Previous |  Up |  Next


multistage stochastic programming problem; individual probability constraints; autoregressive sequence; Wasserstein metric; empirical estimates; multiobjective problems
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.
[1] Birge J. R., Louveuax F.: Introduction to Stochastic Programming. Springer, Berlin 1997 MR 1460264
[2] Dupačová J.: Multistage stochastic programs: the state-of-the-art and selected bibliography. Kybernetika 31 (1995), 2, 151–174 MR 1334507 | Zbl 0860.90093
[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 2162956 | Zbl 1190.90063
[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 2143243 | Zbl 1132.90001
[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 0229453 | Zbl 0181.22806
[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 1650784
[10] Kaňková V.: On the stability in stochastic programming: The case of individual probability constraints. Kybernetika 33 (1997), 5, 525–546 MR 1603961 | Zbl 0908.90198
[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 1944422 | Zbl 1028.90031
[13] Kaňková V., Šmíd M.: On approximation in multistage stochastic programs: Markov dependence. Kybernetika 40 (2004), 5, 625–638 MR 2121001
[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 1162.90528
[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 1209.90282
[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 0957334
[23] Kuhn D.: Generalized Bounds for Convex Multistage Stochastic Programs. (Lecture Notes in Economics and Mathematical Systems 548.) Springer, Berlin 2005 MR 2103400 | Zbl 1103.90069
[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 1843544
[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 2052761
[26] Prékopa A.: Stochastic Programming. Akadémiai Kiadó, Budapest and Kluwer, Dordrecht 1995 MR 1375234 | Zbl 1219.90114
[27] Römisch W., Schulz R.: Stability of solutions for stochastic programs with complete recourse. Math. Oper. Res. 18 (1993), 590–609 MR 1250562
[28] Salinetti G.: Approximations for chance constrained programming problems. Stochastics 10 (1983), 157–179 MR 0727452 | Zbl 0536.90067
[29] Serfling J. R.: Approximation Theorems of Mathematical Statistics. Wiley, New York 1980 MR 0595165 | Zbl 1001.62005
[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 0328982
[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 2052762
[32] Wallace S. W., Ziemba W. T: Applications of Stochastic Programming. SIAM and MPS, Philadelphia 2005 MR 2162941 | Zbl 1068.90002
Partner of
EuDML logo