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Suboptimal Solutions to Dynamic Optimization Problems via Approximations of the Policy Functions

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  • G. Gnecco

    (University of Genoa)

  • M. Sanguineti

    (University of Genoa)

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.

Suggested Citation

  • G. Gnecco & M. Sanguineti, 2010. "Suboptimal Solutions to Dynamic Optimization Problems via Approximations of the Policy Functions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 764-794, September.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:3:d:10.1007_s10957-010-9680-7
    DOI: 10.1007/s10957-010-9680-7
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    References listed on IDEAS

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    1. Araujo, A, 1991. "The Once but Not Twice Differentiability of the Policy Function," Econometrica, Econometric Society, vol. 59(5), pages 1383-1393, September.
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    5. Vēra Kůrková & Marcello Sanguineti, 2008. "Approximate Minimization of the Regularized Expected Error over Kernel Models," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 747-756, August.
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    8. Montrucchio, Luigi, 1998. "Thompson metric, contraction property and differentiability of policy functions," Journal of Economic Behavior & Organization, Elsevier, vol. 33(3-4), pages 449-466, January.
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    Cited by:

    1. Mauro Gaggero & Giorgio Gnecco & Marcello Sanguineti, 2014. "Approximate dynamic programming for stochastic N-stage optimization with application to optimal consumption under uncertainty," Computational Optimization and Applications, Springer, vol. 58(1), pages 31-85, May.
    2. Mauro Gaggero & Giorgio Gnecco & Marcello Sanguineti, 2013. "Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 380-416, February.
    3. Giorgio Gnecco & Berna Tuncay & Fabio Pammolli, 2018. "A Comparison of Game-Theoretic Models for Parallel Trade," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-57, September.
    4. Giorgio Gnecco & Fabio Pammolli & Berna Tuncay, 2022. "Welfare and research and development incentive effects of uniform and differential pricing schemes," Computational Management Science, Springer, vol. 19(2), pages 229-268, June.
    5. Giorgio Gnecco, 2016. "On the Curse of Dimensionality in the Ritz Method," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 488-509, February.
    6. Yuqing Zheng & Guoshan Zhang, 2020. "Suboptimal Control for Nonlinear Systems with Disturbance via Integral Sliding Mode Control and Policy Iteration," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 652-677, May.
    7. Andrea Bacigalupo & Giorgio Gnecco & Marco Lepidi & Luigi Gambarotta, 2020. "Machine-Learning Techniques for the Optimal Design of Acoustic Metamaterials," Journal of Optimization Theory and Applications, Springer, vol. 187(3), pages 630-653, December.

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