Duality-based single-level reformulations of bilevel optimization problems

TL;DR

This paper analyzes duality-based single-level reformulations of bilevel optimization problems, highlighting their equivalence to the original problems and discussing issues with implicit variables and constraint qualifications.

Contribution

It revisits three duality-based reformulations, compares their properties, and clarifies misconceptions about constraint qualifications in these approaches.

Findings

Reformulations are equivalent to original problems under mild assumptions.

Implicit variables pose challenges in solution analysis.

Mangasarian-Fromovitz constraint qualification often fails in these reformulations.

Abstract

Usually, bilevel optimization problems need to be transformed into single-level ones in order to derive optimality conditions and solution algorithms. Among the available approaches, the replacement of the lower-level problem by means of duality relations became popular quite recently. We revisit three realizations of this idea which are based on the lower-level Lagrange, Wolfe, and Mond--Weir dual problem. The resulting single-level surrogate problems are equivalent to the original bilevel optimization problem from the viewpoint of global minimizers under mild assumptions. However, all these reformulations suffer from the appearance of so-called implicit variables, i.e., surrogate variables which do not enter the objective function but appear in the feasible set for modeling purposes. Treating implicit variables as explicit ones has been shown to be problematic when locally optimal solutions, stationary points, and applicable constraint qualifications are compared to the original problem. Indeed, we illustrate that the same difficulties have to be faced when using these duality-based reformulations. Furthermore, we show that the Mangasarian-Fromovitz constraint qualification is likely to be violated at each feasible point of these reformulations, contrasting assertions in some recently published papers.