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<p>The proximal alternating direction method of multipliers (ADMM) is an efficient algorithm for solving the linearly constrained optimization problem. This work studies the second-order convergence for proximal ADMM. By analyzing the Jacobian matrix associated with the proximal ADMM mapping, we show that the proximal ADMM avoids strict saddle points for almost all initial points. Furthermore, combining this result with the existing first-order convergence, we can establish that the proximal ADMM almost always converges to a second-order stationary point. In addition, we provide a numerical example to validate the second-order convergence of the proximal ADMM.</p>

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A Note on the Second-Order Convergence of Proximal ADMM for Non-Convex Optimization

  • 期刊论文

  • 发表日期：

    2026年3月14日

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  • Jingyu Gao,
  • Xiurui Geng

摘要

The proximal alternating direction method of multipliers (ADMM) is an efficient algorithm for solving the linearly constrained optimization problem. This work studies the second-order convergence for proximal ADMM. By analyzing the Jacobian matrix associated with the proximal ADMM mapping, we show that the proximal ADMM avoids strict saddle points for almost all initial points. Furthermore, combining this result with the existing first-order convergence, we can establish that the proximal ADMM almost always converges to a second-order stationary point. In addition, we provide a numerical example to validate the second-order convergence of the proximal ADMM.