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Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results

Author

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  • Mauro Gaggero

    (National Research Council of Italy)

  • Giorgio Gnecco

    (University of Genova)

  • Marcello Sanguineti

    (University of Genova)

Abstract

Value-function approximation is investigated for the solution via Dynamic Programming (DP) of continuous-state sequential N-stage decision problems, in which the reward to be maximized has an additive structure over a finite number of stages. Conditions that guarantee smoothness properties of the value function at each stage are derived. These properties are exploited to approximate such functions by means of certain nonlinear approximation schemes, which include splines of suitable order and Gaussian radial-basis networks with variable centers and widths. The accuracies of suboptimal solutions obtained by combining DP with these approximation tools are estimated. The results provide insights into the successful performances appeared in the literature about the use of value-function approximators in DP. The theoretical analysis is applied to a problem of optimal consumption, with simulation results illustrating the use of the proposed solution methodology. Numerical comparisons with classical linear approximators are presented.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:156:y:2013:i:2:d:10.1007_s10957-012-0118-2
    DOI: 10.1007/s10957-012-0118-2
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    1. Jerome Adda & Russell W. Cooper, 2003. "Dynamic Economics: Quantitative Methods and Applications," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012014, December.
    2. Karp, Larry & Lee, In Ho, 2001. "Learning-by-Doing and the Choice of Technology: The Role of Patience," Journal of Economic Theory, Elsevier, vol. 100(1), pages 73-92, September.
    3. Victoria C. P. Chen & David Ruppert & Christine A. Shoemaker, 1999. "Applying Experimental Design and Regression Splines to High-Dimensional Continuous-State Stochastic Dynamic Programming," Operations Research, INFORMS, vol. 47(1), pages 38-53, February.
    4. Martin L. Puterman & Moon Chirl Shin, 1978. "Modified Policy Iteration Algorithms for Discounted Markov Decision Problems," Management Science, INFORMS, vol. 24(11), pages 1127-1137, July.
    5. R. Zoppoli & M. Sanguineti & T. Parisini, 2002. "Approximating Networks and Extended Ritz Method for the Solution of Functional Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 403-440, February.
    6. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, December.
    7. Wim M. Nawijn, 1990. "Look-Ahead Policies for Admission to a Single Server Loss System," Operations Research, INFORMS, vol. 38(5), pages 854-862, October.
    8. Angelo Alessandri & Giorgio Gnecco & Marcello Sanguineti, 2010. "Minimizing Sequences for a Family of Functional Optimal Estimation Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 243-262, November.
    9. Sharon A. Johnson & Jery R. Stedinger & Christine A. Shoemaker & Ying Li & José Alberto Tejada-Guibert, 1993. "Numerical Solution of Continuous-State Dynamic Programs Using Linear and Spline Interpolation," Operations Research, INFORMS, vol. 41(3), pages 484-500, June.
    10. Boldrin, Michele & Montrucchio, Luigi, 1986. "On the indeterminacy of capital accumulation paths," Journal of Economic Theory, Elsevier, vol. 40(1), pages 26-39, October.
    11. Semmler, Willi & Sieveking, Malte, 2000. "Critical debt and debt dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 24(5-7), pages 1121-1144, June.
    12. 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.
    13. Michael Kopel & Gustav Feichtinger & Herbert Dawid, 1997. "Complex solutions of nonconcave dynamic optimization models (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(3), pages 427-439.
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    Cited by:

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    2. 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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