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

TL;DR

This paper reveals that solving bilevel problems with continuous, nonconvex lower levels is computationally ill-behaved, as approximate solutions can be arbitrarily far from true solutions despite satisfying nonlinear constraints.

Contribution

It demonstrates the inherent computational difficulties in nonconvex bilevel problems, even under favorable conditions, highlighting the need for caution in their solution approaches.

Findings

Approximate solutions can be arbitrarily far from true solutions.

Nonlinearities in the lower level cause bad behavior, unlike linear cases.

Results hold even with unique solvability and regularity conditions.

Abstract

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, 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 $\varepsilon$-feasibility regarding its nonlinear constraints for an arbitrarily small but positive $\varepsilon$, the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater's constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of $\varepsilon$-feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.