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Volume 34, Issue 3
A Parameter-Self-Adjusting Levenberg-Marquardt Method for Solving Nonsmooth Equations

Liyan Qi, Xiantao Xiao & Liwei Zhang

DOI: 10.4208/jcm.1512-m2015-0333

J. Comp. Math., 34 (2016), pp. 317-338.

Published online: 2016-06

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  • Abstract

A parameter-self-adjusting Levenberg-Marquardt method (PSA-LMM) is proposed for solving a nonlinear system of equations $F(x) = 0$, where $F : \mathbb{R}^n$ →$\mathbb{R}^n$ is a semismooth mapping. At each iteration, the LM parameter $μ_k$ is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA-LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.

  • Keywords

Levenberg-Marquardt method, Nonsmooth equations, Nonlinear complementarity problems.

  • AMS Subject Headings

65K05, 90C30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

qiliyan@dlou.edu.cn (Liyan Qi)

xtxiao@dlut.edu.cn (Xiantao Xiao)

lwzhang@dlut.edu.cn (Liwei Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-34-317, author = {Qi , LiyanXiao , Xiantao and Zhang , Liwei}, title = {A Parameter-Self-Adjusting Levenberg-Marquardt Method for Solving Nonsmooth Equations}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {3}, pages = {317--338}, abstract = {

A parameter-self-adjusting Levenberg-Marquardt method (PSA-LMM) is proposed for solving a nonlinear system of equations $F(x) = 0$, where $F : \mathbb{R}^n$ →$\mathbb{R}^n$ is a semismooth mapping. At each iteration, the LM parameter $μ_k$ is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA-LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1512-m2015-0333}, url = {http://global-sci.org/intro/article_detail/jcm/9798.html} }
TY - JOUR T1 - A Parameter-Self-Adjusting Levenberg-Marquardt Method for Solving Nonsmooth Equations AU - Qi , Liyan AU - Xiao , Xiantao AU - Zhang , Liwei JO - Journal of Computational Mathematics VL - 3 SP - 317 EP - 338 PY - 2016 DA - 2016/06 SN - 34 DO - http://doi.org/10.4208/jcm.1512-m2015-0333 UR - https://global-sci.org/intro/article_detail/jcm/9798.html KW - Levenberg-Marquardt method, Nonsmooth equations, Nonlinear complementarity problems. AB -

A parameter-self-adjusting Levenberg-Marquardt method (PSA-LMM) is proposed for solving a nonlinear system of equations $F(x) = 0$, where $F : \mathbb{R}^n$ →$\mathbb{R}^n$ is a semismooth mapping. At each iteration, the LM parameter $μ_k$ is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA-LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.

Qi , LiyanXiao , Xiantao and Zhang , Liwei. (2016). A Parameter-Self-Adjusting Levenberg-Marquardt Method for Solving Nonsmooth Equations. Journal of Computational Mathematics. 34 (3). 317-338. doi:10.4208/jcm.1512-m2015-0333
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