On a Stochastic Bilevel Programming Problem with Knapsack Constraints

In this paper we propose a mixed integer bilevel problem having a probabilistic knapsack constraint in the first level. The problem formulation is mainly motivated by practical pricing and service provision problems as it can be interpreted as a model for the interaction between a service provider and clients. We assume the probability space to be discrete which allows us to reformulate the problem as a deterministic equivalent bilevel problem. Via a reformulation as linear bilevel problem, we obtain a quadratic optimization problem, the so called Global Linear Complementarity Problem. Based on this quadratic problem, we finally propose a procedure to compute upper bounds on the initial problem by using a Lagrangian relaxation and an iterative linear minmax scheme.