• Previous Article
    Reliability optimization of component assignment problem for a multistate network in terms of minimal cuts
  • JIMO Home
  • This Issue
  • Next Article
    A differential equation method for solving box constrained variational inequality problems
January  2011, 7(1): 199-210. doi: 10.3934/jimo.2011.7.199

On the convergence rate of the inexact Levenberg-Marquardt method

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.
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, (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. 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. 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. 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. 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. 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. Google Scholar

[8]

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

[9]

G. W. Stewart and J. G. Sun, "Matrix Perturbation Theory,", Academic Press, (1990). Google Scholar

[10]

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

show all references

References:
[1]

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

[8]

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

[9]

G. W. Stewart and J. G. Sun, "Matrix Perturbation Theory,", Academic Press, (1990). Google Scholar

[10]

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

[1]

Jinyan Fan, Jianyu Pan. Inexact Levenberg-Marquardt method for nonlinear equations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1223-1232. doi: 10.3934/dcdsb.2004.4.1223

[2]

Liyan Qi, Xiantao Xiao, Liwei Zhang. On the global convergence of a parameter-adjusting Levenberg-Marquardt method. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 25-36. doi: 10.3934/naco.2015.5.25

[3]

Jinyan Fan. On the Levenberg-Marquardt methods for convex constrained nonlinear equations. Journal of Industrial & Management Optimization, 2013, 9 (1) : 227-241. doi: 10.3934/jimo.2013.9.227

[4]

Johann Baumeister, Barbara Kaltenbacher, Antonio Leitão. On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations. Inverse Problems & Imaging, 2010, 4 (3) : 335-350. doi: 10.3934/ipi.2010.4.335

[5]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[6]

Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007

[7]

Wen-ling Zhao, Dao-jin Song. A global error bound via the SQP method for constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 775-781. doi: 10.3934/jimo.2007.3.775

[8]

Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008

[9]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1

[10]

Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901

[11]

Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203

[12]

Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333

[13]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic & Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037

[14]

Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial & Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235

[15]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[16]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61

[17]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627

[18]

Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413

[19]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

[20]

Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]