A Surrogate Value Function Formulation for Bilevel Optimization

Abstract: The value function formulation captures the hierarchical nature of bilevel optimization through the optimal value function of the lower level problem, yet its implicit and nonsmooth characteristics pose significant analytical and computational difficulties. We introduce a surrogate value function formulation that replaces the intractable value function with an explicit surrogate derived from lower level stationarity conditions. This surrogate formulation preserves the essential idea of the classical value function model but fundamentally departs from Karush Kuhn Tucker (KKT) formulations, which embed lower level stationary points into the upper level feasible region and obscure the hierarchical dependence. Instead, it enforces the hierarchy through a dominance constraint that remains valid even when lower level constraint qualifications fail at the solution. We establish equivalence with the original bilevel problem, reveal the failure of standard constraint qualifications, and show that its strong stationarity implies that of KKT models. To handle the complementarity constraints in the surrogate formulation, we apply a smoothing barrier augmented Lagrangian method and prove its convergence to solutions and Clarke stationary points. Extensive experiments demonstrate the robustness and high numerical precision of this formulation, especially in nonconvex settings, including the classical Mirrlees problem where KKT models fail.