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arXiv · 2604.23487

Optimality Conditions and Numerical Algorithms for a Class of Minimax Bilevel Optimization Problems

Yaling Hu·Jiani Wang·Yu-hong Dai·Xiaojiao Tong
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Abstract

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.

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Yaling Hu, Jiani Wang, Yu-hong Dai, Xiaojiao Tong. 2026-04-26. Optimality Conditions and Numerical Algorithms for a Class of Minimax Bilevel Optimization Problems. https://arxiv.org/abs/2604.23487

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