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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

 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.
Citation: Liyan Qi, Xiantao Xiao, Liwei Zhang. On the global convergence of a parameter-adjusting Levenberg-Marquardt method. Numerical Algebra, Control and 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. [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. [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. [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. [5] K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-166. [6] D. W. Marquardt, An algorithm for least-squares estimation of nonlinear inequalities, SIAM J. Appl. Math., 11 (1963), 431-441. [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. [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. [9] J. Nocedal and S. J. Wright, Numerical optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. [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.

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##### 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. [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. [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. [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. [5] K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-166. [6] D. W. Marquardt, An algorithm for least-squares estimation of nonlinear inequalities, SIAM J. Appl. Math., 11 (1963), 431-441. [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. [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. [9] J. Nocedal and S. J. Wright, Numerical optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. [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.
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