A modified inexact Levenberg–Marquardt method with the descent property for solving nonlinear equations

Jianghua Yin, Jinbao Jian () and Guodong Ma
Additional contact information
Jianghua Yin: Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University
Jinbao Jian: Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University
Guodong Ma: Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University

Computational Optimization and Applications, 2024, vol. 87, issue 1, No 9, 289-322

Abstract: Abstract In this work, we propose a modified inexact Levenberg–Marquardt method with the descent property for solving nonlinear equations. A novel feature of the proposed method is that one can directly use the search direction generated by the approach to perform Armijo-type line search once the unit step size is not acceptable. We achieve this via properly controlling the level of inexactness such that the resulting search direction is automatically a descent direction for the merit function. Under the local Lipschitz continuity of the Jacobian, the global convergence of the proposed method is established, and an iteration complexity bound of $$O(1/\epsilon ^2)$$ O ( 1 / ϵ 2 ) to reach an $$\epsilon $$ ϵ -stationary solution is proved under some appropriate conditions. Moreover, with the aid of the designed inexactness condition, we establish the local superlinear rate of convergence for the proposed method under the Hölderian continuity of the Jacobian and the Hölderian local error bound condition. For some special parameters, the convergence rate is even quadratic. The numerical experiments on the underdetermined nonlinear equations illustrate the effectiveness and efficiency of the algorithm compared with a previously proposed inexact Levenberg–Marquardt method. Finally, applying it to solve the Tikhonov-regularized logistic regression shows that our proposed method is quite promising.

Keywords: Nonlinear equations; Inexact Levenberg–Marquardt method; Convergence; Hölderian local error bound; Local convergence rate; 65K05; 90C30 (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-023-00513-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:coopap:v:87:y:2024:i:1:d:10.1007_s10589-023-00513-z

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-023-00513-z

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().