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Compactness of the space of non-randomized policies in countable-state sequential decision processes

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  • Richard Chen
  • Eugene Feinberg

Abstract

For sequential decision processes with countable state spaces, we prove compactness of the set of strategic measures corresponding to nonrandomized policies. For the Borel state case, this set may not be compact (Piunovskiy, Optimal control of random sequences in problems with constraints. Kluwer, Boston, p. 170, 1997) in spite of compactness of the set of strategic measures corresponding to all policies (Schäl, On dynamic programming: compactness of the space of policies. Stoch Processes Appl 3(4):345–364, 1975b; Balder, On compactness of the space of policies in stochastic dynamic programming. Stoch Processes Appl 32(1):141–150, 1989). We use the compactness result from this paper to show the existence of optimal policies for countable-state constrained optimization of expected discounted and nonpositive rewards, when the optimality is considered within the class of nonrandomized policies. This paper also studies the convergence of a value-iteration algorithm for such constrained problems. Copyright Springer-Verlag 2010

Suggested Citation

  • Richard Chen & Eugene Feinberg, 2010. "Compactness of the space of non-randomized policies in countable-state sequential decision processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(2), pages 307-323, April.
    • Handle: RePEc:spr:mathme:v:71:y:2010:i:2:p:307-323
    DOI: 10.1007/s00186-009-0298-1
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    References listed on IDEAS

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    1. Balder, Erik J., 1989. "On compactness of the space of policies in stochastic dynamic programming," Stochastic Processes and their Applications, Elsevier, vol. 32(1), pages 141-150, June.
    2. Eugene A. Feinberg & Adam Shwartz, 1996. "Constrained Discounted Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 21(4), pages 922-945, November.
    3. Richard Chen & Eugene Feinberg, 2007. "Non-randomized policies for constrained Markov decision processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(1), pages 165-179, August.
    4. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
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    Cited by:

    1. 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.
    2. 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.

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