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Constrained Markov decision processes in Borel spaces: from discounted to average optimality

Author

Listed:
  • Armando F. Mendoza-Pérez

    (UNACH)

  • Héctor Jasso-Fuentes

    (CINVESTAV–IPN)

  • Omar A. De-la-Cruz Courtois

    (UNACH)

Abstract

In this paper we study discrete-time Markov decision processes in Borel spaces with a finite number of constraints and with unbounded rewards and costs. Our aim is to provide a simple method to compute constrained optimal control policies when the payoff functions and the constraints are of either: infinite-horizon discounted type and average (a.k.a. ergodic) type. To deduce optimality results for the discounted case, we use the Lagrange multipliers method that rewrites the original problem (with constraints) into a parametric family of discounted unconstrained problems. Based on the dynamic programming technique as long with a simple use of elementary differential calculus, we obtain both suitable Lagrange multipliers and a family of control policies associated to these multipliers, this last family becomes optimal for the original problem with constraints. We next apply the vanishing discount factor method in order to obtain, in a straightforward way, optimal control policies associated to the average problem with constraints. Finally, to illustrate our results, we provide a simple application to linear–quadratic systems (LQ-systems).

Suggested Citation

  • Armando F. Mendoza-Pérez & Héctor Jasso-Fuentes & Omar A. De-la-Cruz Courtois, 2016. "Constrained Markov decision processes in Borel spaces: from discounted to average optimality," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(3), pages 489-525, December.
    • Handle: RePEc:spr:mathme:v:84:y:2016:i:3:d:10.1007_s00186-016-0551-3
    DOI: 10.1007/s00186-016-0551-3
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    References listed on IDEAS

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    3. Dutta, P.K., 1991. "What Do Discounted Optima Converge To? A Theory of Discount Rate Asymptotics in Economic Models," RCER Working Papers 264, University of Rochester - Center for Economic Research (RCER).
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