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doi: 10.3934/jimo.2021222
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On convergence properties of the modified trust region method under Hölderian error bound condition

1. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Jinyan Fan

Received  May 2021 Revised  September 2021 Early access December 2021

Fund Project: The authors are supported by the National Natural Science Foundation of China grant 11971309

Trust region method is one of the important methods for nonlinear equations. In this paper, we show that the modified trust region method converges globally under the Hölderian continuity of the Jacobian. The convergence order of the method is also given under the Hölderian error bound condition.

Citation: Jirui Ma, Jinyan Fan. On convergence properties of the modified trust region method under Hölderian error bound condition. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021222
References:
[1]

M. AhookhoshF. J. AragónR. M. T. Fleming and Phan T. Vuong, Local convergence of Levenberg-Marquardt methods under Hölderian metric subregularity, Adv. Comput. Math., 45 (2019), 2771-2806.  doi: 10.1007/s10444-019-09708-7.

[2]

J. Fan, Convergence rate of the trust region method for nonlinear equations under local error bound condition, Comput. Optim. Appl., 34 (2006), 215-227.  doi: 10.1007/s10589-005-3078-8.

[3]

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.

[4]

J. FanJ. Huang and J. Pan, An adaptive multi-step Levenberg-Marquardt method, J. Sci. Comput., 78 (2019), 531-548.  doi: 10.1007/s10915-018-0777-8.

[5]

J. Fan and N. Lu, On the modified trust region algorithm for nonlinear equations, Optim. Methods Softw., 30 (2015), 478-491.  doi: 10.1080/10556788.2014.932943.

[6]

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.

[7]

C. T. Kelley, Solving Nonlinear Equations with Newton's Method, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718898.

[8]

K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-168.  doi: 10.1090/qam/10666.

[9]

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

[10]

J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, in G. A. Watson, ed., Lecture Notes in Mathematics 630: Numerical Analysis, Springer-Verlag, Berlin, (1978), 105–116.

[11]

M. J. D. Powell, Convergence properties of a class of minimization algorithms, Nonlinear Programming, 2 (1974), 1-27. 

[12]

H. Wang and J. Fan, Convergence rate of the Levenberg-Marquardt method under Hölderian error bound, Optim. Methods Softw., 35 (2020), 767-786.  doi: 10.1080/10556788.2019.1694927.

[13]

H. Wang and J. Fan, Convergence properties of inexact Levenberg-Marquardt method under Hölderian error bound, J. Ind. Manag. Optim., 17 (2021), 2265-2275.  doi: 10.3934/jimo.2020068.

[14]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, Topics in Numerical Analysis, 15 (2001), 239-249.  doi: 10.1007/978-3-7091-6217-0_18.

[15]

Y. Yuan, Trust region algorithms for nonlinear equations, Information, 1 (1998), 7-20. 

[16]

Y. Yuan, Subspace methods for large scale nonlinear equations and nonlinear least squares, Optim. Eng., 10 (2009), 207-218.  doi: 10.1007/s11081-008-9064-0.

[17]

Y. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least sqaures, Numer. Algebra Control Optim., 1 (2011), 15-34.  doi: 10.3934/naco.2011.1.15.

[18]

X. Zhu and G.-H. Lin, Improved convergence results for a modified Levenberg-Marquardt method for nonlinear equations and applications in MPCC, Optim. Methods Softw., 31 (2016), 791-804.  doi: 10.1080/10556788.2016.1171863.

show all references

References:
[1]

M. AhookhoshF. J. AragónR. M. T. Fleming and Phan T. Vuong, Local convergence of Levenberg-Marquardt methods under Hölderian metric subregularity, Adv. Comput. Math., 45 (2019), 2771-2806.  doi: 10.1007/s10444-019-09708-7.

[2]

J. Fan, Convergence rate of the trust region method for nonlinear equations under local error bound condition, Comput. Optim. Appl., 34 (2006), 215-227.  doi: 10.1007/s10589-005-3078-8.

[3]

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.

[4]

J. FanJ. Huang and J. Pan, An adaptive multi-step Levenberg-Marquardt method, J. Sci. Comput., 78 (2019), 531-548.  doi: 10.1007/s10915-018-0777-8.

[5]

J. Fan and N. Lu, On the modified trust region algorithm for nonlinear equations, Optim. Methods Softw., 30 (2015), 478-491.  doi: 10.1080/10556788.2014.932943.

[6]

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.

[7]

C. T. Kelley, Solving Nonlinear Equations with Newton's Method, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718898.

[8]

K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-168.  doi: 10.1090/qam/10666.

[9]

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

[10]

J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, in G. A. Watson, ed., Lecture Notes in Mathematics 630: Numerical Analysis, Springer-Verlag, Berlin, (1978), 105–116.

[11]

M. J. D. Powell, Convergence properties of a class of minimization algorithms, Nonlinear Programming, 2 (1974), 1-27. 

[12]

H. Wang and J. Fan, Convergence rate of the Levenberg-Marquardt method under Hölderian error bound, Optim. Methods Softw., 35 (2020), 767-786.  doi: 10.1080/10556788.2019.1694927.

[13]

H. Wang and J. Fan, Convergence properties of inexact Levenberg-Marquardt method under Hölderian error bound, J. Ind. Manag. Optim., 17 (2021), 2265-2275.  doi: 10.3934/jimo.2020068.

[14]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, Topics in Numerical Analysis, 15 (2001), 239-249.  doi: 10.1007/978-3-7091-6217-0_18.

[15]

Y. Yuan, Trust region algorithms for nonlinear equations, Information, 1 (1998), 7-20. 

[16]

Y. Yuan, Subspace methods for large scale nonlinear equations and nonlinear least squares, Optim. Eng., 10 (2009), 207-218.  doi: 10.1007/s11081-008-9064-0.

[17]

Y. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least sqaures, Numer. Algebra Control Optim., 1 (2011), 15-34.  doi: 10.3934/naco.2011.1.15.

[18]

X. Zhu and G.-H. Lin, Improved convergence results for a modified Levenberg-Marquardt method for nonlinear equations and applications in MPCC, Optim. Methods Softw., 31 (2016), 791-804.  doi: 10.1080/10556788.2016.1171863.

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