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<p>The augmented Lagrangian method (ALM) is a fundamental framework for solving convex optimization problems with linear constraints, and its convergence has been extensively studied. Its proximal variant, known as the Proximal ALM, improves stability by adding a positive-definite quadratic proximal term, which regularizes the primal subproblem at each iteration. Recently, an indefinite proximal augmented Lagrangian method (IDP-ALM) was proposed, which relaxes the requirement for a positive-definite quadratic proximal term. This approach allows for a broader range of parameter choices while still guaranteeing convergence. In this paper, we focus on relaxing the parameter range of the IDP-ALM. Specifically, we extend its parameter range under the strong convexity assumption of the objective function and establish its convergence under this condition. With respect to the convergence rate, we demonstrate that IDP-ALM achieves an ergodic convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{O}(1/k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when the objective function is strongly convex. To validate the theoretical findings, numerical experiments are conducted, demonstrating the feasibility of the IDP-ALM algorithm under the relaxed parameter conditions and comparing its performance against other related algorithms.</p>

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Convergence analysis of the indefinite proximal augmented lagrange method under strongly convex conditions

  • 期刊论文

  • 发表日期：

    2025年12月18日

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  • Jinzhuang Xu,
  • Yu Lei,
  • Shibei Xue,
  • Shan Ma

摘要

The augmented Lagrangian method (ALM) is a fundamental framework for solving convex optimization problems with linear constraints, and its convergence has been extensively studied. Its proximal variant, known as the Proximal ALM, improves stability by adding a positive-definite quadratic proximal term, which regularizes the primal subproblem at each iteration. Recently, an indefinite proximal augmented Lagrangian method (IDP-ALM) was proposed, which relaxes the requirement for a positive-definite quadratic proximal term. This approach allows for a broader range of parameter choices while still guaranteeing convergence. In this paper, we focus on relaxing the parameter range of the IDP-ALM. Specifically, we extend its parameter range under the strong convexity assumption of the objective function and establish its convergence under this condition. With respect to the convergence rate, we demonstrate that IDP-ALM achieves an ergodic convergence rate of \(\mathcal{O}(1/k)\) O ( 1 / k ) when the objective function is strongly convex. To validate the theoretical findings, numerical experiments are conducted, demonstrating the feasibility of the IDP-ALM algorithm under the relaxed parameter conditions and comparing its performance against other related algorithms.