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 $\varepsilon$-KKT solution for them and show that an $\varepsilon$-KKT solution leads to an $O(\sqrt{\varepsilon})$- or $O(\varepsilon)$-hypergradient based stionary point under suitable assumptions. We also propose first-order penalty methods for finding an $\varepsilon$-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 \emph{operation complexity} of $O(\varepsilon^{-4}\log\varepsilon^{-1})$ and $O(\varepsilon^{-7}\log\varepsilon^{-1})$, measured by their fundamental operations, is established for the proposed penalty methods for finding an $\varepsilon$-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.

翻译：本文研究一类无约束和约束双层优化问题，其中下层为可能非光滑的凸优化问题，上层为可能非凸的优化问题。我们引入$\varepsilon$-KKT解的概念，并证明在适当假设下，$\varepsilon$-KKT解可导出基于$O(\sqrt{\varepsilon})$或$O(\varepsilon)$超梯度的稳定点。同时提出一阶罚函数方法以寻找$\varepsilon$-KKT解，其子问题转化为结构化的极小极大问题，可通过作者近期发展的适当一阶方法求解。在适当假设下，针对无约束和约束双层优化问题的$\varepsilon$-KKT解，所提罚方法的操作复杂度分别达到$O(\varepsilon^{-4}\log\varepsilon^{-1})$和$O(\varepsilon^{-7}\log\varepsilon^{-1})$（以基本运算次数度量）。初步数值结果验证了所提方法的有效性。据我们所知，本文首次证明双层优化可近似转化为极小极大优化求解，并为这类复杂双层优化问题提供了首个具有复杂度保证的可实现方法。