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Spline Approximations To Value Functions

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  • TRICK, MICHAEL A.
  • ZIN, STANLEY E.

Abstract

We review the properties of algorithms that characterize the solution of the Bellman equation of a stochastic dynamic program, as the solution to a linear program. The variables in this problem are the ordinates of the value function; hence, the number of variables grows with the state space. For situations in which this size becomes computationally burdensome, we suggest the use of low-dimensional cubic-spline approximations to the value function. We show that fitting this approximation through linear programming provides upper and lower bounds on the solution to the original large problem. The information contained in these bounds leads to inexpensive improvements in the accuracy of approximate solutions.

Suggested Citation

  • Trick, Michael A. & Zin, Stanley E., 1997. "Spline Approximations To Value Functions," Macroeconomic Dynamics, Cambridge University Press, vol. 1(1), pages 255-277, January.
    • Handle: RePEc:cup:macdyn:v:1:y:1997:i:01:p:255-277_00
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    2. Jiaqiao Hu & Michael C. Fu & Vahid R. Ramezani & Steven I. Marcus, 2007. "An Evolutionary Random Policy Search Algorithm for Solving Markov Decision Processes," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 161-174, May.
    3. Alejandro Toriello & William B. Haskell & Michael Poremba, 2014. "A Dynamic Traveling Salesman Problem with Stochastic Arc Costs," Operations Research, INFORMS, vol. 62(5), pages 1107-1125, October.
    4. Lars Grüne & Willi Semmler, 2007. "Asset pricing with dynamic programming," Computational Economics, Springer;Society for Computational Economics, vol. 29(3), pages 233-265, May.
    5. Willi Semmler & Lars Grüne, 2004. "Asset Pricing with Delayed Consumption Decisions," Computing in Economics and Finance 2004 59, Society for Computational Economics.
    6. Cai, Yongyang & Judd, Kenneth L. & Lontzek, Thomas S. & Michelangeli, Valentina & Su, Che-Lin, 2017. "A Nonlinear Programming Method For Dynamic Programming," Macroeconomic Dynamics, Cambridge University Press, vol. 21(2), pages 336-361, March.
    7. Nikolov, Boris & Schmid, Lukas & Steri, Roberto, 2021. "The Sources of Financing Constraints," Journal of Financial Economics, Elsevier, vol. 139(2), pages 478-501.
    8. Oleksandr Shlakhter & Chi-Guhn Lee & Dmitry Khmelev & Nasser Jaber, 2010. "Acceleration Operators in the Value Iteration Algorithms for Markov Decision Processes," Operations Research, INFORMS, vol. 58(1), pages 193-202, February.
    9. Yongyang Cai & Kenneth Judd, 2015. "Dynamic programming with Hermite approximation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(3), pages 245-267, June.
    10. Daniela Pucci de Farias & Benjamin Van Roy, 2006. "A Cost-Shaping Linear Program for Average-Cost Approximate Dynamic Programming with Performance Guarantees," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 597-620, August.
    11. Mrkaic, Mico, 2002. "Policy iteration accelerated with Krylov methods," Journal of Economic Dynamics and Control, Elsevier, vol. 26(4), pages 517-545, April.
    12. Grune, Lars & Semmler, Willi, 2004. "Using dynamic programming with adaptive grid scheme for optimal control problems in economics," Journal of Economic Dynamics and Control, Elsevier, vol. 28(12), pages 2427-2456, December.
    13. Qihang Lin & Selvaprabu Nadarajah & Negar Soheili, 2020. "Revisiting Approximate Linear Programming: Constraint-Violation Learning with Applications to Inventory Control and Energy Storage," Management Science, INFORMS, vol. 66(4), pages 1544-1562, April.
    14. Nikolov, Boris & Schmid, Lukas & Steri, Roberto, 2019. "Dynamic corporate liquidity," Journal of Financial Economics, Elsevier, vol. 132(1), pages 76-102.
    15. Selvaprabu Nadarajah & François Margot & Nicola Secomandi, 2015. "Relaxations of Approximate Linear Programs for the Real Option Management of Commodity Storage," Management Science, INFORMS, vol. 61(12), pages 3054-3076, December.
    16. Alemdar, Nedim M. & Sirakaya, Sibel & Husseinov, Farhad, 2006. "Optimal time aggregation of infinite horizon control problems," Journal of Economic Dynamics and Control, Elsevier, vol. 30(4), pages 569-593, April.
    17. Daniela Pucci de Farias & Benjamin Van Roy, 2004. "On Constraint Sampling in the Linear Programming Approach to Approximate Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 462-478, August.
    18. D. P. de Farias & B. Van Roy, 2003. "The Linear Programming Approach to Approximate Dynamic Programming," Operations Research, INFORMS, vol. 51(6), pages 850-865, December.

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