# Risk-Averse Models in Bilevel Stochastic Linear Programming

## TL;DR

This paper studies bilevel stochastic linear programming models where the leader's risk-averse decision-making is analyzed under distributional perturbations, providing stability, continuity, and reformulation results.

## Contribution

It introduces stability and differentiability results for risk measures in bilevel stochastic problems and offers a reformulation approach for finite discrete distributions.

## Key findings

- Qualitative stability under probability distribution perturbations
- Lipschitz continuity and differentiability conditions for risk measures
- Reformulation of finite discrete distribution problems as standard bilevel problems

## Abstract

We consider bilevel linear problems, where some parameters are stochastic, and 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 we evaluate based on some law-invariant convex risk measure. A qualitative stability result under perturbations of the underlying probability distribution is presented. Moreover, for the expectation, the expected excess, and the upper semideviation, we establish Lipschitz continuity as well as sufficient conditions for differentiability. Finally, for finite discrete distributions, we reformulate the bilevel stochastic problems as standard bilevel problems and propose a regularization scheme for bilevel linear problems.