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The Expected Total Cost Criterion for Markov Decision Processes under Constraints: A Convex Analytic Approach

Part of: Markov processes

Published online by Cambridge University Press:  04 January 2016

François Dufour ,
M. Horiguchi  and
A. B. Piunovskiy
François Dufour*
Affiliation:
Université Bordeaux, IMB and INRIA Bordeaux Sud-ouest
M. Horiguchi*
Affiliation:
Kanagawa University
A. B. Piunovskiy*
Affiliation:
University of Liverpool
*
∗ Postal address: INRIA Bordeaux Sud-ouest, CQFD Team, 351 cours de la Libération, F-33400 Talence, France. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan. Email address: [email protected]
∗∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK. Email address: [email protected]
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Abstract

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This paper deals with discrete-time Markov decision processes (MDPs) under constraints where all the objectives have the same form of expected total cost over the infinite time horizon. The existence of an optimal control policy is discussed by using the convex analytic approach. We work under the assumptions that the state and action spaces are general Borel spaces, and that the model is nonnegative, semicontinuous, and there exists an admissible solution with finite cost for the associated linear program. It is worth noting that, in contrast to the classical results in the literature, our hypotheses do not require the MDP to be transient or absorbing. Our first result ensures the existence of an optimal solution to the linear program given by an occupation measure of the process generated by a randomized stationary policy. Moreover, it is shown that this randomized stationary policy provides an optimal solution to this Markov control problem. As a consequence, these results imply that the set of randomized stationary policies is a sufficient set for this optimal control problem. Finally, our last main result states that all optimal solutions of the linear program coincide on a special set with an optimal occupation measure generated by a randomized stationary policy. Several examples are presented to illustrate some theoretical issues and the possible applications of the results developed in the paper.

Keywords

Markov decision processexpected total cost criterionconstraintlinear programmingoccupation measure

MSC classification

Primary: 90C40: Markov and semi-Markov decision processes
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90C90: Applications of mathematical programming
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 774 - 793
Copyright
© Applied Probability Trust 

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