Duality-based single-level reformulations of bilevel optimization problems
Stephan Dempe, Patrick Mehlitz
TL;DR
This paper analyzes three duality-based single-level reformulations of optimistic bilevel optimization problems, using the lower-level Lagrange, Wolfe, and Mond-Weir duals. It shows that, with mild assumptions, these reformulations are globally equivalent to the original bilevel problem with respect to global minimizers, but they introduce artificial implicit variables that complicate the relationship between local minimizers and stationary points compared to the original problem. All three approaches inherit the same fundamental difficulties as the classical KKT-based reformulation, notably the likely failure of constraint qualifications such as the (nonsmooth) Mangasarian-Fromovitz constraint qualification at feasible points, which undermines standard optimality conditions. The work further clarifies that claims of MFCQ holding for Wolfe and Mond-Weir reformulations are unfounded, illustrating a parallel with the Lagrange-based approach, and suggesting that relaxation strategies may be necessary to harness these reformulations computationally. Overall, the paper highlights intrinsic regularity and modeling challenges common to duality-based bilevel reformulations and sets the stage for future numerical evaluations of their practical utility.
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.