您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (8): 118-126.doi: 10.6040/j.issn.1671-9352.0.2022.597

•   • 上一篇    

一种求解非线性方程组的改进Shamanskii-like Levenberg-Marquardt算法

房明磊(),丁德凤*(),王敏,盛雨婷   

  1. 安徽理工大学数学与大数据学院, 安徽 淮南 232001
  • 收稿日期:2022-11-09 出版日期:2023-08-20 发布日期:2023-07-28
  • 通讯作者: 丁德凤 E-mail:fmlmath@sina.com;2021201353@edu.aust.cn
  • 作者简介:房明磊(1978—), 男, 硕士, 副教授, 研究方向为最优化理论与算法. E-mail: fmlmath@sina.com
  • 基金资助:
    安徽省高校自然科学基金资助项目(KJ2021A0451);安徽省自然科学基金资助项目(2008085MA01)

A new modified efficient Shamanskii-like Levenberg-Marquardt method for solving systems of nonlinear equations

Minglei FANG(),Defeng DING*(),Ming WANG,Yuting SHENG   

  1. School of Mathmatics and Big Data, Anhui University of Science and Technology, Huainan 232001, Anhui, China
  • Received:2022-11-09 Online:2023-08-20 Published:2023-07-28
  • Contact: Defeng DING E-mail:fmlmath@sina.com;2021201353@edu.aust.cn

RichHTML

2

PDF (PC)

325

摘要：

在Shamanskii-like Levenberg-Marquardt(SLM)算法中引入参数, 得到一种改进的SLM算法, 在m阶非单调Armijo线搜索下证明所提出的算法具有全局收敛性, 并且有m+1阶收敛率。数值实验表明, 算法对于求解非线性方程组大规模问题有效。

关键词: 无约束优化, Armijo线搜索, Shamanskii-like Levenberg-Marquardt算法, 非线性方程组

Abstract:

A new Shamanskii-like Levenberg-Marquardt method(SLM) to solve systems of nonlinear equations is presented by introducing parameters in this paper. With m order nonmonotone Armijo line search, the global convergence of the new algorithm is proved and the convergence rate of whom is shown to be m+1. Numerical experiments demonstrate that the new algorithm can solve large scale problems effectively.

Key words: unconstrained optimization, Armijo line search, Shamanskii-like Levenberg-Marquardt method, systems of nonlinear equations

中图分类号: 

  • O221.2

表1

数值实验的数据和结果1 $(\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-1, m=4) $"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 76.555 3.19E-06 17 9 81.402 3.21E-06
10 17 9 67.410 3.19E-06 17 9 89.941 3.21E-06
100 17 9 76.029 3.19E-06 17 9 101.496 1.06E-07
Problem2 3 000 1 37 19 13.318 1.06E-07 37 19 16.446 1.06E-07
10 37 19 14.538 1.06E-07 37 19 17.367 1.06E-07
100 37 19 13.652 1.06E-07 37 19 17.522 2.05E-07
10 000 1 37 19 426.502 2.05E-07 37 19 474.798 2.05E-07
10 37 19 429.713 2.05E-07 37 19 521.876 2.05E-07
100 37 19 423.808 2.05E-07 37 19 784.704 3.33E-04
Problem3 3 000 1 23 12 10.889 3.32E-04 23 12 12.365 3.33E-04
10 23 12 11.383 3.32E-04 23 12 13.205 3.33E-04
100 23 12 12.419 3.32E-04 23 12 14.297 1.54E-04
10 000 1 25 13 379.290 1.52E-04 25 13 448.392 1.54E-04
10 25 13 381.506 1.52E-04 25 13 449.274 1.54E-04
100 25 13 402.279 1.52E-04 25 13 457.089 3.59E-05
Problem4 3 000 1 21 11 7.337 3.59E-05 21 11 7.499 4.16E-05
10 29 15 11.570 1.41E-04 31 16 12.383 3.83E-05
100 37 19 21.359 6.29E-05 55 28 32.069 6.55E-05
10 000 1 21 11 235.916 6.55E-05 21 11 297.483 1.08E-04
10 31 16 381.813 6.43E-05 31 16 389.080 2.25E-04
100 37 19 976.738 1.15E-04 89 45 1507.956 3.09E+00

表2

数值实验的数据和结果2 $ (\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-1, m=5)$"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 56.153 3.19E-06 17 9 61.007 3.21E-06
10 17 9 69.121 3.19E-06 17 9 97.596 3.21E-06
100 17 9 71.824 3.19E-06 17 9 97.673 3.21E-06
Problem2 1 37 19 15.059 1.06E-07 39 20 28.859 5.70E-08
3 000 10 37 19 14.791 1.06E-07 39 20 31.826 5.70E-08
100 37 19 14.508 1.06E-07 39 20 33.386 5.70E-08
1 37 19 540.234 2.05E-07 39 20 583.387 2.63E-07
10 000 10 37 19 772.284 2.05E-07 39 20 888.355 2.63E-07
100 37 19 749.794 2.05E-07 39 20 860.357 2.63E-07
Problem3 1 23 12 13.445 3.32E-04 23 12 27.244 3.33E-04
3 000 10 23 12 13.146 3.32E-04 23 12 23.083 3.33E-04
100 23 12 14.902 3.32E-04 23 12 26.641 3.33E-04
1 23 12 12.463 3.32E-04 25 13 700.327 1.54E-04
10 000 10 25 13 501.323 1.52E-04 25 13 562.015 1.54E-04
100 25 13 470.546 1.52E-04 25 13 692.879 1.54E-04
Problem4 1 21 11 12.101 3.59E-05 21 11 17.185 3.59E-05
3 000 10 29 15 16.989 1.41E-04 31 16 27.769 4.16E-05
100 37 19 48.289 6.29E-05 55 28 58.192 3.83E-05
1 21 11 261.971 6.55E-05 21 11 384.457 6.55E-05
10 000 10 31 16 431.609 6.43E-05 31 16 666.508 1.08E-04
100 37 19 2 034.931 1.15E-04 89 45 2 414.532 2.25E-04

表3

数值实验的数据和结果3 $(\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-2, m=4) $"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 56.100 3.19E-06 17 9 81.402 3.21E-06
10 17 9 56.283 3.19E-06 17 9 89.941 3.21E-06
100 17 9 55.768 3.19E-06 17 9 101.496 1.06E-07
Problem2 1 27 14 12.161 1.12E-04 37 19 16.446 1.06E-07
3 000 10 27 14 11.688 1.12E-04 37 19 17.367 1.06E-07
100 27 14 13.202 1.12E-04 37 19 17.522 2.05E-07
1 27 14 423.625 2.04E-04 37 19 474.798 2.05E-07
10 000 10 27 14 650.622 2.04E-04 37 19 521.876 2.05E-07
100 27 14 581.387 2.04E-04 37 19 784.704 3.33E-04
Problem3 1 25 13 12.663 1.17E-04 23 12 12.365 3.33E-04
3 000 10 25 13 14.205 1.17E-04 23 12 13.205 3.33E-04
100 25 13 14.663 1.17E-04 23 12 14.297 1.54E-04
1 25 13 757.739 1.65E-04 25 13 448.392 1.54E-04
10 000 10 25 13 926.005 1.65E-04 25 13 449.274 1.54E-04
100 25 13 488.993 1.65E-04 25 13 457.089 3.59E-05
Problem4 1 21 11 8.214 3.59E-05 21 11 7.498515 4.16E-05
3 000 10 29 15 13.087 1.41E-04 31 16 12.383 3.83E-05
100 37 19 25.485 6.29E-05 55 28 32.069 6.55E-05
1 21 11 268.594 6.55E-05 21 11 297.483 1.08E-04
10 000 10 31 16 441.670 6.43E-05 31 16 389.080 2.25E-04
100 37 19 1 131.421 1.15E-04 89 45 1 507.956 3.09E+00

表4

数值实验的数据和结果4 $ (\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-2, m=5)$"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 55.984 3.19E-06 200 100 531.416 5.10E-04
10 17 9 58.558 3.19E-06 200 100 545.202 5.10E-04
100 17 9 58.388 3.19E-06 200 100 546.315 5.10E-04
Problem2 1 27 14 11.443 1.12E-04 27 14 12.28 1.12E-04
3 000 10 27 14 17.987 1.12E-04 27 14 18.058 1.12E-04
100 27 14 16.723 1.12E-04 27 14 17.898 1.12E-04
1 27 14 598.614 2.04E-04 27 14 634.585 2.07E-04
10 000 10 27 14 503.249 2.04E-04 27 14 616.028 2.07E-04
100 27 14 555.029 2.04E-04 27 14 574.702 2.07E-04
Problem3 1 25 13 11.769 1.17E-04 25 13 13.818 1.17E-04
3 000 10 25 13 12.683 1.17E-04 25 13 19.147 1.17E-04
100 25 13 14.274 1.17E-04 25 13 18.621 1.17E-04
1 25 13 458.903 1.65E-04 25 13 754.511 1.64E-04
10 000 10 25 13 455.095 1.65E-04 25 13 739.808 1.64E-04
100 25 13 434.821 1.65E-04 25 13 2794.36 1.64E-04
Problem4 1 21 11 8.238 3.59E-05 21 11 11.469 3.59E-05
3 000 10 29 15 12.516 1.41E-04 31 16 18.836 4.16E-05
100 37 19 23.106 6.29E-05 55 28 38.185 3.83E-05
1 21 11 257.89 6.55E-05 21 11 320.29 6.55E-05
10 000 10 31 16 551.884 6.43E-05 31 16 568.913 1.08E-04
100 37 19 1 987.197 1.15E-04 89 45 7 778.456 2.25E-04
1 LEVENBERG K . A method for the solution of certain non-linear problems in least squares[J]. Quarterly of Applied Mathematics, 1944, 2 (2): 164- 168.
doi: 10.1090/qam/10666
2 MARQUARDT D W . An algorithm for least-squares estimation of nonlinear parameters[J]. Journal of the Society for Industrial and Applied Mathematics, 1963, 11 (2): 431- 441.
doi: 10.1137/0111030
3 YAMASHITA N , FUKUSHIMA M . On the rate of convergence of the Levenberg-Marquardt method[M]. New York: Springer, 2001: 239- 249.
4 FAN J Y . The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence[J]. Mathematics of Computation, 2012, 81 (277): 447- 466.
doi: 10.1090/S0025-5718-2011-02496-8
5 FAN J Y . A Shamanskii-like Levenberg-Marquardt method for nonlinear equations[J]. Computational Optimization and Applications, 2013, 56 (1): 63- 80.
doi: 10.1007/s10589-013-9549-4
6 ZHOU W . On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order Armijo type line search[J]. Journal of Computational and Applied Mathematics, 2013, 239, 152- 161.
doi: 10.1016/j.cam.2012.09.025
7 AMINI K , ROSTAMI F . Three-steps modified Levenberg-Marquardt method with a new line search for systems of nonlinear equations[J]. Journal of Computational and Applied Mathematics, 2016, 300, 30- 42.
doi: 10.1016/j.cam.2015.12.013
8 CHEN L , MA Y F . Shamanskii-like Levenberg-Marquardt method with a new line search for systems of nonlinear equations[J]. Journal of Systems Science and Complexity, 2020, 33 (5): 1694- 1707.
doi: 10.1007/s11424-020-9043-x
9 AMINI K , ROSTAMI F , CARISTI G . An efficient Levenberg-Marquardt method with a new LM parameter for systems of nonlinear equations[J]. Optimization, 2018, 67 (5): 637- 650.
doi: 10.1080/02331934.2018.1435655
10 MA Changfeng , JIANG Lihua . Some research on Levenberg-Marquardt method for the nonlinear equations[J]. Applied Mathematics and Computation, 2007, 184 (2): 1032- 1040.
doi: 10.1016/j.amc.2006.07.004
11 DENNIS J E , MORÉ J J . A characterization of super linear convergence and its application to quasi-Newton methods[J]. Mathematics of Computation, 1974, 28 (126): 549- 560.
doi: 10.1090/S0025-5718-1974-0343581-1
12 WU Z X , ZHOU T , LI L , et al. A new modified efficient Levenberg-Marquardt method for solving systems of nonlinear equations[J]. Mathematical Problems in Engineering, 2021, 2021, 1- 11.
13 BEHLING R , IUSEM A . The effect of calmness on the solution set of systems of nonlinear equations[J]. Mathematical Programming, 2013, 137 (1): 155- 165.
14 MORÉ J J , GARBOW B S , HILLSTROM K E . Testing unconstrained optimization software[J]. ACM Transactions on Mathematical Software, 1981, 7 (1): 17- 41.
doi: 10.1145/355934.355936
15 SCHNABEL R B , FRANK P D . Tensor methods for nonlinear equations[J]. SIAM Journal on Numerical Analysis, 1984, 21 (5): 815- 843.
doi: 10.1137/0721054
[1] 王松华,罗丹,黎勇. 一类新型的修正WYL共轭梯度算法[J]. 《山东大学学报(理学版)》, 2021, 56(9): 87-95.
[2] 林穗华. Wolfe线搜索下的修正FR谱共轭梯度法[J]. 山东大学学报（理学版）, 2017, 52(4): 6-12.
[3] 郑秀云,史加荣. Armijo型线搜索下的全局收敛共轭梯度法[J]. 山东大学学报（理学版）, 2017, 52(1): 98-101.
[4] 崔建斌,姬安召,鲁洪江,王玉风,何姜毅,许泰. Schwarz Christoffel变换数值解法[J]. 山东大学学报（理学版）, 2016, 51(4): 104-111.
[5] 王开荣,王书敏. 具有充分下降性的修正型混合共轭梯度法[J]. J4, 2013, 48(09): 78-84.
[6] 王洋. 求解一类非线性方程组的Newton-PLHSS方法[J]. J4, 2012, 47(12): 96-102.
[7] 李敏,陈宇,屈爱平. 一种充分下降的DY共轭梯度法及其收敛性[J]. J4, 2011, 46(7): 101-105.
[8] 程李晴1,2, 石巧连2. 一种新的混合共轭梯度算法[J]. J4, 2010, 45(6): 81-85.
[9] 程李晴. 一类共轭梯度法的全局收敛性[J]. J4, 2010, 45(5): 101-105.
[10] . 一类新的Wolfe线性搜索下的记忆梯度法[J]. J4, 2009, 44(7): 33-37.
[11] 陈 蓓 . 求解Toeplitz矩阵特征值反问题的不精确牛顿方法[J]. J4, 2008, 43(9): 89-93 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 冒爱琴1, 2, 杨明君2, 3, 俞海云2, 张品1, 潘仁明1*. 五氟乙烷灭火剂高温热解机理研究[J]. J4, 2013, 48(1): 51 -55 .
[2] 李永明1, 丁立旺2. PA误差下半参数回归模型估计的r-阶矩相合[J]. J4, 2013, 48(1): 83 -88 .
[3] 董丽红1,2,郭双建1. Yetter-Drinfeld模范畴上的弱Hopf模基本定理[J]. J4, 2013, 48(2): 20 -22 .
[4] 唐风琴1,白建明2. 一类带有广义负上限相依索赔额的风险过程大偏差[J]. J4, 2013, 48(1): 100 -106 .
[5] 程智1,2,孙翠芳2,王宁1,杜先能1. 关于Zn的拉回及其性质[J]. J4, 2013, 48(2): 15 -19 .
[6] 汤晓宏1，胡文效2*，魏彦锋2，蒋锡龙2，张晶莹2，. 葡萄酒野生酿酒酵母的筛选及其生物特性的研究[J]. 山东大学学报（理学版）, 2014, 49(03): 12 -17 .
[7] 廖明哲. 哥德巴赫的两个猜想[J]. J4, 2013, 48(2): 1 -14 .
[8] 赵同欣1,刘林德1*,张莉1,潘成臣2,贾兴军1. 紫藤传粉昆虫与花粉多型性研究[J]. 山东大学学报（理学版）, 2014, 49(03): 1 -5 .
[9] 王开荣,高佩婷. 建立在DY法上的两类混合共轭梯度法[J]. 山东大学学报（理学版）, 2016, 51(6): 16 -23 .
[10] 罗斯特,卢丽倩,崔若飞,周伟伟,李增勇*. Monte-Carlo仿真酒精特征波长光子在皮肤中的传输规律及光纤探头设计[J]. J4, 2013, 48(1): 46 -50 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发