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On the reduction of total‐cost and average‐cost MDPs to discounted MDPs

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  • Eugene A. Feinberg
  • Jefferson Huang

Abstract

This article provides conditions under which total‐cost and average‐cost Markov decision processes (MDPs) can be reduced to discounted ones. Results are given for transient total‐cost MDPs with transition rates whose values may be greater than one, as well as for average‐cost MDPs with transition probabilities satisfying the condition that there is a state such that the expected time to reach it is uniformly bounded for all initial states and stationary policies. In particular, these reductions imply sufficient conditions for the validity of optimality equations and the existence of stationary optimal policies for MDPs with undiscounted total cost and average‐cost criteria. When the state and action sets are finite, these reductions lead to linear programming formulations and complexity estimates for MDPs under the aforementioned criteria.© 2017 Wiley Periodicals, Inc. Naval Research Logistics 66:38–56, 2019

Suggested Citation

  • Eugene A. Feinberg & Jefferson Huang, 2019. "On the reduction of total‐cost and average‐cost MDPs to discounted MDPs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(1), pages 38-56, February.
    • Handle: RePEc:wly:navres:v:66:y:2019:i:1:p:38-56
    DOI: 10.1002/nav.21743
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    References listed on IDEAS

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    1. B. Curtis Eaves & Arthur F. Veinott, 2014. "Maximum-Stopping-Value Policies in Finite Markov Population Decision Chains," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 597-606, August.
    2. Eugene A. Feinberg & Uriel G. Rothblum, 2012. "Splitting Randomized Stationary Policies in Total-Reward Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 129-153, February.
    3. Rommert Dekker & Arie Hordijk, 1992. "Recurrence Conditions for Average and Blackwell Optimality in Denumerable State Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 17(2), pages 271-289, May.
    4. Bruno Scherrer, 2016. "Improved and Generalized Upper Bounds on the Complexity of Policy Iteration," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 758-774, August.
    5. K. Hinderer & K.-H. Waldmann, 2003. "The critical discount factor for finite Markovian decision processes with an absorbing set," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(1), pages 1-19, April.
    6. Eugene A. Feinberg & Pavlo O. Kasyanov & Nina V. Zadoianchuk, 2012. "Average Cost Markov Decision Processes with Weakly Continuous Transition Probabilities," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 591-607, November.
    7. R. Dekker & A. Hordijk & F. M. Spieksma, 1994. "On the Relation Between Recurrence and Ergodicity Properties in Denumerable Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 539-559, August.
    8. Yinyu Ye, 2011. "The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 593-603, November.
    9. Uriel G. Rothblum & Peter Whittle, 1982. "Growth Optimality for Branching Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 7(4), pages 582-601, November.
    10. Alexander Zadorojniy & Guy Even & Adam Shwartz, 2009. "A Strongly Polynomial Algorithm for Controlled Queues," Mathematics of Operations Research, INFORMS, vol. 34(4), pages 992-1007, November.
    11. Stanley R. Pliska, 1976. "Optimization of Multitype Branching Processes," Management Science, INFORMS, vol. 23(2), pages 117-124, October.
    12. Uriel G. Rothblum, 1975. "Normalized Markov Decision Chains I; Sensitive Discount Optimality," Operations Research, INFORMS, vol. 23(4), pages 785-795, August.
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