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Provably Near-Optimal Approximation Schemes for Implicit Stochastic and Sample-Based Dynamic Programs

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  • Nir Halman

    (School of Business Administration, Hebrew University of Jerusalem, 91905 Jerusalem, Israel)

Abstract

In this paper, we address two models of nondeterministic discrete time finite-horizon dynamic programs (DPs): implicit stochastic DPs (the information about the random events is given by value oracles to their cumulative distribution functions) and sample-based DPs (the information about the random events is deduced by drawing random samples). Such data-driven models frequently appear in practice, where the cumulative distribution functions of the underlying random variables are either unavailable or too complicated to work with. In both models, the single-period cost functions are accessed via value oracle calls and assumed to possess either monotone or convex structure. We develop the first near-optimal relative approximation schemes for each of the two models. Applications in stochastic inventory control (that is, several variants of the so-called newsvendor problem) are discussed in detail. Our results are achieved by a combination of Bellman equation calculations, density estimation results, and extensions of the technique of K -approximation sets and functions introduced by Halman et al. (2009) [Halman N, Klabjan D, Mostagir M, Orlin J, Simchi-Levi D (2009) A fully polynomial time approximation scheme for single-item stochastic inventory control with discrete demand. Math. Oper. Res. 34(3):674–685.].

Suggested Citation

  • Nir Halman, 2020. "Provably Near-Optimal Approximation Schemes for Implicit Stochastic and Sample-Based Dynamic Programs," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1157-1181, October.
    • Handle: RePEc:inm:orijoc:v:32:y:4:i:2020:p:1157-1181
    DOI: 10.1287/ijoc.2019.0926
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    References listed on IDEAS

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    1. Khouja, Moutaz, 1999. "The single-period (news-vendor) problem: literature review and suggestions for future research," Omega, Elsevier, vol. 27(5), pages 537-553, October.
    2. Nir Halman & James B. Orlin & David Simchi-Levi, 2012. "Approximating the Nonlinear Newsvendor and Single-Item Stochastic Lot-Sizing Problems When Data Is Given by an Oracle," Operations Research, INFORMS, vol. 60(2), pages 429-446, April.
    3. Nir Halman & Diego Klabjan & Mohamed Mostagir & Jim Orlin & David Simchi-Levi, 2009. "A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 674-685, August.
    4. Qin, Yan & Wang, Ruoxuan & Vakharia, Asoo J. & Chen, Yuwen & Seref, Michelle M.H., 2011. "The newsvendor problem: Review and directions for future research," European Journal of Operational Research, Elsevier, vol. 213(2), pages 361-374, September.
    5. Retsef Levi & Georgia Perakis & Joline Uichanco, 2015. "The Data-Driven Newsvendor Problem: New Bounds and Insights," Operations Research, INFORMS, vol. 63(6), pages 1294-1306, December.
    6. Wang Chi Cheung & David Simchi-Levi, 2019. "Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 668-692, May.
    7. Gerhard J. Woeginger, 2000. "When Does a Dynamic Programming Formulation Guarantee the Existence of a Fully Polynomial Time Approximation Scheme (FPTAS)?," INFORMS Journal on Computing, INFORMS, vol. 12(1), pages 57-74, February.
    8. Retsef Levi & Robin O. Roundy & David B. Shmoys, 2007. "Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models," Mathematics of Operations Research, INFORMS, vol. 32(4), pages 821-839, November.
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