Comments on finite termination of the generalized Newton method for absolute value equations

IF 1.3 4区数学 Q2 MATHEMATICS, APPLIED

Optimization Letters Pub Date : 2024-05-20 DOI:10.1007/s11590-024-02121-0

Chun-Hua Guo

引用次数: 0

Abstract

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) \(Ax-|x|=b\). The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever \(\rho (|A^{-1}|)<1/3\). We also present new results for the case where \(A-I\) is a nonsingular M-matrix or an irreducible singular M-matrix. When \(A-I\) is an irreducible singular M-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.

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关于绝对值方程广义牛顿法有限终止的评论

我们考虑了绝对值方程（AVE） \(Ax-|x|=b\)的广义牛顿法（GNM）。无论绝对值方程是否有唯一解，只要该方法收敛，它就具有有限终止特性。我们证明，只要 \(\rho (|A^{-1}|)<1/3\), GNM 就是收敛的。我们还针对 \(A-I\) 是非奇异 M 矩阵或不可还原奇异 M 矩阵的情况提出了新的结果。当 \(A-I\) 是不可还原的奇异 M 矩阵时，AVE 可能有无穷多个解。在这种情况下，我们证明了只要初始猜测至少有一个非正分量，GNM 总是以一个唯一可识别的解结束。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Optimization Letters 管理科学-应用数学

CiteScore

3.40

自引率

6.20%

发文量

116

审稿时长

9 months

期刊介绍： Optimization Letters is an international journal covering all aspects of optimization, including theory, algorithms, computational studies, and applications, and providing an outlet for rapid publication of short communications in the field. Originality, significance, quality and clarity are the essential criteria for choosing the material to be published. Optimization Letters has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time one of the most striking trends in optimization is the constantly increasing interdisciplinary nature of the field. Optimization Letters aims to communicate in a timely fashion all recent developments in optimization with concise short articles (limited to a total of ten journal pages). Such concise articles will be easily accessible by readers working in any aspects of optimization and wish to be informed of recent developments.