# Article

Full entry | PDF   (0.4 MB)
Keywords:
the optimal solution; $\varepsilon$-minimal solutions; level-minimal solutions; randomness
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.
References:
[1] Lachout, P.: Stability of stochastic optimization problem – nonmeasurable case. Kybernetika 44 (2008), 2, 259–276. MR 2428223
[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.
[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. DOI 10.1007/BF02507027 | MR 2160652
[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
[5] Robinson, S. M.: Analysis of sample-path optimization. Math. Oper. Res. 21 (1996), 3, 513–528. DOI 10.1287/moor.21.3.513 | MR 1403302 | Zbl 0868.90087
[6] Rockafellar, T., Wets, R. J.-B.: Variational Analysis. Springer-Verlag, Berlin 1998. MR 1491362 | Zbl 0888.49001
[7] Vajda, I., Janžura, M.: On asymptotically optimal estimates for general observations. Stoch. Process. Appl. 72 (1997), 1, 27–45. DOI 10.1016/S0304-4149(97)00082-3 | MR 1483610
[8] Vaart, A. W. van der, Wellner, J. A.: Weak Convergence and Empirical Processes. Springer, New York 1996. MR 1385671

Partner of