On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

Abstract

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this short note, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to ε-feasibility regarding its nonlinear constraints for an arbitrarily small but positive ε, the obtained bilevel solution can be arbitrarily far away from the unique bilevel solution that one obtains for exactly feasible points in the lower level. Since the consideration of ε-feasibility cannot be avoided for nonconvex problems, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care.