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Perturbation Theory and Undiscounted Markov Renewal Programming

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  • Paul J. Schweitzer

    (Institute for Defense Analyses, Arlington, Virginia)

Abstract

A recently-developed perturbation formalism for finite Markov chains is used here to analyze the policy iteration algorithm for undiscounted, single-chain Markov renewal programming. The relative values are shown to be essentially partial derivatives of the gain rate with respect to the transition probabilities, and they rank the states by indicating desirable changes in the probabilistic structure. This both implies the optimality of nonrandomized policies and suggests a gradient technique for optimizing the gain rate with respect to a parameter. The policy iteration algorithm is shown to be a steepest-ascent technique in policy space: the successor to a given policy is chosen in a direction that maximizes the directional derivative of the gain rate. The occurrence during policy improvement of the gain and relative values of the original policy is explained by their essentially determining the gradient of the gain rate.

Suggested Citation

  • Paul J. Schweitzer, 1969. "Perturbation Theory and Undiscounted Markov Renewal Programming," Operations Research, INFORMS, vol. 17(4), pages 716-727, August.
    • Handle: RePEc:inm:oropre:v:17:y:1969:i:4:p:716-727
    DOI: 10.1287/opre.17.4.716
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    Cited by:

    1. Dijk, N.M. van, 1991. "An improved error bound theorem for approximate Markov chains," Serie Research Memoranda 0084, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    2. Dijk, N.M. van, 1991. "On error bound analysis for transient continuous-time Markov reward structures," Serie Research Memoranda 0005, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.

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