Operations Research Letters 29 (2001) 171–179 www.elsevier.com/locate/dsw A trust region algorithm for nonlinear bilevel programming P. Marcotte a ; ∗ , G. Savard b , D.L. Zhu c a DIRO and CRT, Universit e de Montr eal, CP 6128, succursale Centre-ville, Montr eal, QC, Canada H3C 3J7 b GERAD and D epartement de math ematiques et g enie industriel,  Ecole Polytechnique, C.P. 6079, succursale Centre-ville, Montr eal, QC, Canada H3C 3A7 c CRT, Universit e de Montr eal, CP 6128, succursale Centre-ville, Montr eal, QC, Canada H3C 3J7 and Fudan University, Shanghai, China Received 5 October 2000; received in revised form 21 June 2001; accepted 5 July 2001 Abstract We propose to solve generalized bilevel programs by a trust region approach where the “model” takes the form of a bilevel program involving a linear program at the upper level and a linear variational inequality at the lower level. By coupling the concepts of trust region and linesearch in a novel way, we obtain an implementable algorithm that converges to a strong stationary point of the original bilevel program. c  2001 Elsevier Science B.V. All rights reserved. Keywords: Bilevel programming; Trust region; Linesearch 1. A trust region approach to bilevel programming In this paper we consider bilevel programs (or MPECs, i.e., mathematical programs with equilibrium con- straints, see [5]) of the form min x∈X;y∈Y (x) f(x;y) s:t : 〈F (x;y);y - y ′ 〉 6 0 ∀y ′ ∈ Y (x); where the mapping F is strongly monotone with respect to the lower level variable y and where the sets X and Y (x)= {y: Ax + By ¿ b} are polyhedral. We propose for its solution a trust region approach where the model is itself a bilevel program of a combinatorial nature that can be solved for a global optimum. This approach, which mixes continuous and discrete optimization, trust regions and linesearches, is in stark contrast with more traditional descent methods whose convergence properties are frequently weak and dicult to analyze (see the works of Outrata et al. [7] or, more recently, the sequential programming scheme developed by Fukushima and Tseng [3]). Although the underlying motivation is quite dierent, it is related to a trust ∗ Corresponding author. E-mail address: marcotte@IRO.umontreal.ca (P. Marcotte). 0167-6377/01/$-see front matter c  2001 Elsevier Science B.V. All rights reserved. PII:S0167-6377(01)00092-X