A sensitivity-based method for bilevel optimization problems: Theoretical analysis and computational performance

Abstract

Bilevel optimization provides a powerful framework for modelling hierarchical decision-making systems. This work presents a sensitivity-based algorithm that addresses the bilevel structure directly by treating the lower-level optimal solution as an implicit, locally differentiable function of the upper-level variables, thereby avoiding classical single-level reformulations. Under standard regularity assumptions on the lower level, an adjoint-based representation of the reduced upper-level gradient is derived, replacing explicit construction of the sensitivity Jacobian with a single linear adjoint solve per iteration and reducing gradient evaluation cost by a factor equal to the upper-level dimension. The reduced problem is solved within an Augmented Lagrangian framework, with inner subproblems managed by an L-BFGS-B quasi-Newton solver. Convergence to KKT points of the reduced problem is established, and these points are shown to be equivalent to S-stationary solutions of the associated mathematical programme with complementarity constraints under MPEC-LICQ. Computational experiments on benchmark bilevel problems validate the method's correctness and robustness, and demonstrate the effectiveness of a pragmatic dual-criterion stopping condition in handling the asymmetric primal-dual convergence rates characteristic of augmented Lagrangian methods.