A Sensitivity-Based Method for Bilevel Optimization Problems: Theoretical Analysis and Computational Performance

Abstract: Bilevel optimization provides a powerful framework for modeling hierarchical decision-making systems. This work presents a novel sensitivity-based algorithm that directly addresses the bilevel structure by treating the lower-level optimal solution as an implicit function of the upper-level variables, thus avoiding classical single-level reformulations. This implicit problem is solved within a robust Augmented Lagrangian framework, where the inner subproblems are managed by a quasi-Newton (L-BFGS-B) solver to handle the ill-conditioned and non-smooth landscapes that can arise. The validity of the proposed method is established through both theoretical convergence guarantees and extensive computational experiments. These experiments demonstrate the algorithm's efficiency and robustness and validate the use of a pragmatic dual-criterion stopping condition to address the practical challenge of asymmetric primal-dual convergence rates.