Abstract
In this paper, bilevel stochastic programming problems with probabilistic and quantile criteria are considered. The lower level problem is assumed to be linear for fixed leader’s (upper level) variables and fixed realizations of the random parameters. The objective function and the constraints of the lower level problem depend on the leader’s strategy and random parameters. The objective function of the upper level problem is defined as the value of the probabilistic or quantile functional of the random losses on the upper level. We suggest conditions guaranteeing that the objective function of the upper level is a normal integrand. It is shown that these conditions are satisfied for a class of problems with positive coefficients of the lower level problem. This allows us to suggest sufficient conditions of the existence of a solution to the considered problem. We construct sample approximations of these problems. These approximations reduce to mixed integer nonlinear programming problems. We describe sufficient conditions of the convergence of the sample approximations to the original problems.
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The reported study was funded by Russian Foundation for Basic Research (RFBR) according to the research project № 20-37-70022.
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Ivanov, S.V., Ignatov, A.N. (2021). Sample Approximations of Bilevel Stochastic Programming Problems with Probabilistic and Quantile Criteria. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham. https://doi.org/10.1007/978-3-030-77876-7_15
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