Approximative solutions of stochastic optimization problems

Petr Lachout

Kybernetika (2010)

  • Volume: 46, Issue: 3, page 513-523
  • ISSN: 0023-5954

Abstract

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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, -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.

How to cite

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Lachout, Petr. "Approximative solutions of stochastic optimization problems." Kybernetika 46.3 (2010): 513-523. <http://eudml.org/doc/196783>.

@article{Lachout2010,
abstract = {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.},
author = {Lachout, Petr},
journal = {Kybernetika},
keywords = {the optimal solution; $\varepsilon $-minimal solutions; level-minimal solutions; randomness; -minimal solution; level-minimal solution; randomness},
language = {eng},
number = {3},
pages = {513-523},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Approximative solutions of stochastic optimization problems},
url = {http://eudml.org/doc/196783},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Lachout, Petr
TI - Approximative solutions of stochastic optimization problems
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 3
SP - 513
EP - 523
AB - 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.
LA - eng
KW - the optimal solution; $\varepsilon $-minimal solutions; level-minimal solutions; randomness; -minimal solution; level-minimal solution; randomness
UR - http://eudml.org/doc/196783
ER -

References

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  1. Lachout, P., Stability of stochastic optimization problem – nonmeasurable case, Kybernetika 44 (2008), 2, 259–276. MR2428223
  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., 10.1007/BF02507027, Ann. Inst. Statist. Math. 57 (2005), 2, 291–313. MR2160652DOI10.1007/BF02507027
  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. MR1980125
  5. Robinson, S. M., 10.1287/moor.21.3.513, Math. Oper. Res. 21 (1996), 3, 513–528. Zbl0868.90087MR1403302DOI10.1287/moor.21.3.513
  6. Rockafellar, T., Wets, R. J.-B., Variational Analysis, Springer-Verlag, Berlin 1998. Zbl0888.49001MR1491362
  7. Vajda, I., Janžura, M., 10.1016/S0304-4149(97)00082-3, Stoch. Process. Appl. 72 (1997), 1, 27–45. MR1483610DOI10.1016/S0304-4149(97)00082-3
  8. Vaart, A. W. van der, Wellner, J. A., Weak Convergence and Empirical Processes, Springer, New York 1996. MR1385671

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