×

Optimal and suboptimal solutions to stochastically uncertain problems of quintile optimization. (English. Russian original) Zbl 1143.93027

Autom. Remote Control 68, No. 7, 1145-1157 (2007); translation from Avtom. Telemekh. 2007, No. 7, 31-43 (2007).
Summary: Problems of stochastic optimization under incomplete information on distribution of random perturbations with the quintile and probability criteria are considered. The minimax approach is used when optimal solutions are chosen. Conditions for equivalency of direct and inverse problems of stochastic optimization under incomplete statistical information are studied. The solution method for statistically uncertain problems of optimization with the quintile criterion basing on the use of generalized confidence sets for statistically uncertain random quantities is proposed. The use of confidence sets for finding suboptimal solutions to the problem of stochastic optimization under incomplete information is considered. Examples of the application of obtained relations are represented.

MSC:

93E20 Optimal stochastic control
93C41 Control/observation systems with incomplete information
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Krasovskii, N.N., Igrovye zadachi o vstreche dvizhenii (Game Problems of Motion Contact), Moscow: Nauka, 1970.
[2] Kats, I.Ya. and Kurzhanskii, A.B., Minimax Estimation in Multistep Systems, Dokl. Akad. Nauk SSSR, 1975, vol. 221, no. 3, pp. 535–538.
[3] Raik, E., Problems of Stochastic Programming with Decision Functions, Izv. Akad. Nauk ESSR, 1972, vol. 21, pp. 258–263.
[4] Malyshev, V.V. and Kibzun, A.I., Analiz i sintez vysokotochnogo upravleniya letatel’nymi apparatami (Analysis and Synthesis of High-Accuracy Control of Flight Vehicles), Moscow: Mashinostroenie, 1987.
[5] Kibzun, A.I. and Kan, Yu.S., Stochastic Programming Problems with Probability and Quantile Functions, Chichester: Wiley, 1996. · Zbl 0885.90088
[6] Kan, Yu.S. and Kibzun, A.I., Convex Properties of Probability and Quantile Functions in Optimization Problems, Avtom. Telemekh., 1996, no. 3, pp. 82–102. · Zbl 0927.90089
[7] Kan, Yu.S. and Mistryukov, A.A., Qualitative Investigation on Functions of Probability and Quantile, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 1996, no. 3, pp. 36–40. · Zbl 0925.93953
[8] Kan, Yu.S., Control Optimization by the Quantile Criterion, Avtom. Telemekh., 2001, no. 5, pp. 82–102.
[9] Precopa, A., Stochastic Programming, Amsterdam: Kluwer, 1995.
[10] Yudin, D.B., Zadachi i metody stokhasticheskogo programmirovaniya (Problems and Methods of Stochastic Programming), Moscow: Sovetskoe Radio, 1979. · Zbl 0486.90058
[11] Bakhshiyan, B.Ts., Nazirov, R.R., and El’yasberg, P.E., Opredelenie i korrektsiya dvizheniya (Determinition and Correction of Motion), Moscow: Nauka, 1980. · Zbl 0743.49008
[12] Timofeeva, G.A., Generalized Confidence Sets for a Statistically Indeterminate Random Vector, Avtom. Telemekh., 2002, no. 6, pp. 44–56. · Zbl 1106.62317
[13] Shiryaev, A.N., Veroyatnost’ (Probability), Moscow: Nauka, 1980.
[14] Tomifeeva, G.A., Problems of Quintile Control and Generalized Confidence Sets under Incomplete Information, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2002, no. 6, pp. 48–54.
[15] Ermoliev, Yu., Norkin, V., and Wets, R.J.-B., The Minimization of the Semi-Continious Functions: Mollifier Subgradients, SIAM J. Control Optim., 1995, vol. 1, pp. 149–167. · Zbl 0822.49016 · doi:10.1137/S0363012992238369
[16] Szantai, T., A Computer Code for Solution of Probabilistic Constrained Stochastic Programming Problems, in Numerical Techniques for Stochastic Optimization, Ermoliev, Yu. and West, R., Eds., New York: Springer, 1988, pp. 229–235.
[17] Gaivoronski, A.A., Linearization Methods for Optimization Functionals which Depend on Probability Measures, Math. Program. Study, 1986, vol. 28, pp. 157–181. · Zbl 0596.90071 · doi:10.1007/BFb0121130
[18] Dupacova, J., Stochastic Programming with Incomplete Information, Laxenburg: IIASA, 1986, working paper-86-008.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.