Scholarlink

<p>This paper introduces a novel augmented Lagrangian function that incorporates the functions and lower-level gradients of bilevel programs to address constrained bilevel optimization problems. We explore the rate of convergence of the augmented Lagrangian method (ALM) when applied to bilevel programs with equality constraints. Under the linear independence constraint qualification (LICQ) and a set of second-order sufficient optimality conditions (SOSC) for the bilevel program, we demonstrate that the method achieves a linear convergence rate with respect to the distance between the current and optimal Lagrange multipliers, proportional to 1/c, where c is the penalty parameter exceeding a predefined threshold. Additionally, we estimate the Q-linear convergence rate of the multiplier sequence when the penalty parameter sequence remains bounded, and prove that the convergence rate becomes superlinear if the penalty parameter sequence increases unboundedly. A detailed analysis is provided for bilevel programs with quadratic objective functions and linear equality constraints. The numerical experiments further confirm the efficiency and robustness of the proposed method.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

The rate of convergence of augmented Lagrangian method for bilevel optimization problems with equality constraints

  • 期刊论文

  • 发表日期：

    2025年12月8日

查看全文
  • Xiang Liu,
  • Mengwei Xu,
  • Liwei Zhang

摘要

This paper introduces a novel augmented Lagrangian function that incorporates the functions and lower-level gradients of bilevel programs to address constrained bilevel optimization problems. We explore the rate of convergence of the augmented Lagrangian method (ALM) when applied to bilevel programs with equality constraints. Under the linear independence constraint qualification (LICQ) and a set of second-order sufficient optimality conditions (SOSC) for the bilevel program, we demonstrate that the method achieves a linear convergence rate with respect to the distance between the current and optimal Lagrange multipliers, proportional to 1/c, where c is the penalty parameter exceeding a predefined threshold. Additionally, we estimate the Q-linear convergence rate of the multiplier sequence when the penalty parameter sequence remains bounded, and prove that the convergence rate becomes superlinear if the penalty parameter sequence increases unboundedly. A detailed analysis is provided for bilevel programs with quadratic objective functions and linear equality constraints. The numerical experiments further confirm the efficiency and robustness of the proposed method.