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<p>Minimax optimization plays a crucial role in fields such as adversarial learning, robust optimization, and game theory, where competing objectives must be balanced. However, solving constrained minimax problems introduces considerable difficulties, particularly due to the complexity of the constraints. In this paper, we introduce a gradient-based algorithm that addresses such difficulty by leveraging the augmented Lagrangian method to solve the inner-level problem and efficiently updating the outer variable based on gradient information. By ensuring the Lipschitz continuity of the solution mapping, the algorithm is designed to adapt to various convexity conditions. Specially, for strongly convex objective functions, the algorithm achieves R-linear convergence rate of the objective value. For convex objective functions, the algorithm achieves a local convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}\left( \frac{1}{k}\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> toward a local minimax value, and for nonconvex objectives, it attains the same rate for convergence to a first-order stationary point. Extensive numerical experiments demonstrate its effectiveness in comparison to existing methods, especially for problems with nonlinear constraints.</p>

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Convergence Properties of Gradient-Based Methods for Minimax Problems with Nonlinear Constraints

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Qingying Hu,
  • Bo Wang,
  • Mengwei Xu

摘要

Minimax optimization plays a crucial role in fields such as adversarial learning, robust optimization, and game theory, where competing objectives must be balanced. However, solving constrained minimax problems introduces considerable difficulties, particularly due to the complexity of the constraints. In this paper, we introduce a gradient-based algorithm that addresses such difficulty by leveraging the augmented Lagrangian method to solve the inner-level problem and efficiently updating the outer variable based on gradient information. By ensuring the Lipschitz continuity of the solution mapping, the algorithm is designed to adapt to various convexity conditions. Specially, for strongly convex objective functions, the algorithm achieves R-linear convergence rate of the objective value. For convex objective functions, the algorithm achieves a local convergence rate of \(\mathcal {O}\left( \frac{1}{k}\right) \) O 1 k toward a local minimax value, and for nonconvex objectives, it attains the same rate for convergence to a first-order stationary point. Extensive numerical experiments demonstrate its effectiveness in comparison to existing methods, especially for problems with nonlinear constraints.