Article

 Title: Approximative solutions of stochastic optimization problems (English) Author: Lachout, Petr Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 46 Issue: 3 Year: 2010 Pages: 513-523 Summary lang: English . Category: math . Summary: The aim of this paper is to present some ideas how to relax the notion of the optimal solution of the stochastic optimization problem. In the deterministic case, $\varepsilon$-minimal solutions and level-minimal solutions are considered as desired relaxations. We call them approximative solutions and we introduce some possibilities how to combine them with randomness. Relations among random versions of approximative solutions and their consistency are presented in this paper. No measurability is assumed, therefore, treatment convenient for nonmeasurable objects is employed. (English) Keyword: the optimal solution Keyword: $\varepsilon$-minimal solutions Keyword: level-minimal solutions Keyword: randomness MSC: 60F99 MSC: 62F12 MSC: 90C15 MSC: 90C31 idZBL: Zbl 1229.90110 idMR: MR2676087 . Date available: 2010-09-13T17:01:29Z Last updated: 2013-09-21 Stable URL: http://hdl.handle.net/10338.dmlcz/140765 . Reference: [1] Lachout, P.: Stability of stochastic optimization problem – nonmeasurable case.Kybernetika 44 (2008), 2, 259–276. MR 2428223 Reference: [2] Lachout, P.: Stochastic optimization sensitivity without measurability.In: Proc. 15th MMEI held in Herlány, Slovakia (K. Cechlárová, M. Halická, V. Borbelóvá, and V. Lacko, eds.) 2007, pp. 131–136. Reference: [3] Lachout, P., Liebscher, E., Vogel, S.: Strong convergence of estimators as $\varepsilon _n$-minimizers of optimization problems.Ann. Inst. Statist. Math. 57 (2005), 2, 291–313. MR 2160652, 10.1007/BF02507027 Reference: [4] Lachout, P., Vogel, S.: On continuous convergence and epi-convergence of random functions.Part I: Theory and relations. Kybernetika 39 (2003), 1, 75–98. MR 1980125 Reference: [5] Robinson, S. M.: Analysis of sample-path optimization.Math. Oper. Res. 21 (1996), 3, 513–528. Zbl 0868.90087, MR 1403302, 10.1287/moor.21.3.513 Reference: [6] Rockafellar, T., Wets, R. J.-B.: Variational Analysis.Springer-Verlag, Berlin 1998. Zbl 0888.49001, MR 1491362 Reference: [7] Vajda, I., Janžura, M.: On asymptotically optimal estimates for general observations.Stoch. Process. Appl. 72 (1997), 1, 27–45. MR 1483610, 10.1016/S0304-4149(97)00082-3 Reference: [8] Vaart, A. W. van der, Wellner, J. A.: Weak Convergence and Empirical Processes.Springer, New York 1996. MR 1385671 .

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