Sample path average optimality of Markov control processes with strictly unbounded cost

Oscar Vega-Amaya

Applicationes Mathematicae (1999)

  • Volume: 26, Issue: 4, page 363-381
  • ISSN: 1233-7234

Abstract

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We study the existence of sample path average cost (SPAC-) optimal policies for Markov control processes on Borel spaces with strictly unbounded costs, i.e., costs that grow without bound on the complement of compact subsets. Assuming only that the cost function is lower semicontinuous and that the transition law is weakly continuous, we show the existence of a relaxed policy with 'minimal' expected average cost and that the optimal average cost is the limit of discounted programs. Moreover, we show that if such a policy induces a positive Harris recurrent Markov chain, then it is also sample path average (SPAC-) optimal. We apply our results to inventory systems and, in a particular case, we compute explicitly a deterministic stationary SPAC-optimal policy.

How to cite

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Vega-Amaya, Oscar. "Sample path average optimality of Markov control processes with strictly unbounded cost." Applicationes Mathematicae 26.4 (1999): 363-381. <http://eudml.org/doc/219246>.

@article{Vega1999,
abstract = {We study the existence of sample path average cost (SPAC-) optimal policies for Markov control processes on Borel spaces with strictly unbounded costs, i.e., costs that grow without bound on the complement of compact subsets. Assuming only that the cost function is lower semicontinuous and that the transition law is weakly continuous, we show the existence of a relaxed policy with 'minimal' expected average cost and that the optimal average cost is the limit of discounted programs. Moreover, we show that if such a policy induces a positive Harris recurrent Markov chain, then it is also sample path average (SPAC-) optimal. We apply our results to inventory systems and, in a particular case, we compute explicitly a deterministic stationary SPAC-optimal policy.},
author = {Vega-Amaya, Oscar},
journal = {Applicationes Mathematicae},
keywords = {strictly unbounded costs; sample path average cost criterion; inventory systems; Markov control processes},
language = {eng},
number = {4},
pages = {363-381},
title = {Sample path average optimality of Markov control processes with strictly unbounded cost},
url = {http://eudml.org/doc/219246},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Vega-Amaya, Oscar
TI - Sample path average optimality of Markov control processes with strictly unbounded cost
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 4
SP - 363
EP - 381
AB - We study the existence of sample path average cost (SPAC-) optimal policies for Markov control processes on Borel spaces with strictly unbounded costs, i.e., costs that grow without bound on the complement of compact subsets. Assuming only that the cost function is lower semicontinuous and that the transition law is weakly continuous, we show the existence of a relaxed policy with 'minimal' expected average cost and that the optimal average cost is the limit of discounted programs. Moreover, we show that if such a policy induces a positive Harris recurrent Markov chain, then it is also sample path average (SPAC-) optimal. We apply our results to inventory systems and, in a particular case, we compute explicitly a deterministic stationary SPAC-optimal policy.
LA - eng
KW - strictly unbounded costs; sample path average cost criterion; inventory systems; Markov control processes
UR - http://eudml.org/doc/219246
ER -

References

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  1. A. Arapostathis et al. (1993), Discrete time controlled Markov processes with an average cost criterion: A survey, SIAM J. Control Optim. 31, 282-344. Zbl0770.93064
  2. D. P. Bertsekas (1987), Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall, Englewood Cliffs, NJ. Zbl0649.93001
  3. D. P. Bertsekas and S. E. Shreve (1978), Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York. Zbl0471.93002
  4. P. Billingsley (1968), Convergence of Probability Measures, Wiley. Zbl0172.21201
  5. V. S. Borkar (1991), Topics in Controlled Markov Chains, Pitman Res. Notes Math. Ser. 240, Longman Sci. Tech. Zbl0725.93082
  6. R. Cavazos-Cadena and E. Fernández-Gaucherand (1995), Denumerable controlled Markov chains with average reward criterion : sample path optimality, Z. Oper. Res. 41, 89-108. Zbl0835.90116
  7. R. M. Dudley (1989), Real Analysis and Probability, Wadsworth & Brooks. Zbl0686.60001
  8. P. Hall and C. C. Heyde (1980), Martingale Limit Theory and Its Application, Academic Press. Zbl0462.60045
  9. O. Hernández-Lerma (1993), Existence of average optimal policies in Markov control processes with strictly unbounded costs, Kybernetika 29, 1-17. Zbl0792.93120
  10. O. Hernández-Lerma and J. B. Lasserre (1995), Invariant probabilities for Feller-Markov chains, J. Appl. Math. Stochastic Anal. 8, 341-345. Zbl0870.60061
  11. O. Hernández-Lerma and J. B. Lasserre (1996), Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer, New York. Zbl0840.93001
  12. O. Hernández-Lerma and J. B. Lasserre (1997), Policy iteration for average cost Markov control processes on Borel spaces, Acta Appl. Math., to appear. Zbl0872.93080
  13. O. Hernández-Lerma and M. Muñoz-de-Osak (1992), Discrete-time Markov con- trol processes with discounted unbounded cost: optimality criteria Kybernetika 28, 191-212. Zbl0771.93054
  14. O. Hernández-Lerma, O. Vega-Amaya and G. Carrasco (1998), Sample-path optimality and variance-minimization of average cost Markov control processes, Reporte Interno #236, Departamento de Matemáticas, CINVESTAV-IPN, México City. Zbl0951.93074
  15. K. Hinderer (1970), Foundations of Non-Stationary Dynamic Programming with Discrete Time Parameters, Lecture Notes in Oper. Res. and Math. Systems 33, Springer, Berlin. Zbl0202.18401
  16. J. B. Lasserre (1997), Sample-path average optimality for Markov control processes, Report No. 97102, LAAS-CNRS, Toulouse. Zbl0956.93066
  17. H. L. Lee and S. Nahmias (1993), Single-product, single-location models, in: Logistic of Production and Inventory, S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin (eds.), Handbooks in Operations Research and Management Science, Vol. 4, North-Holland, 3-51. 
  18. P. Mandl and M. Lausmanová (1991), Two extensions of asymptotic methods in controlled Markov chains, Ann. Oper. Res. 28, 67-80. Zbl0754.60081
  19. S. P. Meyn (1989), Ergodic theorems for discrete time stochastic systems using a stochastic Lyapunov function, SIAM J. Control Optim. 27, 1409-1439. Zbl0681.60067
  20. S. P. Meyn (1995), The policy iteration algorithm for average reward Markov decision processes with general state space, preprint, Coordinated Science Laboratory, University of Illinois, Urbana, IL. 
  21. S. P. Meyn and R. L. Tweedie (1993), Markov Chains and Stochastic Stability, Springer, London. Zbl0925.60001
  22. M. Parlar and R. Rempała (1992), Stochastic inventory problem with piecewise quadratic holding cost function containing a cost-free interval, J. Optim. Theory Appl. 75, 133-153. Zbl0795.90014
  23. O. Vega-Amaya and R. Montes-de-Oca (1998), Application of average dynamic programming to inventory systems, Math. Methods Oper. Res. 47, 451-471. Zbl0940.90007

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