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Diskretn. Anal. Issled. Oper., 2018, Volume 25, Number 4, Pages 27–45 (Mi da907)  

A bilevel stochastic programming problem with random parameters in the Follower's objective function

S. V. Ivanov

Moscow Aviation Institute (National Research University), 4 Volokolamskoe Shosse, 125993 Moscow, Russia

Abstract: Under study is a bilevel stochastic linear programming problem with quantile criterion. Bilevel programming problems can be considered as formalization of the process of interaction between two parties. The first party is a Leader making a decision first; the second is a Follower making a decision knowing the Leader's strategy and the realization of the random parameters. It is assumed that the Follower's problem is linear if the realization of the random parameters and the Leader's strategy are given. The aim of the Leader is the minimization of the quantile function of a loss function that depends on his own strategy and the optimal Follower's strategy. It is shown that the Follower's problem has a unique solution with probability 1 if the distribution of the random parameters is absolutely continuous. The lower-semicontinuity of the loss function is proved and some conditions are obtained of the solvability of the problem under consideration. Some example shows that the continuity of the quantile function cannot be provided. The sample average approximation of the problem is formulated. The conditions are given to provide that, as the sample size increases, the sample average approximation converges to the original problem with respect to the strategy and the objective value. It is shown that the convergence conditions hold for almost all values of the reliability level. A model example is given of determining the tax rate, and the numerical experiments are executed for this example. Tab. 1, illustr. 2, bibliogr. 13.

Keywords: stochastic programming, bilevel problem, quantile criterion, value-at-risk, sample average approximation.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-07-00203а
The author was supported by the Russian Foundation for Basic Research (project no. 17-07-00203a).


DOI: https://doi.org/10.17377/daio.2018.25.596

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English version:
Journal of Applied and Industrial Mathematics, 2018, 12:4, 658–667

Document Type: Article
UDC: 519.8
Received: 16.10.2017
Revised: 19.04.2018

Citation: S. V. Ivanov, “A bilevel stochastic programming problem with random parameters in the Follower's objective function”, Diskretn. Anal. Issled. Oper., 25:4 (2018), 27–45; J. Appl. Industr. Math., 12:4 (2018), 658–667

Citation in format AMSBIB
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\by S.~V.~Ivanov
\paper A bilevel stochastic programming problem with random parameters in the Follower's objective function
\jour Diskretn. Anal. Issled. Oper.
\yr 2018
\vol 25
\issue 4
\pages 27--45
\mathnet{http://mi.mathnet.ru/da907}
\crossref{https://doi.org/10.17377/daio.2018.25.596}
\elib{http://elibrary.ru/item.asp?id=36449709}
\transl
\jour J. Appl. Industr. Math.
\yr 2018
\vol 12
\issue 4
\pages 658--667
\crossref{https://doi.org/10.1134/S1990478918040063}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85058101176}


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