AI 中文总结
本文提出基于格拉姆矩阵近似的随机拟高斯-牛顿方法，可处理欠定与超定一般非线性方程，欠定情形下局部超线性收敛到最优解，超定情形下无依赖条件数收敛到驻点，优于现有拟牛顿方法。
AI 中文摘要
本文研究求解一般非线性方程的局部收敛性，基于格拉姆矩阵的近似建立了随机拟高斯-牛顿方法，同时处理欠定与超定两种情形。对于欠定情形，证明该方法能以局部超线性收敛到最优解；对于超定情形，证明该方法能以无条件数依赖的局部收敛性达到问题非线性最小二乘形式的驻点。相比之下，现有拟牛顿方法大多聚焦于平方系统，且要求初始雅可比估计与精确雅可比足够接近。
英文摘要
This paper considers the local convergence for solving general nonlinear equations. We establish randomized quasi-Gauss--Newton methods based on the approximation of the Gram matrix, addressing both the underdetermined and the overdetermined settings. For the underdetermined case, we show that our methods achieve local superlinear convergence to the optimal solution. For the overdetermined case, we show that our methods achieve local condition-number-free convergence to the stationary point of the nonlinear least-square formulation of the problem. In contrast, existing quasi-Newton methods mostly focus on square systems and additionally require the initial Jacobian estimate to be sufficiently close to the exact one.
CommentsThe conference version was in the proceedings of NeurIPS 2022, where we focus on the special case $n=d$. The results that are generalized to general nonlinear equations were presented at ICCOPT 2025 and ICOTA 2026
