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Volume 35, Issue 4
Extended Levenberg-Marquardt Method for Composite Function Minimization

Jianchao Huang, Zaiwen Wen & Xiantao Xiao

DOI: 10.4208/jcm.1702-m2016-0699

J. Comp. Math., 35 (2017), pp. 529-546.

Published online: 2017-08

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  • Abstract

In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min $ρ(r(x))$, where $r$ : $\mathbb{R}^n$ → $\mathbb{R}^m$ and $ρ$ : $\mathbb{R}^m$ → $\mathbb{R}$. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.

  • Keywords

Unconstrained minimization, Composite function, Levenberg-Marquardt method.

  • AMS Subject Headings

65K05, 90C30, 90C53.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

huangjc@sjtu.edu.cn (Jianchao Huang)

wenzw@pku.edu.cn (Zaiwen Wen)

xtxiao@dlut.edu.cn (Xiantao Xiao)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-529, author = {Huang , JianchaoWen , Zaiwen and Xiao , Xiantao}, title = {Extended Levenberg-Marquardt Method for Composite Function Minimization}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {4}, pages = {529--546}, abstract = {

In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min $ρ(r(x))$, where $r$ : $\mathbb{R}^n$ → $\mathbb{R}^m$ and $ρ$ : $\mathbb{R}^m$ → $\mathbb{R}$. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1702-m2016-0699}, url = {http://global-sci.org/intro/article_detail/jcm/10029.html} }
TY - JOUR T1 - Extended Levenberg-Marquardt Method for Composite Function Minimization AU - Huang , Jianchao AU - Wen , Zaiwen AU - Xiao , Xiantao JO - Journal of Computational Mathematics VL - 4 SP - 529 EP - 546 PY - 2017 DA - 2017/08 SN - 35 DO - http://doi.org/10.4208/jcm.1702-m2016-0699 UR - https://global-sci.org/intro/article_detail/jcm/10029.html KW - Unconstrained minimization, Composite function, Levenberg-Marquardt method. AB -

In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min $ρ(r(x))$, where $r$ : $\mathbb{R}^n$ → $\mathbb{R}^m$ and $ρ$ : $\mathbb{R}^m$ → $\mathbb{R}$. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.

Jianchao Huang, Zaiwen Wen & Xiantao Xiao. (2019). Extended Levenberg-Marquardt Method for Composite Function Minimization. Journal of Computational Mathematics. 35 (4). 529-546. doi:10.4208/jcm.1702-m2016-0699
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