On the convergence of the algorithm for bilevel programming problems by Clegg and Smith

In Clegg and Smith [Transportation Research B 35 (2001) 71–82] and Battye et al. [in: M. Patriksson, M. Labbe (Eds.), Transportation Planning – State of the Art, Proceedings of the 6th Meeting of the EURO Working Group on Transportation, Gothenburg, Sweden, September 9–11, 1998, Kluwer Academic Publishers, Dordrecht], Smith and colleagues present an algorithm for solving the bilevel programming problem. We show that the points reached by the algorithm are not stationary points of bilevel programs in general. We further show that, with a minor modification, this method can be expressed as an inexact penalty method for gap function-constrained bilevel programs.