Multistage stochastic programs via autoregressive sequences and individual probability constraints

Kaňková, Vlasta

Kybernetika, Tome 44 (2008) no. 2, pp. 151-170

Abstract (VO)

The paper deals with a special case of multistage stochastic programming problems. In particular, the paper deals with multistage stochastic programs in which a random element follows an autoregressive sequence and constraint sets correspond to the individual probability constraints. The aim is to investigate a stability (considered with respect to a probability measures space) and empirical estimates. To achieve new results the Wasserstein metric determined by ${\cal L}_{1}$ norm and results of multiobjective optimization theory are employed.

Détail

Classification : 90C15
Keywords: multistage stochastic programming problem; individual probability constraints; autoregressive sequence; Wasserstein metric; empirical estimates; multiobjective problems

Bibliographie

[1] Birge J. R., Louveuax F.: Introduction to Stochastic Programming. Springer, Berlin 1997 | MR

[2] Dupačová J.: Multistage stochastic programs: the state-of-the-art and selected bibliography. Kybernetika 31 (1995), 2, 151–174 | MR | Zbl

[3] Dupačová J., Hurt, J., Štěpán J.: Stochastic Modelling in Economics and Finance. Kluwer, Dordrecht 2002

[4] Dupačová J., Popela P.: Melt control: Charge optimization via stochastic programming. In: Applications of Stochastic Programming (S. W. Wallace and W. T. Ziemba, eds.), Philadelphia, SIAM and MPS 2005, pp. 277–298 | MR | Zbl

[5] Chovanec P.: Multistage Stochastic Programming Problems – Application to Unemployment Problem and Restructuralization (in Czech). Diploma Work. Faculty of Mathematics and Physics, Charles University, Prague 2004

[6] Ehrgott M.: Multicriteria Optimization. Second edition. Springer, Berlin 2005 | MR | Zbl

[7] Frauendorfer K., Schürle M.: Term structure in multistage stochastic programming: estimation and approximation. Ann. Oper. Res. 100 (2000), 1–4, 185–209

[8] Geoffrion A. M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22 (1968), 618–630 | MR | Zbl

[9] Kaňková V.: A Note on multifunction in stochastic programming. In: Stochastic Programming Methods and Technical Applications (K. Marti and P. Kall, eds.), Springer, Berlin 1996 | MR

[10] Kaňková V.: On the stability in stochastic programming: The case of individual probability constraints. Kybernetika 33 (1997), 5, 525–546 | MR | Zbl

[11] Kaňková V.: A note on multistage stochastic programming. In: Proc. 11th Joint Czech–Germany–Slovak Conference: Mathematical Methods in Economy and Industry. University of Technology, Liberec 1998, pp. 45–52

[12] Kaňková V.: A remark on the analysis of multistage stochastic programs, Markov dependence. Z. Angew. Math. Mech. 82 (2002), 781–793 | MR | Zbl

[13] Kaňková V., Šmíd M.: On approximation in multistage stochastic programs: Markov dependence. Kybernetika 40 (2004), 5, 625–638 | MR

[14] Kaňková V., Šmíd M.: A Remark on Approximation in Multistage Stochastic Programs; Markov Dependence. Research Report No. 2101, Institute of Information Theory and Automation, Prague 2002

[15] Kaňková V.: Multistage stochastic decision and economic proceses. Acta Oeconomica Pragensia 13 (2005), 1, 119–127

[16] Kaňková V., Houda M.: Empirical estimates in stochastic programming. In: Prague Stochastics 2006 (M. Hušková and M. Janžura, eds.), Matfyzpress, Prague 2006, pp. 426–436 | Zbl

[17] Kaňková V., Chovanec P.: Unemployment problem via multistage stochastic programming. In: Proc. Quantitative methods in Economics (Multiple Criteria Decision making XIII) (J. Pekár and M. Lukáčik, eds.), The Slovak Society for Operations Research and University of Economics in Bratislava, Bratislava 2006, pp. 69–76

[20] Kaňková V.: Empirical Estimates via Stability in Stochastic programming. Research Report No. 2192, Institute of Information Theory and Automation, Prague 2007

[21] Kaňková V.: Multistage stochastic programs via stochastic parametric optimization. In: Operations Research Proceedings 2007 (J. Kalcsics and S. Nickel, eds.), Springer, Berlin – Heidelberg 2008, pp. 63–68 | Zbl

[22] King A. J.: Stochastic programming problems: examples from the literature. In: Numerical Techniques for Stochastic Optimization Problems (Yu. Ermoliev and J.-B. Wets, eds.), Springer, Berlin 1968, pp. 255–266 | MR

[23] Kuhn D.: Generalized Bounds for Convex Multistage Stochastic Programs. (Lecture Notes in Economics and Mathematical Systems 548.) Springer, Berlin 2005 | MR | Zbl

[24] Nowak M. P., Römisch W.: Stochastic Lagrangian relaxation applied to power scheduling in hydro-termal system under uncertainty. Ann. Oper. Res. 100 (2000), 251–272 | MR

[25] Powell W. B., Topaloglu H.: Stochastic programming in transportation and logistics. In: Stochastic Programming, Handbooks in Operations Research and Management Science, Volume 10 (A. Ruszczyński and A. Shapiro, eds.), Elsevier, Amsterdam 2003, pp. 555–636 | MR

[26] Prékopa A.: Stochastic Programming. Akadémiai Kiadó, Budapest and Kluwer, Dordrecht 1995 | MR | Zbl

[27] Römisch W., Schulz R.: Stability of solutions for stochastic programs with complete recourse. Math. Oper. Res. 18 (1993), 590–609 | MR

[28] Salinetti G.: Approximations for chance constrained programming problems. Stochastics 10 (1983), 157–179 | MR | Zbl

[29] Serfling J. R.: Approximation Theorems of Mathematical Statistics. Wiley, New York 1980 | MR | Zbl

[30] Vallander S. S.: Calculation of the Wasserstein distance between probability distributions on the line (in Russian). Theory Probab. Appl. 18 (1973), 784–786 | MR

[31] Wallace S. W., Fleten S. E.: Stochastic programming models in energy. In: Stochastic Programming, Handbooks in Operations Research and Management Science, Volume 10 (A. Ruszczyński and A. Shapiro, eds.), Elsevier, Amsterdam 2003, pp. 637–677 | MR

[32] Wallace S. W., Ziemba W. T: Applications of Stochastic Programming. SIAM and MPS, Philadelphia 2005 | MR | Zbl