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<p>This work leverages the connection between dynamical systems and optimization algorithms to propose a Güler-type accelerated augmented Lagrangian method (GAALM) for solving convex optimization problems with linear equality constraints. The proposed GAALM is developed by formulating a second-order dual dynamical system associated with the dual formulation of the linearly constrained convex optimization problem and then discretizing it with a tunable parameter that balances implicit and explicit schemes. We establish a convergence rate of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(o(1/k^{2})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the objective residual of the Lagrangian function (Lagrangian residual). Numerical experiments are also presented to illustrate the efficacy and advantages of the proposed method.</p>

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Achieving \(o(1/k^{2})\) Convergence Rate with Güler-Type Accelerated Augmented Lagrangian Method

  • 期刊论文

  • 发表日期：

    2026年7月19日

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  • Tan-Xing Wang,
  • De-Ren Han,
  • Qi Gao,
  • Xing-Ju Cai

摘要

This work leverages the connection between dynamical systems and optimization algorithms to propose a Güler-type accelerated augmented Lagrangian method (GAALM) for solving convex optimization problems with linear equality constraints. The proposed GAALM is developed by formulating a second-order dual dynamical system associated with the dual formulation of the linearly constrained convex optimization problem and then discretizing it with a tunable parameter that balances implicit and explicit schemes. We establish a convergence rate of \(o(1/k^{2})\) o ( 1 / k 2 ) for the objective residual of the Lagrangian function (Lagrangian residual). Numerical experiments are also presented to illustrate the efficacy and advantages of the proposed method.