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Keywords:
stochastic programming problems; stability; Wasserstein metric; ${\cal L}_{1}$ norm; Lipschitz property; empirical estimates; convergence rate; linear and nonlinear dependence; probability and risk constraints; stochastic dominance
stochastic programming problems; stability; Wasserstein metric; ${\cal L}_{1}$ norm; Lipschitz property; empirical estimates; convergence rate; linear and nonlinear dependence; probability and risk constraints; stochastic dominance
Summary:
Optimization problems depending on a probability measure correspond to many applications. These problems can be static (single-stage), dynamic with finite (multi-stage) or infinite horizon, single- or multi-objective. It is necessary to have complete knowledge of the “underlying” probability measure if we are to solve the above-mentioned problems with precision. However this assumption is very rarely fulfilled (in applications) and consequently, problems have to be solved mostly on the basis of data. Stochastic estimates of an optimal value and an optimal solution can only be obtained using this approach. Properties of these estimates have been investigated many times. In this paper we intend to study one-stage problems under unusual (corresponding to reality, however) assumptions. In particular, we try to compare the achieved results under the assumptions of thin and heavy tails in the case of problems with linear and nonlinear dependence on the probability measure, problems with probability and risk measure constraints, and the case of stochastic dominance constraints. Heavy-tailed distributions quite often appear in financial problems [26] while nonlinear dependence frequently appears in problems with risk measures [22, 30]. The results we introduce follow mostly from the stability results based on the Wasserstein metric with the ``underlying" $ {\cal L}_{1}$ norm. Theoretical results are completed by a simulation investigation.
References:
[1] Barrio, E., Giné, E., Matrán, E.: Central limit theorems for a Wasserstein distance between empirical and the true distributions. Ann. Probab. 27 (1999), 2, 1009-1071. DOI 10.1214/aop/1022677394 | MR 1698999
[2] Billingsley, P.: Ergodic Theory and Information. John Wiley and Sons, New York 1965. MR 0192027 | Zbl 0184.43301
[3] Birge, J. R., Louveaux, F.: Introduction in Stochastic Programming. Springer, Berlin 1992.
[4] Dai, L., Chen, C.-H., Birge, J. R.: Convergence properties of two-stage stochastic programming. J. Optim. Theory Appl. 106 (2000), 489-509. DOI 10.1023/a:1004649211111 | MR 1797371 | Zbl 0980.90057
[5] Dentcheva, D., Ruszczynski, A.: Porfolio optimization with stochastic dominance constraints. J. Banking and Finance 30 (2006), 433-451. DOI 10.1016/j.jbankfin.2005.04.024
[6] Dupačová, J., B.Wets, R. J.: Asymptotic behaviour of statistical estimates and optimal solutions of stochastic optimization problems. Ann. Statist. 16 (1984), 1517-1549. DOI 10.1214/aos/1176351052 | MR 0964937
[7] Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and the classical multinomial estimate. Ann. Math. Statist. 56 (1956), 642-669. DOI 10.1214/aoms/1177728174 | MR 0083864
[8] Ermoliev, Y. M., Norkin, V.: Sample everage approximation method for compound stochastic optimization problems. SIAM J. Optim. 23 (2013), 4, 2231-2263. DOI 10.1137/120863277 | MR 3129765
[9] Gut, A.: Probability: A Graduate Course. Springer, New York 2005. MR 2125120 | Zbl 1267.60001
[10] Houda, M.: Stability and Approximations for Stochastic Programs. Doctoral Thesis, Faculty of Mathematics and Physics, Charles University Prague, Prague 2009.
[11] Houda, M., Kaňková, V.: Empirical estimates in economic and financial optimization problems. Bull. Czech Econometr. Soc. 19 (2012), 29, 50-69.
[12] Kaniovski, Y. M., King, A. J., Wets, R. J.-B.: Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems. Ann. Oper. Res. 56 (1995), 189-208. DOI 10.1007/bf02031707 | MR 1339792 | Zbl 0835.90055
[13] Kaňková, V.: Optimum solution of a stochastic optimization problem. In: Trans. 7th Prague Conf. 1974, Academia, Prague 1977, pp. 239-244. MR 0519478 | Zbl 0408.90060
[14] Kaňková, V.: An approximative solution of stochastic optimization problem. In: Trans. 8th Prague Conf., Academia, Prague 1978, pp. 349-353. MR 0536792
[15] Kaňková, V., Lachout, P.: Convergence rate of empirical estimates in stochastic programming. Informatica 3 (1992), 4, 497-523. MR 1243755 | Zbl 0906.90133
[16] Kaňková, V.: Stability in stochastic programming - the case of unknown location parameter. Kybernetika 29 (1993), 1, 97-112. MR 1227744 | Zbl 0803.90096
[17] Kaňková, V.: A note on estimates in stochastic programming. J. Comput. Appl. Math. 56 (1994), 97-112. DOI 10.1016/0377-0427(94)90381-6 | MR 1338638 | Zbl 0824.90104
[18] Kaňková, V.: On the stability in stochastic programming: the case of individual probability constraints. Kybernetika 33 (1997), 5, 525-544. MR 1603961 | Zbl 0908.90198
[19] Kaňková, V., Houda, M.: Empirical estimates in stochastic programming. In: Proc. Prague Stochastics 2006 (M. Hušková and M. Janžura, eds.), MATFYZPRESS, Prague 2006, pp. 426-436. Zbl 1162.90528
[20] Kaňková, V., Houda, M.: Dependent samples in empirical estimation of stochastic programming problems. Austrian J. Statist. 35 (2006), 2 - 3, 271-279.
[21] Kaňková, V.: Empirical estimates in stochastic programming via distribution tails. Kybernetika 46 (2010), 3, 459-471. MR 2676083
[22] Kaňková, V.: Empirical estimates in optimization problems: survey with special regard to heavy tails and dependent samples. Bull. Czech Econometr. Soc. 19 (2012), 30, 92-111.
[23] Kaňková, V.: Risk measures in optimization problems via empirical estimates. Czech Econom. Rev. VII (2013), 3, 162-177.
[24] Klebanov, L. B.: Heavy Tailed Distributions. MATFYZPRESS, Prague 2003.
[25] Meerschaert, M. M., H.-P.Scheffler: Limit Distributions for Sums of Independent Random Vectors (Heavy Tails in Theory and Practice). John Wiley and Sons, New York 2001. MR 1840531
[26] Meerschaert, M. M., H.-P.Scheffler: Portfolio Modelling with Heavy Tailed Random Vectors. In: Handbook of Heavy Tailed Distributions in Finance (S. T. Rachev, ed.), Elsevier, Amsterdam 2003, pp. 595-640.
[27] Meerschaert, M. M., H.-P.Scheffler: Portfolio Modeling with Heavy Tailed Random Vectors. In: Handbook of Heavy Tailed Distributions in Finance (S. T. Rachev, ed.), Elsevier, Amsterdam 2003, pp. 595-640.
[28] Pflug, G. Ch.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. Ser. B 89 (2001), 251-271. DOI 10.1007/pl00011398 | MR 1816503
[29] Pflug, G. Ch.: Stochastic Optimization and Statistical Inference. In: Handbooks in Operations Research and Managemennt 10, Stochastic Programming (A. Ruszczynski and A. A. Shapiro, eds.) Elsevier, Amsterdam 2003, pp. 427-480. MR 2052759
[30] Pflug, G. Ch., Römisch, W.: Modeling Measuring and Managing Risk. World Scientific Publishing Co. Pte. Ltd, New Jersey 2007. MR 2424523 | Zbl 1153.91023
[31] Rachev, S. T., Römisch, W.: Quantitative stability and stochastic programming: the method of probabilistic metrics. Math. Oper. Res. 27 (2002), 792-818. DOI 10.1287/moor.27.4.792.304 | MR 1939178
[32] Rockafellar, R., Wets, R. J. B.: Variational Analysis. Springer, Berlin 1983. Zbl 0888.49001
[33] Römisch, W., Wakolbinger, A.: Obtaining Convergence Rate for Approximation in Stochastic Programming. In: Parametric Optimization and Related Topics (J. Guddat, H. Th. Jongen, B. Kummer and F. Nožička, eds.), Akademie-Verlag, Berlin 1987, pp. 327-343. MR 0909737
[34] Römisch, W.: Stability of Stochastic Programming Problems. In: Handbooks in Operations Research and Managemennt Science 10, Stochastic Programming (A. Ruszczynski and A. A. Shapiro, eds.) Elsevier, Amsterdam 2003, pp. 483-554. MR 2052760
[35] Salinetti, G., Wets, R. J.-B.: On the convergence of sequence of convex sets in finite dimensions. SIAM Rev. 21 (1979), 16-33. DOI 10.1137/1021002 | MR 0516381
[36] Samarodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, New York 1994.
[37] Schulz, R.: Rates of convergence in stochastic programs with complete integer recourse. SIAM J. Optim. 6 (1996), 4, 1138-1152. DOI 10.1137/s1052623494271655 | MR 1416533
[38] Shapiro, A.: Quantitative stability in stochastic programming. Math. Program. 67 (1994), 99-108. DOI 10.1007/bf01582215 | MR 1300821 | Zbl 0828.90099
[39] Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation. Optimization 57 (2008), 395-418. DOI 10.1080/02331930801954177 | MR 2412074 | Zbl 1145.90047
[40] Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming (Modeling and Theory). Published by Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia 2009. MR 2562798 | Zbl 1302.90003
[41] Shiryaev, A. N.: Essential of Stochastic Finance (Facts, Models, Theory). World Scientific, New Jersey 2008. MR 1695318
[42] Shorack, G. R., Wellner, J. A.: Empirical Processes and Applications to Statistics. Wiley, New York 1986. MR 0838963
[43] Šmíd, M.: The expected loss in the discretezation of multistage stochastic programming problems - estimation and convergence rate. Ann. Oper. Res. 165 (2009), 1, 29-45. DOI 10.1007/s10479-008-0355-9 | MR 2470981
[44] Wets, R. J.-B.: A Statistical Approach to the Solution of Stochastic Programs with (Convex) Simple Recourse. Research Report, University of Kentucky 1974.
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