Benders Decomposition for Discrete–Continuous Linear Bilevel Problems with application to traffic network design

• Mixed-integer formulation for the DNDP without auxiliary binary variables.
• Benders Decomposition method for the Discrete–Continuous Linear Bilevel Problems.
• Acceleration of solving the Slave Problem.
• Solving large instances of the DNDP problem to global optimality.

We propose a new fast solution method for linear Bilevel Problems with binary leader and continuous follower variables under the partial cooperation assumption. We reformulate the Bilevel Problem into a single-level problem by using the Karush–Kuhn–Tucker conditions. This non-linear model can be linearized because of the special structure achieved by the binary leader decision variables and subsequently solved by a Benders Decomposition Algorithm to global optimality. We illustrate the capability of the approach on the Discrete Network Design Problem which adds arcs to an existing road network at the leader stage and anticipates the traffic equilibrium for the follower stage. Because of the non-linear objective functions of this problem, we use a linearization method for increasing, convex and non-linear functions based on continuous variables. Numerical tests show that this algorithm can solve even large instances of Bilevel Problems.