On the Complexity of Robust Bilevel Optimization With Uncertain Follower's Objective

TL;DR

This paper analyzes the computational complexity of robust bilevel optimization problems with uncertain follower objectives, revealing high complexity in interval uncertainty cases and limited complexity increase in discrete uncertainty cases.

Contribution

It establishes the complexity classifications of robust bilevel problems under different uncertainty models, highlighting the computational challenges involved.

Findings

Interval uncertainty leads to $ ext{Sigma}_2^P$-hard problems.

Discrete uncertainty cases are at most one level harder than the follower's problem.

Robust bilevel problems can be highly complex even when the follower's problem is tractable.

Abstract

We investigate the complexity of bilevel combinatorial optimization with uncertainty in the follower's objective, in a robust optimization approach. We show that the robust counterpart of the bilevel problem under interval uncertainty can be $\Sigma^{\text P}_2$-hard, even when the certain bilevel problem is NP-equivalent and the follower's problem is tractable. On the contrary, in the discrete uncertainty case, the robust bilevel problem is at most one level harder than the follower's problem.