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 $ε$-KKT point with $\tilde{O}(ε^{-4})$ 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 $\tilde{O}(ε^{-4})$ complexity bound that improves upon the existing $\tilde{O}(ε^{-7})$ 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 $ε$-KKT point with $\tilde{O}(ε^{-9})$ oracle complexity.

翻译：我们研究了一类上下层问题均具有极小极大结构的双层优化问题。该设定涵盖了大量新兴应用场景。尽管关于双层优化和极小极大优化的文献已相当丰富，但现有方法主要针对下层为极小化问题（通常需满足强凸性假设）的双层优化，无法直接适用于本文所考虑的极小极大下层设定。为填补这一空白，我们提出了基于罚函数的一阶方法，该方法不要求下层问题具有强凸性。在确定性设定下，我们证明所提方法能以$\tilde{O}(ε^{-4})$的预言复杂度找到$ε$-KKT点。进一步地，通过拉格朗日对偶性，我们证明含凸约束下层极小化问题的双层优化可转化为本文框架的特例，从而得到$\tilde{O}(ε^{-4})$的复杂度界，优于现有$\tilde{O}(ε^{-7})$的结果。最后，我们将方法推广至仅能获得随机梯度预言的随机设定，并证明所提随机方法能以$\tilde{O}(ε^{-9})$的预言复杂度找到近$ε$-KKT点。