First-Order Penalty Methods for Bilevel Optimization
Zhaosong Lu, Sanyou Mei · SIAM Journal on Optimization · 2024
Abstract. In this paper, we study a class of unconstrained and constrained bilevel optimization problems in which the lower level is a possibly nonsmooth convex optimization problem, while the upper level is a possibly nonconvex optimization problem. We introduce a notion of [Formula: see text]-KKT solution for them and show that an [Formula: see text]-KKT solution leads to an [Formula: see text]- or [Formula: see text]-hypergradient–based stationary point under suitable assumptions. We also propose first-order penalty methods for finding an [Formula: see text]-KKT solution of them, whose subproblems turn out to be a structured minimax problem and can be suitably solved by a first-order method recently developed by the authors. Under suitable assumptions, an operation complexity of [Formula: see text] and [Formula: see text], measured by their fundamental operations, is established for the proposed penalty methods for finding an [Formula: see text]-KKT solution of the unconstrained and constrained bilevel optimization problems, respectively. Preliminary numerical results are presented to illustrate the performance of our proposed methods. To the best of our knowledge, this paper is the first work to demonstrate that bilevel optimization can be approximately solved as minimax optimization, and moreover, it provides the first implementable method with complexity guarantees for such sophisticated bilevel optimization.