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Estimation of the Necessary Sample Size for Approximation of Stochastic Optimization Problems with Probabilistic Criteria

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Mathematical Optimization Theory and Operations Research (MOTOR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

Stochastic optimization problems with probabilistic and quantile objective functions are considered. The probability objective function is defined as the probability that the value of losses does not exceed a fixed level. The quantile function is defined as the minimal value of losses that cannot be exceeded with a fixed probability. Sample approximations of the considered problems are formulated. A method to estimate the accuracy of the approximation of the probability maximization and quantile minimization is described for the case of a finite set of feasible strategies. Based on this method, we estimate the necessary sample size to obtain (with a given probability) an epsilon-optimal strategy to the original problems by solving their approximations in the cases of finite set of feasible strategies. Also, the necessary sample size is obtained for the probability maximization in the case of a bounded infinite set of feasible strategies and a Lipschitz continuous probability function.

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Acknowledgements

The work is supported by the Russian Foundation for Basic Research (project 19-07-00436).

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), Volokolamskoe Shosse, 4, A-80, GSP-3, 125993, Moscow, Russia

    Sergey V. Ivanov & Irina D. Zhenevskaya

Authors
  1. Sergey V. Ivanov
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  2. Irina D. Zhenevskaya
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Correspondence to Sergey V. Ivanov .

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Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Ivanov, S.V., Zhenevskaya, I.D. (2019). Estimation of the Necessary Sample Size for Approximation of Stochastic Optimization Problems with Probabilistic Criteria. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_39

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  • DOI: https://doi.org/10.1007/978-3-030-22629-9_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22628-2

  • Online ISBN: 978-3-030-22629-9

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