Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems

We study a class of bilevel optimization problems in which both the upper- and lower-level problems have minimax structures. This setting captures a broad range of emerging applications. Despite the extensive literature on bilevel optimization and minimax optimization separately, existing methods mainly focus on bilevel optimization with lower-level minimization problems, often under strong convexity assumptions, and are not directly applicable to the minimax lower-level setting considered here. To address this gap, we develop penalty-based first-order methods for bilevel minimax optimization without requiring strong convexity of the lower-level problem. In the deterministic setting, we establish that the proposed method finds an $\epsilon$-KKT point with $\tilde{O}(\epsilon^{-4})$ oracle complexity. We further show that bilevel problems with convex constrained lower-level minimization can be reformulated as special cases of our framework via Lagrangian duality, leading to an $\tilde{O}(\epsilon^{-4})$ complexity bound that improves upon the existing $\tilde{O}(\epsilon^{-7})$ result. Finally, we extend our approach to the stochastic setting, where only stochastic gradient oracles are available, and prove that the proposed stochastic method finds a nearly $\epsilon$-KKT point with $\tilde{O}(\epsilon^{-9})$ oracle complexity.