  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.
