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doi: 10.3934/jimo.2021020
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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

* Corresponding author: Weijun Zhou

Received  June 2020 Revised  October 2020 Early access January 2021

Fund Project: The author is supported by National Natural Science Foundation of China (11371073 and 61972055)

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.

Citation: Weijun Zhou. A globally convergent BFGS method for symmetric nonlinear equations. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021020
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##### References:
Test results on Problem 1 with initial point $x_0 = \beta\hat{x}$
 GN-BFGS Algorithm 2.1 $\beta$ $n$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ 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 $\beta$ $n$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ 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
Test results on Problem 2 with initial point $x_0 = \beta\hat{x}$
 GN-BFGS Algorithm 2.1 $\beta$ $n$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ 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 $\beta$ $n$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ $N_{iter}$ $N_F$ $\|F_k\|$ $C_{B_k}$ 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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