Monotonicity of minimizers in optimization problems with applications to Markov control processes

Flores–Hernández, Rosa M.   ; Montes-de-Oca, Raúl

Kybernetika, Tome 43 (2007) no. 3, pp. 347-368

Abstract (VO)

Firstly, in this paper there is considered a certain class of possibly unbounded optimization problems on Euclidean spaces, for which conditions that permit to obtain monotone minimizers are given. Secondly, the theory developed in the first part of the paper is applied to Markov control processes (MCPs) on real spaces with possibly unbounded cost function, and with possibly noncompact control sets, considering both the discounted and the average cost as optimality criterion. In the context described, conditions to obtain monotone optimal policies are provided. For the conditions of MCPs presented in the article, several controlled models including, in particular, two inventory/production systems and the linear regulator problem are supplied.

Détail

Classification : 90C40, 93E20
Keywords: monotone minimizer in an optimization problem; Markov control process; total discounted cost; average cost; monotone optimal policy

Bibliographie

[1] Ash R. B.: Real Variables with Basic Metric Space Topology. IEEE Press, New York 1993 | MR | Zbl

[2] Cruz-Suárez D., Montes-de-Oca, R., Salem-Silva F.: Conditions for the uniqueness of optimal policies of discounted Markov decision processes. Math. Methods Oper. Res. 60 (2004), 415–436 | MR | Zbl

[3] Cruz-Suárez D., Montes-de-Oca, R., Salem-Silva F.: Pointwise approximations of discounted Markov decision processes to optimal policies. Internat. J. Pure Appl. Math. 28 (2006), 265–281 | MR | Zbl

[4] Fu M. C., Marcus S. I., Wang, I-J: Monotone optimal policies for a transient queueing staffing problem. Oper. Res. 48 (2000), 327–331

[5] Gallish E.: On monotone optimal policies in a queueing model of M/G/1 type with controllable service time distribution. Adv. in Appl. Probab. 11 (1979), 870–887 | MR

[6] Hernández-Lerma O., Lasserre J. B.: Discrete-Time Markov Control Processes. Springer-Verlag, New York 1996 | MR | Zbl

[7] Heyman D. P., Sobel M. J.: Stochastic Models in Operations Research, Vol. II. Stochastic Optimization. McGraw-Hill, New York 1984 | Zbl

[8] Hinderer K., Stieglitz M.: Increasing and Lipschitz continuous minimizers in one-dimensional linear-convex systems without constraints: The continuous and the discrete case. Math. Methods Oper. Res. 44 (1996), 189–204 | MR | Zbl

[9] Kalin D.: A note on ‘monotone optimal policies for Markov decision processes’. Math. Programming 15 (1978), 220–222 | MR | Zbl

[10] Mendelssohn R., Sobel M.: Capital accumulation and the optimization of renewable resource models. J. Econom. Theory 23 (1980), 243–260 | Zbl

[11] Pittenger A. O.: Monotonicity in a Markov decision process. Math. Oper. Res. 13 (1988), 65–73 | MR | Zbl

[12] Porteus E. L.: Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford, Calif. 2002

[13] Puterman M. L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York 1994 | MR | Zbl

[14] Rieder U.: Measurable selection theorems for optimization problems. Manuscripta Math. 24 (1978), 115–131 | MR | Zbl

[15] Ross S. M.: Introduction to Stochastic Dynamic Programming. Academic Press, San Diego 1983 | MR | Zbl

[16] Serfozo R. F.: Monotone optimal policies for Markov decision processes. Math. Programming Stud. 6 (1976), 202–215 | MR | Zbl

[17] Stidham, Sh., Weber R. R.: Monotonic and insensitive optimal policies for control of queues with undiscounted costs. Oper. Res. 37 (1989), 611–625 | MR | Zbl

[18] Stromberg K. R.: An Introduction to Classical Real Analysis. Wadsworth International Group, Belmont 1981 | MR | Zbl

[19] Sundaram R. K.: A First Course in Optimization Theory. Cambridge University Press, Cambridge 1996 | MR | Zbl

[20] Topkis D. M.: Minimizing a submodular function on a lattice. Oper. Res. 26 (1978), 305–321 | MR | Zbl

[21] Topkis D. M.: Supermodularity and Complementarity. Princeton University Press, Princeton, N. J. 1988 | MR