# Article

 Title: Thin and heavy tails in stochastic programming (English) Author: Kaňková, Vlasta Author: Houda, Michal Language: English Journal: Kybernetika ISSN: 0023-5954 (print) ISSN: 1805-949X (online) Volume: 51 Issue: 3 Year: 2015 Pages: 433-456 Summary lang: English . Category: math . 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. (English) Keyword: stochastic programming problems Keyword: stability Keyword: Wasserstein metric Keyword: ${\cal L}_{1}$ norm Keyword: Lipschitz property Keyword: empirical estimates Keyword: convergence rate Keyword: linear and nonlinear dependence Keyword: probability and risk constraints Keyword: stochastic dominance MSC: 90C15 idZBL: Zbl 06487089 idMR: MR3391678 DOI: 10.14736/kyb-2015-3-0433 . Date available: 2015-09-01T09:13:03Z Last updated: 2016-01-03 Stable URL: http://hdl.handle.net/10338.dmlcz/144379 . Reference: [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. MR 1698999, 10.1214/aop/1022677394 Reference: [2] Billingsley, P.: Ergodic Theory and Information..John Wiley and Sons, New York 1965. Zbl 0184.43301, MR 0192027 Reference: [3] Birge, J. 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