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.
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References
J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications (Kluwer Acad. Publ.,Dordrecht, 1998).
S. Dempe, Foundations of Bilevel Programming (Kluwer Acad. Publ., Dordrecht, 2002).
S. Dempe, V. Kalashnikov, G. A. Pérez-Valdés, and N. Kalashnykova, Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Network (Springer, Heidelberg, 2015).
A. I. Kibzun and Yu. S. Kan, Stochastic Programming Problems with Probabilistic Criteria (Fizmatlit, Moscow, 2009) [in Russian].
A. Chen, J. Kim, Z. Zhou, and P. Chootinan, “Alpha Reliable Network Design Problem,” Transp. Res. Rec. J. Transp. Res. Board 2029, 49–57 (2007).
H. Katagiri, T. Uno, K. Kato, H. Tsuda, and H. Tsubaki, “Random Fuzzy Bilevel Linear Programming Through Possibility-Based Value at Risk Model,” Int. J. Mach. Learn. Cybern. 5 (2), 211–224 (2014).
S. V. Ivanov and M. V. Morozova, “Stochastic Problem of Competitive Location of Facilities with Quantile Criterion,” Avtom. Telemekh. No. 3, 109–122 (2016) [Autom. Remote Control 77 (3), 451–461 (2016)].
A. Melnikov and V. Beresnev, “Upper Bound for the Competitive Facility Location Problem with Quantile Criterion,” in Discrete Optimization and Operations Research (Proceedings of 9th International Conference DOOR, Vladivostok, Russia, September 19–23, 2016) (Springer, Cham, 2016), pp. 373–387.
S. Dempe, S. V. Ivanov, and A. Naumov, “Reduction of the Bilevel Stochastic Optimization Problem with Quantile Objective Function to a Mixed-Integer Problem,” Appl. Stoch. Models Bus. Ind. 33 (5), 544–554 (2017).
B. K. Pagnoncelli, S. Ahmed, and A. Shapiro, “Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications,” J. Optim. Theory Appl. 142, 399–416 (2009).
S. V. Ivanov and A. I. Kibzun, “On the Convergence of Sample Approximations for Stochastic Programming Problems with Probabilistic Criteria,” Avtom. Telemekh. No. 2, 19–35 (2018) [Autom. Remote Control 79 (2), 216–228 (2018)].
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis (Springer, Heidelberg, 2009).
A. N. Shiryaev, Probability-1 (MTsNMO,Moscow, 2007; Springer, New York, 2016).
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Original Russian Text © S.V. Ivanov, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 4, pp. 27–45.
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Ivanov, S.V. A Bilevel Stochastic Programming Problem with Random Parameters in the Follower’s Objective Function. J. Appl. Ind. Math. 12, 658–667 (2018). https://doi.org/10.1134/S1990478918040063
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DOI: https://doi.org/10.1134/S1990478918040063