# Bilevel Optimization under Uncertainty

## TL;DR

This paper studies bilevel linear optimization problems with stochastic elements, analyzing their properties, existence conditions, and stability, and explores deterministic reformulations for finite distributions to address computational challenges.

## Contribution

It introduces a comprehensive analysis of bilevel problems under uncertainty, including stability and existence results, and proposes deterministic reformulations for finite distributions.

## Key findings

- Lipschitzian properties of the models
- Conditions for existence and optimality
- Deterministic reformulations for finite distributions

## Abstract

We consider bilevel linear problems, where the right-hand side of the lower level problems is stochastic. The leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, which gives rise to a broad spectrum of models involving coherent or convex risk measures and stochastic dominance constraints. We outline Lipschitzian properties, conditions for existence and optimality, as well as stability results. Moreover, for finite discrete distributions, we discuss the special structure of equivalent deterministic bilevel programs and its potential use to mitigate the curse of dimensionality.