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Characterization of the Optimal Risk-Sensitive Average Cost in Denumerable Markov Decision Chains

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  • Rolando Cavazos-Cadena

    (Departamento de Estadística y Cálculo, Universidad Autónoma Agraria Antonio Narro, Buenavista, Saltillo, Coahuila 25315, México)

Abstract

This work is concerned with Markov decision chains on a denumerable state space. The controller has a positive risk-sensitivity coefficient, and the performance of a control policy is measured by a risk-sensitive average cost criterion. Besides standard continuity-compactness conditions, it is assumed that the state process is communicating under any stationary policy, and that the simultaneous Doeblin condition holds. In this context, it is shown that if the cost function is bounded from below, and the superior limit average index is finite at some point, then (i) the optimal superior and inferior limit average value functions coincide and are constant, (ii) the optimal average cost is characterized via an extended version of the Collatz-Wielandt formula in the theory of positive matrices, and (iii) an optimality inequality is established, from which a stationary optimal policy is obtained. Moreover, an explicit example is given to show that, even if the cost function is bounded, the strict inequality may occur in the optimality relation.

Suggested Citation

  • Rolando Cavazos-Cadena, 2018. "Characterization of the Optimal Risk-Sensitive Average Cost in Denumerable Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 1025-1050, August.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:3:p:1025-1050
    DOI: 10.1287/moor.2017.0893
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    References listed on IDEAS

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    Cited by:

    1. Qingda Wei & Xian Chen, 2023. "Continuous-Time Markov Decision Processes Under the Risk-Sensitive First Passage Discounted Cost Criterion," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 309-333, April.
    2. Gustavo Portillo-Ramírez & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2023. "Contractive approximations in average Markov decision chains driven by a risk-seeking controller," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 98(1), pages 75-91, August.
    3. Julio Saucedo-Zul & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2020. "A Discounted Approach in Communicating Average Markov Decision Chains Under Risk-Aversion," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 585-606, November.

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