Exploiting the polyhedral geometry of stochastic linear bilevel programming

TL;DR

This paper explores the geometry of stochastic linear bilevel programming problems with a focus on vertex-supported beliefs, proposing two algorithms—one enumerative and one Monte Carlo-based—for solving these complex problems.

Contribution

It introduces a novel geometric reformulation of stochastic bilevel problems using chamber complexes and develops two algorithms, including a scalable Monte Carlo method, for solving them.

Findings

The vertex-supported belief reformulation is piecewise affine over the chamber complex.

The enumerative algorithm provides a theoretical baseline but is not scalable.

The Monte Carlo approach is effective and scalable for large problems.

Abstract

We study linear bilevel programming problems whose lower-level objective is given by a random cost vector with known distribution. We consider the case where this distribution is nonatomic, allowing to reformulate the problem of the leader using the Bayesian approach in the sense of Salas and Svensson (2023), with a decision-dependent distribution that concentrates on the vertices of the feasible set of the follower's problem. We call this a vertex-supported belief. We prove that this formulation is piecewise affine over the so-called chamber complex of the feasible set of the high-point relaxation. We propose two algorithmic approaches to solve general problems enjoying this last property. The first one is based on enumerating the vertices of the chamber complex. This approach is not scalable, but we present it as a computational baseline and for its theoretical interest. The second one is a Monte-Carlo approximation scheme based on the fact that randomly drawn points of the domain lie, with probability 1, in the interior of full-dimensional chambers, where the problem (restricted to this chamber) can be reduced to a linear program. Finally, we evaluate these methods through computational experiments showing both approaches' advantages and challenges.