# American Institute of Mathematical Sciences

June  2020, 10(2): 227-235. doi: 10.3934/naco.2019049

## A new type of quasi-newton updating formulas based on the new quasi-newton equation

 Department of Mathematics, College of Computers Sciences and Mathematics, University of Mosul, Iraq

* Corresponding author: Basim A. Hassan

Received  February 2019 Revised  July 2019 Published  September 2019

The quasi-Newton equation is the very foundation of an assortment of the quasi-Newton methods. Therefore, by using the offered alternative equation, we derive the modified BFGS quasi-Newton updating formulas. In this paper, a new y-technique has been introduced to modify the secant equation of the quasi-Newton methods. Prove the global convergence of this algorithm is associated with a line search rule. The numerical results explain that the offered method is effectual for the known test problems.

Citation: Basim A. Hassan. A new type of quasi-newton updating formulas based on the new quasi-newton equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 227-235. doi: 10.3934/naco.2019049
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Some modifications of QN-equations
 Author(s) QN conditions Ref. Powell $B_{k+1}s_k=\tilde{y}_k= \varphi_k y_k +(1-\varphi_k)B_ks_k$ [8] Li and Fukushima $B_{k+1}s_k=\tilde{y}_k= y_k +t_ks_k, t_k \le 10^{-6}$ [5] Wei, Li, and Qi $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{2(f_k-f_{k+1})+(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [9] Zhang, Deng, and Chen $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{6(f_k-f_{k+1})+3(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [18] Yuan and Wei $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{max(0, 2(f_k-f_{k+1})+(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [14] Yuan, Wei and Wu $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{max(0, 6(f_k-f_{k+1})+3(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [17]
 Author(s) QN conditions Ref. Powell $B_{k+1}s_k=\tilde{y}_k= \varphi_k y_k +(1-\varphi_k)B_ks_k$ [8] Li and Fukushima $B_{k+1}s_k=\tilde{y}_k= y_k +t_ks_k, t_k \le 10^{-6}$ [5] Wei, Li, and Qi $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{2(f_k-f_{k+1})+(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [9] Zhang, Deng, and Chen $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{6(f_k-f_{k+1})+3(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [18] Yuan and Wei $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{max(0, 2(f_k-f_{k+1})+(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [14] Yuan, Wei and Wu $B_{k+1}s_k=\tilde{y}_k= y_k +\frac{max(0, 6(f_k-f_{k+1})+3(g_{k+1}+g_k)^Ts_k}{\Vert s_k \Vert^2} s_k$ [17]
Comparison of different BFGS-algorithms with different test functions and different dimensions
 P.No. n BFGS algorithm BBFGS with $u_k=y_k$ BBFGS with $u_k=g_{k+1}$ NI NF NI NF NI NF 1 2 35 140 36 124 8 29 2 2 9 26 8 23 5 16 3 2 43 166 34 123 3 12 4 2 3 30 3 30 3 30 5 2 15 50 15 48 5 17 6 2 2 27 2 27 2 27 7 3 34 113 26 86 7 20 8 3 16 54 15 51 6 18 9 3 2 4 2 4 2 4 10 3 2 27 2 27 2 27 11 3 2 27 2 27 2 27 12 4 20 60 20 60 5 17 13 4 19 61 24 73 4 13 14 4 21 65 23 72 4 10 15 4 17 54 16 49 5 17 16 5 2 27 2 27 2 27 17 6 25 72 33 101 4 12 18 11 3 31 3 31 3 31 19 20 31 102 33 103 4 13 20 400 64 209 91 297 5 17 21 400 2 27 2 27 2 27 22 200 2 5 2 5 2 5 23 100 2 27 2 27 2 27 24 500 9 33 8 28 10 31 25 500 2 4 2 4 2 4 26 500 6 16 7 19 5 14 27 500 57 281 16 114 5 17 28 500 2 4 2 4 2 4 29 500 3 7 3 7 3 7 30 500 3 7 3 7 3 7 Total 453 1756 437 1625 117 527
 P.No. n BFGS algorithm BBFGS with $u_k=y_k$ BBFGS with $u_k=g_{k+1}$ NI NF NI NF NI NF 1 2 35 140 36 124 8 29 2 2 9 26 8 23 5 16 3 2 43 166 34 123 3 12 4 2 3 30 3 30 3 30 5 2 15 50 15 48 5 17 6 2 2 27 2 27 2 27 7 3 34 113 26 86 7 20 8 3 16 54 15 51 6 18 9 3 2 4 2 4 2 4 10 3 2 27 2 27 2 27 11 3 2 27 2 27 2 27 12 4 20 60 20 60 5 17 13 4 19 61 24 73 4 13 14 4 21 65 23 72 4 10 15 4 17 54 16 49 5 17 16 5 2 27 2 27 2 27 17 6 25 72 33 101 4 12 18 11 3 31 3 31 3 31 19 20 31 102 33 103 4 13 20 400 64 209 91 297 5 17 21 400 2 27 2 27 2 27 22 200 2 5 2 5 2 5 23 100 2 27 2 27 2 27 24 500 9 33 8 28 10 31 25 500 2 4 2 4 2 4 26 500 6 16 7 19 5 14 27 500 57 281 16 114 5 17 28 500 2 4 2 4 2 4 29 500 3 7 3 7 3 7 30 500 3 7 3 7 3 7 Total 453 1756 437 1625 117 527
Relative efficiency of the new Algorithms
 BFGS algorithm BBFGS with $u_k=y_k$ BBFGS with $u_k=g_{k+1}$ NI 100% 96.70% 25.82% NF 100% 92.53% 30.01%
 BFGS algorithm BBFGS with $u_k=y_k$ BBFGS with $u_k=g_{k+1}$ NI 100% 96.70% 25.82% NF 100% 92.53% 30.01%
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