial Neural Networks. pp. 295–302, EC2 & Cie, Paris (1995)
Pinkus, A.: Approximation theory of the MLP model in neural networks. Acta Numer. 8, 143–195 (1999)
Kůrková, V., Sanguineti, M.: Comparison of worst-case errors in linear and neural network approximation. IEEE Trans. Inform. Theory 48, 264–275 (2002)
Mhaskar, H.N.: On the tractability of multivariate integration and approximation by neural networks. J. Complex. 20, 561–590 (2004)
Kainen, P.C., Kůrková, V., Sanguineti, M.: Complexity of Gaussian radial basis networks approximating smooth functions. J. Complex. 25, 63–74 (2009)
Kainen, P.C., Kůrková, V., Sanguineti, M.: Estimates of approximation rates by Gaussian radial-basis functions. In: Lecture Notes in Computer Science, vol. 4432, pp. 11–18. Springer, Berlin (2007)
Kůrková, V., Sanguineti, M.: Geometric upper bounds on rates of variable-basis approximation. IEEE Trans. Inform. Theory 54, 5681–5688 (2008)
Kůrková, V., Sanguineti, M.: Approximate minimization of the regularized expected error over kernel models. Math. Oper. Res. 33, 747–756 (2008)
Alessandri, A., Parisini, T., Sanguineti, M., Zoppoli, R.: Neural strategies for nonlinear optimal filtering. In: Proc. IEEE Int. Conf. Syst. Eng., pp. 44–49, Kobe, Japan, 1992
Zoppoli, R., Parisini, T.: Learning techniques and neural networks for the solution of N-stage nonlinear nonquadratic optimal control problems. In: Isidori, A., Tarn, T.J. (eds.) Systems, Models and Feedback: Theory and Applications, pp. 193–210. Birkhäuser, Boston (1992)
Zolezzi, T.: Condition numbers and Ritz type methods in unconstrained optimization. Control Cybern. 36, 811–822 (2007)
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)
Alessandri, A., Sanguineti, M., Maggiore, M.: Optimization-based learning with bounded error for feedforward neural networks. IEEE Trans. Neural Netw. 13, 261–273 (2002)
Chow, T.W.S., Cho, S.-Y.: Neural Networks and Computing: Learning Algorithms and Applications. World Scientific, Singapore (2007)
Dacorogna, B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004)
Rudin, W.: Real and Complex Analysis. McGraw-Hill, Singapore (1987)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Cugno, F., Montrucchio, L.: Scelte Intertemporali. Teoria e modelli. Carocci Editore, Rome (1998)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)
Montrucchio, L.: Thompson metric, contraction property, and differentiability of policy functions. J. Econ. Behav. Organ. 33, 449–466 (1998)
Gnecco, G., Sanguineti, M.: Value and policy function approximations in infinite-horizon optimization problems. J. Dyn. Syst. Geom. Theories 6, 123–147 (2008)
Stokey, N.L., Lucas, R.E., Prescott, E.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge (1989)
Pinkus, A.: n-Widths in Approximation Theory. Springer, Berlin (1985)
Giulini, S., Sanguineti, M.: On dimension-independent approximation by neural networks and linear approximators. In: Proc. Int. Joint Conference on Neural Networks, pp. I283–I288 (2000)
Gnecco, G., Sanguineti, M.: Approximation error bounds via Rademacher’s complexity. Appl. Math. Sci. 2, 153–176 (2008)
Traub, J.F., Werschulz, A.G.: Complexity and Information. Cambridge University Press, Cambridge (1999)
Gnecco, G., Sanguineti, M.: Estimates of the approximation error via Rademacher complexity: Learning vector-valued functions. J. Inequal. Appl. 2008, 640758 (2008)
Ljungqvist, L., Sargent, T.: Recursive Macroeconomic Theory. MIT Press, Cambridge (2004)
Pervozvanskii, A.A., Gaitsgori, V.G.: Theory of Suboptimal Decisions: Decomposition and Aggregation. Kluwer Academic, Norwell (1988)
Li, D., Sun, X.: Nonlinear Integer Programming. Springer, New York (2006)
Burger, M., Neubauer, A.: Error bounds for approximation with neural networks. J. Approx. Theory 112, 235–250 (2001)
Araujo, A.: The once but not twice differentiability of the policy function. Econometrica 59, 1383–1393 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Zirilli.
The authors were partially supported by a grant “Progetti di Ricerca di Ateneo 2008” of the University of Genoa, Project “Solution of Functional Optimization Problems by Nonlinear Approximators and Learning from Data”.
Rights and permissions
About this article
Cite this article
Gnecco, G., Sanguineti, M. Suboptimal Solutions to Dynamic Optimization Problems via Approximations of the Policy Functions. J Optim Theory Appl 146, 764–794 (2010). https://doi.org/10.1007/s10957-010-9680-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-010-9680-7