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Fast Bellman Iteration: An Application of Legendre-Fenchel Duality to Deterministic Dynamic Programming in Discrete Time

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Listed:
  • Ronaldo Carpio

    (School of Business and Finance, University of International Business and Economics)

  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

Abstract

We propose an algorithm, which we call " Fast Bellman Iteration " (FBI), to compute the value function of a deterministic infinite-horizon dynamic programming problem in discrete time. FBI is an efficient algorithm applicable to a class of multidimensional dynamic programming problems with concave return (or convex cost) functions and linear constraints. In this algorithm, a sequence of functions is generated starting from the zero function by repeatedly applying a simple algebraic rule involving the Legendre-Fenchel transform of the return function. The resulting sequence is guaranteed to converge, and the Legendre-Fenchel transform of the limiting function coincides with the value function.

Suggested Citation

  • Ronaldo Carpio & Takashi Kamihigashi, 2016. "Fast Bellman Iteration: An Application of Legendre-Fenchel Duality to Deterministic Dynamic Programming in Discrete Time," Discussion Paper Series DP2016-26, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:dp2016-26
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2016-26.pdf
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    References listed on IDEAS

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    1. Takashi Kamihigashi, 2014. "An order-theoretic approach to dynamic programming: an exposition," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 13-21, April.
    2. Takashi Kamihigashi & Kevin Reffett & Masayuki Yao, 2015. "An application of Kleene's fixed point theorem to dynamic programming," International Journal of Economic Theory, The International Society for Economic Theory, vol. 11(4), pages 429-434, December.
    3. Takashi Kamihigashi, 2008. "On the principle of optimality for nonstationary deterministic dynamic programming," International Journal of Economic Theory, The International Society for Economic Theory, vol. 4(4), pages 519-525, December.
    4. Klein, C. M. & Morin, T. L., 1991. "Conjugate duality and the curse of dimensionality," European Journal of Operational Research, Elsevier, vol. 50(2), pages 220-228, January.
    5. Takashi Kamihigashi, 2014. "Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 251-273, June.
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    Keywords

    Dynamic programming; Legendre-Fenchel transform; Bellman operator; Convex analysis;
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