Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

Abstract

The paper is devoted to the stochastic optimistic bilevel optimization problem with quantile criterion in the upper level problem. If the probability distribution is finite, the problem can be transformed into a mixed-integer nonlinear optimization problem. We formulate assumptions guaranteeing that an optimal solution exists. A production planning problem is used to illustrate usefulness of the model.

Risk-Averse Models in Bilevel Stochastic Linear Programming

“…The results include continuity of the objective function, the existence of a solution, and equivalence to a mixed-integer linear program, if the underlying distribution is finite discrete. The latter result has been extended to the fully random case in [12].…”

“…As ϕ o is Lipschitz continuous by Theorem 7.1 in the Appendix, Lipschitz continuity of M follows from the same result. Suppose that u is not a local minimizer of (12), then there exist a sequence {u n } n∈N such that and u n ∈ U and (15) w − w = 0.…”

Bilevel Optimization under Uncertainty

“…For the other models, a similar decomposition is possible after Lagrangean relaxation of the coupling constraints involving different scenarios. c. For R = CVaR α , every evaluation the objective function in the standard bilevel linear program corresponds to solving a bilevel linear problem with scalar upper level variable η. d. Alternate models for R = VaR α are given in [17] and [48], where the considered bilevel stochastic linear problem is reduced to a mixed-integer nonlinear program and a mathematical programming problem with equilibrium constraints, respectively. A mean-risk model with R = CVaR α is used in [5,Sect.…”