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January  2011, 7(1): 199-210. doi: 10.3934/jimo.2011.7.199

On the convergence rate of the inexact Levenberg-Marquardt method

Jinyan Fan 1, and  Jianyu Pan 2,

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
2. 

Department of Mathematics, East China Normal University, Shanghai 200062

Received  July 2010 Revised  November 2010 Published  January 2011

In this paper we study the convergence rate of the inexact Levenberg-Marquardt method for nonlinear equations. Under the local error bound condition which is weaker than nonsingularity, we derive an explicit formula of the convergence order of the inexact LM method, which is a continuous function with respect to not only the LM parameter but also the perturbation vector. The new formula includes many convergence rate results in the literature as its special cases.
Keywords: local error bound condition, convergence rate., Nonlinear equations, inexact Levenberg-Marquardt method.
Mathematics Subject Classification: 90C30, 65K05, 34A3.
Citation: Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199
References:
[1]

Y. H. Dai and Y. X. Yuan, "Nonlinear Conjugate Gradient Methods," Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[2]

H. Dan, N. Yamashita and M. Fukushima, Convergence properties of the inexact Levenberg-Marquardt method under local error bound, Optimization Methods and Software, 17 (2002), 605-626. doi: 10.1080/1055678021000049345.  Google Scholar

[3]

F. Facchinei and C. Kanzow, A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Mathematical Programming, 76 (1997), 493-512. doi: 10.1007/BF02614395.  Google Scholar

[4]

J. Y. Fan and J. Y. Pan, Inexact Levenberg-Marquardt method for nonlinear equations, Discrete Continuous Dynamical System-Series B, 4 (2004), 1223-1232. doi: 10.3934/dcdsb.2004.4.1223.  Google Scholar

[5]

J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23-39. doi: 10.1007/s00607-004-0083-1.  Google Scholar

[6]

A. Fischera, P. K. Shuklaa and M. Wang, On the inexactness level of robust Levenberg-Marquardt methods, Optimization, 59 (2010), 273-287. doi: 10.1080/02331930801951256.  Google Scholar

[7]

K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-166. Google Scholar

[8]

D. W. Marquardt, An algorithm for least-squares estimation of nonlinear inequalities, SIAM J. Appl. Math., 11 (1963), 431-441. doi: 10.1137/0111030.  Google Scholar

[9]

G. W. Stewart and J. G. Sun, "Matrix Perturbation Theory," Academic Press, San Diego, CA, 1990.  Google Scholar

[10]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, Computing, (Supplement 15), (2001), 237-249.  Google Scholar

show all references

References:
[1]

Y. H. Dai and Y. X. Yuan, "Nonlinear Conjugate Gradient Methods," Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[2]

H. Dan, N. Yamashita and M. Fukushima, Convergence properties of the inexact Levenberg-Marquardt method under local error bound, Optimization Methods and Software, 17 (2002), 605-626. doi: 10.1080/1055678021000049345.  Google Scholar

[3]

F. Facchinei and C. Kanzow, A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Mathematical Programming, 76 (1997), 493-512. doi: 10.1007/BF02614395.  Google Scholar

[4]

J. Y. Fan and J. Y. Pan, Inexact Levenberg-Marquardt method for nonlinear equations, Discrete Continuous Dynamical System-Series B, 4 (2004), 1223-1232. doi: 10.3934/dcdsb.2004.4.1223.  Google Scholar

[5]

J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23-39. doi: 10.1007/s00607-004-0083-1.  Google Scholar

[6]

A. Fischera, P. K. Shuklaa and M. Wang, On the inexactness level of robust Levenberg-Marquardt methods, Optimization, 59 (2010), 273-287. doi: 10.1080/02331930801951256.  Google Scholar

[7]

K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-166. Google Scholar

[8]

D. W. Marquardt, An algorithm for least-squares estimation of nonlinear inequalities, SIAM J. Appl. Math., 11 (1963), 431-441. doi: 10.1137/0111030.  Google Scholar

[9]

G. W. Stewart and J. G. Sun, "Matrix Perturbation Theory," Academic Press, San Diego, CA, 1990.  Google Scholar

[10]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, Computing, (Supplement 15), (2001), 237-249.  Google Scholar

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