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We study linear bilevel programming problems whose lower-level objective isgiven by a random cost vector with known distribution. We consider the casewhere this distribution is nonatomic, allowing to reformulate the problem ofthe leader using the Bayesian approach in the sense of Salas and Svensson(2023) with a vertex-supported belief. We prove that this formulation ispiecewise affine over the so-called chamber complex of the feasible set of thehigh point relaxation. We propose two algorithmic approaches to solve generalproblems enjoying this last property. The first one is based on enumerating thevertices of the chamber complex. The second one is a Monte-Carlo approximationscheme based on the fact that randomly drawn points of the domain lie, withprobability 1, in the interior of full-dimensional chambers, where the problem(restricted to this chamber) can be reduced to a linear program. Finally, weevaluate these methods through computational experiments showing bothapproaches' advantages and challenges.

Exploiting the polyhedral geometry of stochastic linear bilevel programming