The mixed integer linear bilevel programming problem
Operations Research
An algorithm for the mixed-integer nonlinear bilevel programming problem
Annals of Operations Research - Special issue on hierarchical optimization
Interval analysis: theory and applications
Journal of Computational and Applied Mathematics - Special issue on numerical analysis in the 20th century vol. 1: approximation theory
Convexification and Global Optimization in Continuous And
Convexification and Global Optimization in Continuous And
Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs
Mathematical Programming: Series A and B
Global solution of semi-infinite programs
Mathematical Programming: Series A and B
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications)
Constraint Qualifications and KKT Conditions for Bilevel Programming Problems
Mathematics of Operations Research
The Adaptive Convexification Algorithm: A Feasible Point Method for Semi-Infinite Programming
SIAM Journal on Optimization
Relaxation-Based Bounds for Semi-Infinite Programs
SIAM Journal on Optimization
Global solution of bilevel programs with a nonconvex inner program
Journal of Global Optimization
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An algorithm for the global optimization of nonlinear bilevel mixed-integer programs is presented, based on a recent proposal for continuous bilevel programs by Mitsos et al. (J Glob Optim 42(4):475---513, 2008). The algorithm relies on a convergent lower bound and an optional upper bound. No branching is required or performed. The lower bound is obtained by solving a mixed-integer nonlinear program, containing the constraints of the lower-level and upper-level programs; its convergence is achieved by also including a parametric upper bound to the optimal solution function of the lower-level program. This lower-level parametric upper bound is based on Slater-points of the lower-level program and subsets of the upper-level host sets for which this point remains lower-level feasible. Under suitable assumptions the KKT necessary conditions of the lower-level program can be used to tighten the lower bounding problem. The optional upper bound to the optimal solution of the bilevel program is obtained by solving an augmented upper-level problem for fixed upper-level variables. A convergence proof is given along with illustrative examples. An implementation is described and applied to a test set comprising original and literature problems. The main complication relative to the continuous case is the construction of the parametric upper bound to the lower-level optimal objective value, in particular due to the presence of upper-level integer variables. This challenge is resolved by performing interval analysis over the convex hull of the upper-level integer variables.