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On sets of countable non-negative matrices and Markov decision processes

Published online by Cambridge University Press:  01 July 2016

Douglas P. Kennedy
Douglas P. Kennedy*
Affiliation:
University of Cambridge
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Abstract

Consider a set S of countable non-negative matrices satisfying the property that for any two indices i, j, for some n ≧ 1 there are matrices M1, M2, · · ·, Mn in S with (M1M2 · · · Mn)ij >0. For non-negative vectors x set Tx = supM∈SMx, where the supremum is taken separately in each coordinate. Assume that for each x with Tx finite in each coordinate there is a matrix in S which achieves the supremum simultaneously for all coordinates. With these two assumptions on S, the R-theory for a countable irreducible matrix is extended to the operator T. The results are used to consider the existence of stationary optimal policies for Markov decision processes with multiplicative rewards.

Keywords

NON-NEGATIVE MATRICESR-THEORYPERRON–FROBENIUS THEORYDYNAMIC PROGRAMMINGMARKOV DECISION PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 3 , September 1978 , pp. 633 - 646
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Bather, J. (1973) Optimal decision procedures for finite Markov chains, II. Adv. Appl. Prob. 5, 521–540.CrossRefGoogle Scholar
[2] Bellman, R. (1956) On a class of quasi-linear equations. Canadian J. Maths 8, 198–202.CrossRefGoogle Scholar
[3] Bellman, R. (1957) Dynamic Programming. Princeton University Press, Princeton, N.J. Google ScholarPubMed
[4] Hordijk, A. (1974) Dynamic Programming and Markov Potential Theory. Mathematisch Centrum, Amsterdam.Google Scholar
[5] Howard, P. A. and Matheson, J. E. (1972) Risk-sensitive Markov decision processes. Management Sci. 18, 356–369.CrossRefGoogle Scholar
[6] Mandl, P. and Seneta, E. (1969) The theory of non-negative matrices in a dynamic programming problem. Austral. J. Statist. 11, 85–96.CrossRefGoogle Scholar
[7] Seneta, E. (1973) Non-Negative Matrices. Allen and Unwin, London.Google Scholar
[8] Vere-Jones, D. (1967) Ergodic properties of non-negative matrices I. Pacific J. Maths 22, 361–386.CrossRefGoogle Scholar