Approximate Gauss-Newton methods for solving underdetermined nonlinear least squares problems

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• 作者：Ji-Feng Bao^a ; ^b ; ¹ ; ^{feelwinter@163.com" class="auth_mail" title="E-mail the corresponding author} ; Chong Li^c ; ² ; ^{cli@zju.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wei-Ping Shen^d ; ^{shenweiping@zjnu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jen-Chih Yao^e ; ³ ; ^{yaojc@mail.cmu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Sy-Ming Guu^f ; ⁴ ; ^{iesmguu@mail.cgu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; ^{iesmguu@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear least squares problems ; Approximate Gauss&ndash ; Newton methods ; Lipschitz condition
• 刊名：Applied Numerical Mathematics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：111
• 期：Complete
• 页码：92-110
• 全文大小：1299 K

文摘

We propose several approximate Gauss–Newton methods, i.e., the truncated, perturbed, and truncated-perturbed GN methods, for solving underdetermined nonlinear least squares problems. Under the assumption that the Fréchet derivatives are Lipschitz continuous and of full row rank, Kantorovich-type convergence criteria of the truncated GN method are established and local convergence theorems are presented with the radii of convergence balls obtained. As consequences of the convergence results for the truncated GN method, convergence theorems of the perturbed and truncated-perturbed GN methods are also presented. Finally, numerical experiments are presented where the comparisons with the standard inexact Gauss–Newton method and the inexact trust-region method for bound-constrained least squares problems [23] are made.