Optimality Conditions and Numerical Algorithms for a Class of Minimax Bilevel Optimization Problems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915958941024256 |
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| author | Hu, Yaling Wang, Jiani Dai, Yu-hong Tong, Xiaojiao |
| author_facet | Hu, Yaling Wang, Jiani Dai, Yu-hong Tong, Xiaojiao |
| contents | In many applications, including Stackelberg games, machine learning, and power systems \cite{Mackay2018Selftuning,Heinrich1952The,Wang2021Bi-Level}, the decisions in a minimax optimization problem can be constrained by a solution to an optimization problem. In this paper, we introduce optimality conditions of this novel minimax bilevel optimization problem and develops efficient first-order algorithms for this class of problems. Firstly, we establish the optimality conditions for minimax bilevel problems by reconstructing the lower-level problem through its Karush-Kuhn-Tucker (KKT) conditions and value function. Secondly, we develop a penalty method framework to approximately solve the minimax bilevel problem by transforming it into a single-level minimax problem. Thirdly, we design a projected gradient multi-step ascent descent method to solve the resulting minimax problem, which can find an $ε$-KKT solution for the original minimax bilevel problem within $\mathcal{O}(ε^{-3} \log(ε^{-1}))$ iterations. To improve {the convergence rate} of the algorithm, we provide its Nesterov accelerated extension with $\mathcal{O}(ε^{-3} \log(ε^{-1}))$ iteration complexity. Finally, we demonstrate the effectiveness of our model and algorithms through numerical experiments on various minimax bilevel optimization problems and a bilevel economic dispatch in the power system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23487 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimality Conditions and Numerical Algorithms for a Class of Minimax Bilevel Optimization Problems Hu, Yaling Wang, Jiani Dai, Yu-hong Tong, Xiaojiao Optimization and Control 65K05, 90C30, 90C47, 90C90, 90C99 In many applications, including Stackelberg games, machine learning, and power systems \cite{Mackay2018Selftuning,Heinrich1952The,Wang2021Bi-Level}, the decisions in a minimax optimization problem can be constrained by a solution to an optimization problem. In this paper, we introduce optimality conditions of this novel minimax bilevel optimization problem and develops efficient first-order algorithms for this class of problems. Firstly, we establish the optimality conditions for minimax bilevel problems by reconstructing the lower-level problem through its Karush-Kuhn-Tucker (KKT) conditions and value function. Secondly, we develop a penalty method framework to approximately solve the minimax bilevel problem by transforming it into a single-level minimax problem. Thirdly, we design a projected gradient multi-step ascent descent method to solve the resulting minimax problem, which can find an $ε$-KKT solution for the original minimax bilevel problem within $\mathcal{O}(ε^{-3} \log(ε^{-1}))$ iterations. To improve {the convergence rate} of the algorithm, we provide its Nesterov accelerated extension with $\mathcal{O}(ε^{-3} \log(ε^{-1}))$ iteration complexity. Finally, we demonstrate the effectiveness of our model and algorithms through numerical experiments on various minimax bilevel optimization problems and a bilevel economic dispatch in the power system. |
| title | Optimality Conditions and Numerical Algorithms for a Class of Minimax Bilevel Optimization Problems |
| topic | Optimization and Control 65K05, 90C30, 90C47, 90C90, 90C99 |
| url | https://arxiv.org/abs/2604.23487 |