Enhanced Barrier-Smoothing Technique for Bilevel Optimization with Nonsmooth Mappings

Abstract: Bilevel optimization problems, encountered in fields such as economics, engineering, and machine learning, pose significant computational challenges due to their hierarchical structure and constraints at both upper and lower levels. Traditional gradient-based methods are effective for unconstrained bilevel programs with unique lower level solutions, but struggle with constrained bilevel problems due to the nonsmoothness of lower level solution mappings. To overcome these challenges, this paper introduces the Enhanced Barrier-Smoothing Algorithm (EBSA), a novel approach that integrates gradient-based techniques with an augmented Lagrangian framework. EBSA utilizes innovative smoothing functions to approximate the primal-dual solution mapping of the lower level problem, and then transforms the bilevel problem into a sequence of smooth single-level problems. This approach not only addresses the nonsmoothness but also enhances convergence properties. Theoretical analysis demonstrates its superiority in achieving Clarke and, under certain conditions, Bouligand stationary points for bilevel problems. Both theoretical analysis and preliminary numerical experiments confirm the robustness and efficiency of EBSA.