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2015, 5(1): 25-36. doi: 10.3934/naco.2015.5.25

On the global convergence of a parameter-adjusting Levenberg-Marquardt method

Liyan Qi 1, , Xiantao Xiao 1, and  Liwei Zhang 1,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China, China

Received  January 2015 Revised  March 2015 Published  March 2015

The Levenberg-Marquardt (LM) method is a classical but popular method for solving nonlinear equations. Based on the trust region technique, we propose a parameter-adjusting LM (PALM) method, in which the LM parameter $\mu_k$ is self-adjusted at each iteration based on the ratio between actual reduction and predicted reduction. Under the level-bounded condition, we prove the global convergence of PALM. We also propose a modified parameter-adjusting LM (MPALM) method. Numerical results show that the two methods are very efficient.
Keywords: Nonlinear equations, global convergence., Levenberg-Marquardt method.
Mathematics Subject Classification: Primary: 90C30, 65K0.
Citation: 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
References:
[1]

J. Fan, The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence, Math. Comp., 81 (2012), 447-466. doi: 10.1090/S0025-5718-2011-02496-8.  Google Scholar

[2]

J. Fan, Accelerating the modified Levenberg-Marquardt method for nonlinear equations, Math. Comp., 83 (2014), 1173-1187. doi: 10.1090/S0025-5718-2013-02752-4.  Google Scholar

[3]

J. Fan and J. Pan, Convergence properties of a self-adaptive Levenberg-Marquardt algorithm under local error bound condition, Comput. Optim. Appl., 34 (2006), 47-62. doi: 10.1007/s10589-005-3074-z.  Google Scholar

[4]

J. Fan and Y. 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

[5]

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

[6]

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

[7]

J. J. Moré, Recent developments in algorithms and software for trust region methods, in Mathematical Programming: the state of the art (Bonn, 1982), Springer, Berlin, (1983), 258-287.  Google Scholar

[8]

J. J. Moré, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41. doi: 10.1145/355934.355936.  Google Scholar

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J. Nocedal and S. J. Wright, Numerical optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[10]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, in Topics in numerical analysis, Comput. Suppl., Springer, Vienna, 15 (2001), 239-249. doi: 10.1007/978-3-7091-6217-0_18.  Google Scholar

show all references

References:
[1]

J. Fan, The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence, Math. Comp., 81 (2012), 447-466. doi: 10.1090/S0025-5718-2011-02496-8.  Google Scholar

[2]

J. Fan, Accelerating the modified Levenberg-Marquardt method for nonlinear equations, Math. Comp., 83 (2014), 1173-1187. doi: 10.1090/S0025-5718-2013-02752-4.  Google Scholar

[3]

J. Fan and J. Pan, Convergence properties of a self-adaptive Levenberg-Marquardt algorithm under local error bound condition, Comput. Optim. Appl., 34 (2006), 47-62. doi: 10.1007/s10589-005-3074-z.  Google Scholar

[4]

J. Fan and Y. 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

[5]

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

[6]

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

[7]

J. J. Moré, Recent developments in algorithms and software for trust region methods, in Mathematical Programming: the state of the art (Bonn, 1982), Springer, Berlin, (1983), 258-287.  Google Scholar

[8]

J. J. Moré, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41. doi: 10.1145/355934.355936.  Google Scholar

[9]

J. Nocedal and S. J. Wright, Numerical optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[10]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, in Topics in numerical analysis, Comput. Suppl., Springer, Vienna, 15 (2001), 239-249. doi: 10.1007/978-3-7091-6217-0_18.  Google Scholar

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