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A globally convergent BFGS method for symmetric nonlinear equations
Department of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China |
A BFGS type method is presented to solve symmetric nonlinear equations, which is shown to be globally convergent under suitable conditions. Compared with some existing Gauss-Newton-based BFGS methods whose iterative matrix approximates the Gauss-Newton matrix, an important feature of the proposed method lies in that the iterative matrix is an approximation of the Jacobian, which greatly reduces condition number of the iterative matrix. Numerical results are reported to support the theory.
References:
[1] |
S. Bojari and M. R. Eslahchi,
Global convergence of a family of modified BFGS methods under a modified weak Wolfe-Powell line search for nonconvex functions, 4OR, 18 (2020), 219-244.
doi: 10.1007/s10288-019-00412-2. |
[2] |
R. H. Byrd and J. Nocedal,
A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer. Anal., 26 (1989), 727-739.
doi: 10.1137/0726042. |
[3] |
H. Cao and D. Li,
Adjoint Broyden methods for symmetric nonlinear equations, Pac. J. Optim., 13 (2017), 645-663.
|
[4] |
J. E. Dennis and J. J. Moré,
A characterization of superlinear convergence and its applications to quasi-Newton methods, Math. Comput., 28 (1974), 549-560.
doi: 10.1090/S0025-5718-1974-0343581-1. |
[5] |
Y.-H. Dai,
Convergence properties of the BFGS algorithm, SIAM J. Optim., 13 (2002), 693-701.
doi: 10.1137/S1052623401383455. |
[6] |
G. Gu, D. Li, L. Qi and S. Zhou,
Descent directions of quasi-Newton method for symmetric nonlinear equations, SIAM J. Numer. Anal., 40 (2002), 1763-1774.
doi: 10.1137/S0036142901397423. |
[7] |
D. Li and M. Fukushima,
A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM J. Numer. Anal., 37 (1999), 152-172.
doi: 10.1137/S0036142998335704. |
[8] |
D. Li and M. Fukushima,
A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.
doi: 10.1016/S0377-0427(00)00540-9. |
[9] |
D. Li and M. Fukushima,
On the global convergence of the BFGS method for nonconvex unconstrained optimization problems, SIAM J. Optim., 11 (2001), 1054-1064.
doi: 10.1137/S1052623499354242. |
[10] |
W. F. Mascarenhas,
The BFGS method with exact line searches fails for nonconvex objective functions, Math. Program., 99 (2004), 49-61.
doi: 10.1007/s10107-003-0421-7. |
[11] |
W. Sun and Y. Yuan, Optimization Theory and Methods, Springer Science and Business Media, LLC, New York, 2006. |
[12] |
Z. Wang, Y. Chen, S. Huang and D. Feng,
A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations, Optim. Lett., 8 (2014), 1845-1860.
doi: 10.1007/s11590-013-0678-6. |
[13] |
Z. Wan, K. Teo, X. Chen and C. Hu,
New BFGS method for unconstrained optimization problem based on modified Armijo line search, Optimization, 63 (2014), 285-304.
doi: 10.1080/02331934.2011.644284. |
[14] |
G. Yuan and X. Lu,
A new backtracking inexact BFGS method for symmetric nonlinear equations, Comput. Math. Appl., 55 (2008), 116-129.
doi: 10.1016/j.camwa.2006.12.081. |
[15] |
G. Yuan, Z. Sheng, B. Wang, W. Hu and C. Li,
The global convergence of a modified BFGS method for nonconvex functions, J. Comput. Appl. Math., 327 (2018), 274-294.
doi: 10.1016/j.cam.2017.05.030. |
[16] |
G. Yuan, Z. Wei and X. Lu,
Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search, Appl. Math. Model., 47 (2017), 811-825.
doi: 10.1016/j.apm.2017.02.008. |
[17] |
G. Yuan and S. Yao,
A BFGS algorithm for solving symmetric nonlinear equations, Optimization, 62 (2013), 85-99.
doi: 10.1080/02331934.2011.564621. |
[18] |
L. Zhang,
A derivative-free conjugate residual method using secant condition for general large-scale nonlinear equations, Numer. Algo., 83 (2020), 1277-1293.
doi: 10.1007/s11075-019-00725-7. |
[19] |
L. Zhang and H. Tang,
A hybrid MBFGS and CBFGS method for nonconvex minimization with a global complexity bound, Pac. J. Optim., 14 (2018), 693-702.
|
[20] |
W. Zhou,
A Gauss-Newton-based BFGS method for symmetric nonlinear least squares problems, Pac. J. Optim., 9 (2013), 373-389.
|
[21] |
W. Zhou, A modified BFGS type quasi-Newton method with line search for symmetric nonlinear equations problems, J. Comput. Appl. Math., 367 (2020), 112454, 8 pp.
doi: 10.1016/j.cam.2019.112454. |
[22] |
W. Zhou and X. Chen,
Global convergence of a new hybrid Gauss-Newton structured BFGS methods for nonlinear least squares problems, SIAM J. Optim., 20 (2010), 2422-2441.
doi: 10.1137/090748470. |
[23] |
W. Zhou and D. Li,
On the Q-linear convergence rate of a class of methods for monotone nonlinear equations, Pac. J. Optim., 14 (2018), 723-737.
|
[24] |
W. Zhou and D. Shen,
Convergence properties of an iterative method for solving symmetric nonlinear equations, J. Optim. Theory Appl., 164 (2015), 277-289.
doi: 10.1007/s10957-014-0547-1. |
[25] |
W. Zhou and D. Shen,
An inexact PRP conjugate gradient method for symmetric nonlinear equations, Numer. Funct. Anal. Optim., 35 (2014), 370-388.
doi: 10.1080/01630563.2013.871290. |
[26] |
W. Zhou and F. Wang,
A PRP-based residual method for large-scale monotone nonlinear equations, Appl. Math. Comput., 261 (2015), 1-7.
doi: 10.1016/j.amc.2015.03.069. |
[27] |
W. Zhou and L. Zhang, A modified Broyden-like quasi-Newton method for nonlinear equations, J. Comput. Appl. Math., 372 (2020), 112744, 10 pp.
doi: 10.1016/j.cam.2020.112744. |
show all references
References:
[1] |
S. Bojari and M. R. Eslahchi,
Global convergence of a family of modified BFGS methods under a modified weak Wolfe-Powell line search for nonconvex functions, 4OR, 18 (2020), 219-244.
doi: 10.1007/s10288-019-00412-2. |
[2] |
R. H. Byrd and J. Nocedal,
A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer. Anal., 26 (1989), 727-739.
doi: 10.1137/0726042. |
[3] |
H. Cao and D. Li,
Adjoint Broyden methods for symmetric nonlinear equations, Pac. J. Optim., 13 (2017), 645-663.
|
[4] |
J. E. Dennis and J. J. Moré,
A characterization of superlinear convergence and its applications to quasi-Newton methods, Math. Comput., 28 (1974), 549-560.
doi: 10.1090/S0025-5718-1974-0343581-1. |
[5] |
Y.-H. Dai,
Convergence properties of the BFGS algorithm, SIAM J. Optim., 13 (2002), 693-701.
doi: 10.1137/S1052623401383455. |
[6] |
G. Gu, D. Li, L. Qi and S. Zhou,
Descent directions of quasi-Newton method for symmetric nonlinear equations, SIAM J. Numer. Anal., 40 (2002), 1763-1774.
doi: 10.1137/S0036142901397423. |
[7] |
D. Li and M. Fukushima,
A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM J. Numer. Anal., 37 (1999), 152-172.
doi: 10.1137/S0036142998335704. |
[8] |
D. Li and M. Fukushima,
A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.
doi: 10.1016/S0377-0427(00)00540-9. |
[9] |
D. Li and M. Fukushima,
On the global convergence of the BFGS method for nonconvex unconstrained optimization problems, SIAM J. Optim., 11 (2001), 1054-1064.
doi: 10.1137/S1052623499354242. |
[10] |
W. F. Mascarenhas,
The BFGS method with exact line searches fails for nonconvex objective functions, Math. Program., 99 (2004), 49-61.
doi: 10.1007/s10107-003-0421-7. |
[11] |
W. Sun and Y. Yuan, Optimization Theory and Methods, Springer Science and Business Media, LLC, New York, 2006. |
[12] |
Z. Wang, Y. Chen, S. Huang and D. Feng,
A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations, Optim. Lett., 8 (2014), 1845-1860.
doi: 10.1007/s11590-013-0678-6. |
[13] |
Z. Wan, K. Teo, X. Chen and C. Hu,
New BFGS method for unconstrained optimization problem based on modified Armijo line search, Optimization, 63 (2014), 285-304.
doi: 10.1080/02331934.2011.644284. |
[14] |
G. Yuan and X. Lu,
A new backtracking inexact BFGS method for symmetric nonlinear equations, Comput. Math. Appl., 55 (2008), 116-129.
doi: 10.1016/j.camwa.2006.12.081. |
[15] |
G. Yuan, Z. Sheng, B. Wang, W. Hu and C. Li,
The global convergence of a modified BFGS method for nonconvex functions, J. Comput. Appl. Math., 327 (2018), 274-294.
doi: 10.1016/j.cam.2017.05.030. |
[16] |
G. Yuan, Z. Wei and X. Lu,
Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search, Appl. Math. Model., 47 (2017), 811-825.
doi: 10.1016/j.apm.2017.02.008. |
[17] |
G. Yuan and S. Yao,
A BFGS algorithm for solving symmetric nonlinear equations, Optimization, 62 (2013), 85-99.
doi: 10.1080/02331934.2011.564621. |
[18] |
L. Zhang,
A derivative-free conjugate residual method using secant condition for general large-scale nonlinear equations, Numer. Algo., 83 (2020), 1277-1293.
doi: 10.1007/s11075-019-00725-7. |
[19] |
L. Zhang and H. Tang,
A hybrid MBFGS and CBFGS method for nonconvex minimization with a global complexity bound, Pac. J. Optim., 14 (2018), 693-702.
|
[20] |
W. Zhou,
A Gauss-Newton-based BFGS method for symmetric nonlinear least squares problems, Pac. J. Optim., 9 (2013), 373-389.
|
[21] |
W. Zhou, A modified BFGS type quasi-Newton method with line search for symmetric nonlinear equations problems, J. Comput. Appl. Math., 367 (2020), 112454, 8 pp.
doi: 10.1016/j.cam.2019.112454. |
[22] |
W. Zhou and X. Chen,
Global convergence of a new hybrid Gauss-Newton structured BFGS methods for nonlinear least squares problems, SIAM J. Optim., 20 (2010), 2422-2441.
doi: 10.1137/090748470. |
[23] |
W. Zhou and D. Li,
On the Q-linear convergence rate of a class of methods for monotone nonlinear equations, Pac. J. Optim., 14 (2018), 723-737.
|
[24] |
W. Zhou and D. Shen,
Convergence properties of an iterative method for solving symmetric nonlinear equations, J. Optim. Theory Appl., 164 (2015), 277-289.
doi: 10.1007/s10957-014-0547-1. |
[25] |
W. Zhou and D. Shen,
An inexact PRP conjugate gradient method for symmetric nonlinear equations, Numer. Funct. Anal. Optim., 35 (2014), 370-388.
doi: 10.1080/01630563.2013.871290. |
[26] |
W. Zhou and F. Wang,
A PRP-based residual method for large-scale monotone nonlinear equations, Appl. Math. Comput., 261 (2015), 1-7.
doi: 10.1016/j.amc.2015.03.069. |
[27] |
W. Zhou and L. Zhang, A modified Broyden-like quasi-Newton method for nonlinear equations, J. Comput. Appl. Math., 372 (2020), 112744, 10 pp.
doi: 10.1016/j.cam.2020.112744. |
GN-BFGS | Algorithm 2.1 | ||||||||
0.01 | 9 | 22 | 63 | 1.9e-007 | 1578 | 17 | 33 | 1.27e-008 | 46 |
49 | 288 | 2041 | 7.85e-007 | 1117816 | 135 | 626 | 9.47e-007 | 1419 | |
99 | 1000 | 8307 | 3.2e-006 | 18348874 | 1000 | 8072 | 0.000741 | 8672 | |
0.1 | 9 | 22 | 64 | 4.36e-007 | 1663 | 17 | 36 | 5.04e-007 | 42 |
49 | 298 | 2130 | 8.31e-007 | 1137498 | 118 | 487 | 8.26e-007 | 969 | |
99 | 1000 | 8144 | 0.000850 | 1642370 | 1000 | 7800 | 0.000692 | 10814 | |
1 | 9 | 23 | 67 | 4.76e-007 | 1787 | 19 | 40 | 2.31e-007 | 45 |
49 | 191 | 947 | 1.13e-007 | 1061079 | 160 | 697 | 4.13e-007 | 1255 | |
99 | 1000 | 9113 | 0.000756 | 6266525 | 1000 | 7474 | 0.000590 | 10730 | |
10 | 9 | 24 | 69 | 7.03e-007 | 1781 | 20 | 39 | 1.73e-007 | 45 |
49 | 181 | 861 | 2.21e-007 | 1110234 | 148 | 609 | 4.41e-007 | 1292 | |
99 | 1000 | 8591 | 0.000122 | 1664830 | 1000 | 6719 | 5.27e-005 | 5989 | |
50 | 9 | 25 | 71 | 2.51e-007 | 1713 | 20 | 39 | 4.5e-007 | 43 |
49 | 185 | 883 | 1.11e-007 | 1098144 | 133 | 531 | 8.71e-007 | 1217 | |
99 | 823 | 6041 | 4.55e-008 | 18835000 | 548 | 3361 | 8.72e-007 | 4638 |
GN-BFGS | Algorithm 2.1 | ||||||||
0.01 | 9 | 22 | 63 | 1.9e-007 | 1578 | 17 | 33 | 1.27e-008 | 46 |
49 | 288 | 2041 | 7.85e-007 | 1117816 | 135 | 626 | 9.47e-007 | 1419 | |
99 | 1000 | 8307 | 3.2e-006 | 18348874 | 1000 | 8072 | 0.000741 | 8672 | |
0.1 | 9 | 22 | 64 | 4.36e-007 | 1663 | 17 | 36 | 5.04e-007 | 42 |
49 | 298 | 2130 | 8.31e-007 | 1137498 | 118 | 487 | 8.26e-007 | 969 | |
99 | 1000 | 8144 | 0.000850 | 1642370 | 1000 | 7800 | 0.000692 | 10814 | |
1 | 9 | 23 | 67 | 4.76e-007 | 1787 | 19 | 40 | 2.31e-007 | 45 |
49 | 191 | 947 | 1.13e-007 | 1061079 | 160 | 697 | 4.13e-007 | 1255 | |
99 | 1000 | 9113 | 0.000756 | 6266525 | 1000 | 7474 | 0.000590 | 10730 | |
10 | 9 | 24 | 69 | 7.03e-007 | 1781 | 20 | 39 | 1.73e-007 | 45 |
49 | 181 | 861 | 2.21e-007 | 1110234 | 148 | 609 | 4.41e-007 | 1292 | |
99 | 1000 | 8591 | 0.000122 | 1664830 | 1000 | 6719 | 5.27e-005 | 5989 | |
50 | 9 | 25 | 71 | 2.51e-007 | 1713 | 20 | 39 | 4.5e-007 | 43 |
49 | 185 | 883 | 1.11e-007 | 1098144 | 133 | 531 | 8.71e-007 | 1217 | |
99 | 823 | 6041 | 4.55e-008 | 18835000 | 548 | 3361 | 8.72e-007 | 4638 |
GN-BFGS | Algorithm 2.1 | ||||||||
0.01 | 50 | 4 | 37 | NaN | Inf | 60 | 235 | 8.9382e-007 | 14 |
100 | 5 | 53 | NaN | NaN | 79 | 321 | 8.7896e-007 | 28 | |
200 | 5 | 53 | NaN | Inf | 91 | 362 | 8.363e-007 | 29 | |
0.1 | 50 | 67 | 306 | 7.6196e-007 | 72 | 59 | 234 | 5.3211e-007 | 13 |
100 | 132 | 623 | 6.1482e-007 | 133 | 83 | 341 | 6.9712e-007 | 29 | |
200 | 170 | 879 | 8.2061e-007 | 356 | 88 | 358 | 8.8805e-007 | 51 | |
1 | 50 | 69 | 312 | 9.7115e-007 | 618 | 59 | 225 | 7.0356e-007 | 13 |
100 | 134 | 619 | 9.3146e-007 | 458 | 79 | 322 | 9.6653e-007 | 33 | |
200 | 186 | 956 | 8.5496e-007 | 595 | 101 | 401 | 9.152e-007 | 43 | |
10 | 50 | 18 | 241 | NaN | NaN | 65 | 265 | 7.7764e-007 | 15 |
100 | 15 | 194 | NaN | NaN | 88 | 385 | 9.7307e-007 | 49 | |
200 | 15 | 188 | NaN | NaN | 111 | 461 | 7.7243e-007 | 50 | |
-0.1 | 50 | 77 | 341 | 4.4278e-007 | 59 | 60 | 236 | 9.0422e-007 | 26 |
100 | 120 | 585 | 9.5748e-007 | 136 | 73 | 305 | 9.8125e-007 | 39 | |
200 | 221 | 1011 | 8.6068e-007 | 456 | 90 | 366 | 8.6948e-007 | 40 | |
500 | 296 | 1684 | 9.1214e-007 | 1095 | 88 | 359 | 9.9204e-007 | 79 |
GN-BFGS | Algorithm 2.1 | ||||||||
0.01 | 50 | 4 | 37 | NaN | Inf | 60 | 235 | 8.9382e-007 | 14 |
100 | 5 | 53 | NaN | NaN | 79 | 321 | 8.7896e-007 | 28 | |
200 | 5 | 53 | NaN | Inf | 91 | 362 | 8.363e-007 | 29 | |
0.1 | 50 | 67 | 306 | 7.6196e-007 | 72 | 59 | 234 | 5.3211e-007 | 13 |
100 | 132 | 623 | 6.1482e-007 | 133 | 83 | 341 | 6.9712e-007 | 29 | |
200 | 170 | 879 | 8.2061e-007 | 356 | 88 | 358 | 8.8805e-007 | 51 | |
1 | 50 | 69 | 312 | 9.7115e-007 | 618 | 59 | 225 | 7.0356e-007 | 13 |
100 | 134 | 619 | 9.3146e-007 | 458 | 79 | 322 | 9.6653e-007 | 33 | |
200 | 186 | 956 | 8.5496e-007 | 595 | 101 | 401 | 9.152e-007 | 43 | |
10 | 50 | 18 | 241 | NaN | NaN | 65 | 265 | 7.7764e-007 | 15 |
100 | 15 | 194 | NaN | NaN | 88 | 385 | 9.7307e-007 | 49 | |
200 | 15 | 188 | NaN | NaN | 111 | 461 | 7.7243e-007 | 50 | |
-0.1 | 50 | 77 | 341 | 4.4278e-007 | 59 | 60 | 236 | 9.0422e-007 | 26 |
100 | 120 | 585 | 9.5748e-007 | 136 | 73 | 305 | 9.8125e-007 | 39 | |
200 | 221 | 1011 | 8.6068e-007 | 456 | 90 | 366 | 8.6948e-007 | 40 | |
500 | 296 | 1684 | 9.1214e-007 | 1095 | 88 | 359 | 9.9204e-007 | 79 |
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