An unbounded Berge's minimum theorem with applications to discounted Markov decision processes

Raúl Montes-de-Oca; Enrique Lemus-Rodríguez

Kybernetika (2012)

  • Volume: 48, Issue: 2, page 268-286
  • ISSN: 0023-5954

Abstract

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This paper deals with a certain class of unbounded optimization problems. The optimization problems taken into account depend on a parameter. Firstly, there are established conditions which permit to guarantee the continuity with respect to the parameter of the minimum of the optimization problems under consideration, and the upper semicontinuity of the multifunction which applies each parameter into its set of minimizers. Besides, with the additional condition of uniqueness of the minimizer, its continuity is given. Some examples of nonconvex optimization problems that satisfy the conditions of the article are supplied. Secondly, the theory developed is applied to discounted Markov decision processes with unbounded cost functions and with possibly noncompact actions sets in order to obtain continuous optimal policies. This part of the paper is illustrated with two examples of the controlled Lindley's random walk. One of these examples has nonconstant action sets.

How to cite

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Montes-de-Oca, Raúl, and Lemus-Rodríguez, Enrique. "An unbounded Berge's minimum theorem with applications to discounted Markov decision processes." Kybernetika 48.2 (2012): 268-286. <http://eudml.org/doc/247030>.

@article{Montes2012,
abstract = {This paper deals with a certain class of unbounded optimization problems. The optimization problems taken into account depend on a parameter. Firstly, there are established conditions which permit to guarantee the continuity with respect to the parameter of the minimum of the optimization problems under consideration, and the upper semicontinuity of the multifunction which applies each parameter into its set of minimizers. Besides, with the additional condition of uniqueness of the minimizer, its continuity is given. Some examples of nonconvex optimization problems that satisfy the conditions of the article are supplied. Secondly, the theory developed is applied to discounted Markov decision processes with unbounded cost functions and with possibly noncompact actions sets in order to obtain continuous optimal policies. This part of the paper is illustrated with two examples of the controlled Lindley's random walk. One of these examples has nonconstant action sets.},
author = {Montes-de-Oca, Raúl, Lemus-Rodríguez, Enrique},
journal = {Kybernetika},
keywords = {Berge's minimum theorem; moment function; discounted Markov decision process; uniqueness of the optimal policy; continuous optimal policy; Berge's minimum theorem; moment function; discounted Markov decision process; uniqueness of the optimal policy; continuous optimal policy},
language = {eng},
number = {2},
pages = {268-286},
publisher = {Institute of Information Theory and Automation AS CR},
title = {An unbounded Berge's minimum theorem with applications to discounted Markov decision processes},
url = {http://eudml.org/doc/247030},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Montes-de-Oca, Raúl
AU - Lemus-Rodríguez, Enrique
TI - An unbounded Berge's minimum theorem with applications to discounted Markov decision processes
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 2
SP - 268
EP - 286
AB - This paper deals with a certain class of unbounded optimization problems. The optimization problems taken into account depend on a parameter. Firstly, there are established conditions which permit to guarantee the continuity with respect to the parameter of the minimum of the optimization problems under consideration, and the upper semicontinuity of the multifunction which applies each parameter into its set of minimizers. Besides, with the additional condition of uniqueness of the minimizer, its continuity is given. Some examples of nonconvex optimization problems that satisfy the conditions of the article are supplied. Secondly, the theory developed is applied to discounted Markov decision processes with unbounded cost functions and with possibly noncompact actions sets in order to obtain continuous optimal policies. This part of the paper is illustrated with two examples of the controlled Lindley's random walk. One of these examples has nonconstant action sets.
LA - eng
KW - Berge's minimum theorem; moment function; discounted Markov decision process; uniqueness of the optimal policy; continuous optimal policy; Berge's minimum theorem; moment function; discounted Markov decision process; uniqueness of the optimal policy; continuous optimal policy
UR - http://eudml.org/doc/247030
ER -

References

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  1. C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis., Third Edition. Springer-Verlag, Berlin 2006. Zbl1156.46001MR2378491
  2. R. B. Ash, Real Variables with Basic Metric Space Topology., IEEE Press, New York 1993. Zbl0920.26002MR1193687
  3. J. P. Aubin, I. Ekeland, Applied Nonlinear Analysis., John Wiley, New York 1984. Zbl1115.47049MR0749753
  4. L. M. Ausubel, R. J. Deneckere, 10.1007/BF01213694, Econom. Theory 3 (1993), 99-107. Zbl1002.49500MR1211955DOI10.1007/BF01213694
  5. C. Berge, Topological Spaces., Oliver and Boyd, Edinburgh and London 1963 (reprinted by Dover Publications, Inc., Mineola, New York 1997). Zbl0114.38602MR1464690
  6. D. Cruz-Suárez, R. Montes-de-Oca, F. Salem-Silva, 10.1007/s001860400372, Math. Methods Oper. Res. 60 (2004), 415-436. Zbl1104.90053MR2106092DOI10.1007/s001860400372
  7. J. Dugundji, Topology., Allyn and Bacon, Inc., Boston 1966. Zbl0397.54003MR0193606
  8. P. K. Dutta, M. K.Majumdar, R. K. Sundaram, 10.1016/0165-1889(94)90048-5, J. Econom. Dynam. Control 18 (1994), 1069-1092. Zbl0875.90096MR1298092DOI10.1016/0165-1889(94)90048-5
  9. P. K. Dutta, T. Mitra, 10.1016/0304-4068(89)90006-2, J. Math. Econom. 18 (1989), 77-86. MR0985949DOI10.1016/0304-4068(89)90006-2
  10. O. Hernández-Lerma, J. B. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria., Springer-Verlag, New York 1996. Zbl0840.93001MR1363487
  11. O. Hernández-Lerma, W. J. Runggaldier, Monotone approximations for convex stochastic control problems., J. Math. Systems Estim. Control 4 (1994), 99-140. Zbl0812.93078MR1298550
  12. K. Hinderer, Lipschitz continuity of value functions in Markovian decision Processes., Math. Methods Oper. Res. 60 (2005), 3-22. Zbl1093.90075MR2226965
  13. K. Hinderer, M. Stieglitz, 10.1007/BF01194330, Math. Methods Oper. Res. 44 (1996), 189-204. Zbl0860.90126MR1409065DOI10.1007/BF01194330
  14. A. Horsley, A. J. Wrobel, T. Van Zandt, 10.1016/S0165-1765(98)00177-3, Econom. Lett. 61 (1998), 285-291. Zbl0913.90079MR1676329DOI10.1016/S0165-1765(98)00177-3
  15. J. S. Jordan, 10.2307/1912305, Econometrica 45 (1977), 1365-1376. Zbl0363.90035MR0456573DOI10.2307/1912305
  16. T. Kamihigashi, 10.1016/j.jmateco.2006.05.007, J. Math. Econom. 43 (2007), 477-500. Zbl1154.91032MR2317118DOI10.1016/j.jmateco.2006.05.007
  17. T. Kamihigashi, S. Roy, 10.1016/j.jet.2005.06.007, J. Econom. Theory 132 (2007), 435-460. Zbl1142.91667MR2285614DOI10.1016/j.jet.2005.06.007
  18. R. B. King, Beyond Quartic Equation., Birkhauser, Boston 1996. MR1401346
  19. M. Kitayev, Semi-Markov and jump Markov control models: average cost criterion., Theory Probab. Appl. 30 (1985), 272-288. MR0792619
  20. D. V. Lindley, The theory of queues with a single server., Proc. Cambridge Philos. Soc. 48 (1952), 277-289. Zbl0046.35501MR0046597
  21. M. Majumdar, R. Radner, 10.2307/1912118, Econometrica 51 (1983), 1821-1837. MR0720089DOI10.2307/1912118
  22. S. P. Meyn, 10.1137/0327073, SIAM J. Control Optim. 27 (1989), 1409-1439. MR1022436DOI10.1137/0327073
  23. E. A. Ok, Real Analysis with Economic Applications., Princeton University Press, Princeton 2007. Zbl1119.26001MR2275400
  24. A. L. Peressini, F. E. Sullivan, J. J. Uhl, The Mathematics of Nonlinear Programming., Springer-Verlag, New York 1988. Zbl0663.90054MR0932726
  25. M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming., John Wiley, New York 1994. Zbl1184.90170MR1270015
  26. U. Rieder, 10.1007/BF01168566, Manuscripta Math. 24 (1978), 115-131. Zbl0385.28005MR0493590DOI10.1007/BF01168566
  27. H. L. Royden, Real Analysis., Third Edition. Macmillan, New York 1988. Zbl1191.26002MR1013117
  28. R. H. Stockbridge, Time-average control of martingale problems: a linear programming formulation., Ann. Probab. 18 (1990), 291-314. Zbl0699.49019MR1043944
  29. R. Sundaram, A First Course in Optimization Theory., Cambridge University Press, Cambridge 1996. Zbl0885.90106MR1402910
  30. G. Tian, J. Zhou, 10.1016/0022-247X(92)90302-T, J. Math. Anal. Appl. 166 (1992), 351-364. Zbl0761.90110MR1160931DOI10.1016/0022-247X(92)90302-T
  31. G. Tian, J. Zhou, 10.1016/0304-4068(94)00687-6, J. Math. Econom. 24 (1995), 281-303. MR1320200DOI10.1016/0304-4068(94)00687-6
  32. M. Walker, 10.2307/2526431, Internat. Econom. Rev. 20 (1979), 267-272. Zbl0406.90001MR0525439DOI10.2307/2526431
  33. A. Yushkevich, 10.1137/S0363012995292469, SIAM J. Control Optim. 35 (1997), 2157-2182. Zbl0892.93059MR1478659DOI10.1137/S0363012995292469

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