A variant of the Levenberg-Marquardt method with adaptive parameters for systems of nonlinear equations

• Published in: AIMS mathematics Vol. 7; no. 1; pp. 1241 - 1256
• Main Authors: Zheng, Lin, Chen, Liang, Ma, Yanfang
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2022
• Subjects: error bound condition; global convergence; levenberg-marquardt method; systems of nonlinear equations; wolfe line search
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2022073

The Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations and nonlinear least-squares problems. It enjoys a quadratic convergence rate under the local error bound condition. Recently, to solve nonzero-residue nonlinear least-squares problem, Behling et al. propose a modified Levenberg-Marquardt method with at least superlinearly convergence under a new error bound condtion [<xref ref-type="bibr" rid="b3">3 ] . To extend their results for systems of nonlinear equations, by choosing the LM parameters adaptively, we propose an efficient variant of the Levenberg-Marquardt method and prove its quadratic convergence under the new error bound condition. We also investigate its global convergence by using the Wolfe line search. The effectiveness of the new method is validated by some numerical experiments.