# 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
##### References:
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##### References:
 [1] R. Byrd and J. Nocedal, A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer., 26 (1989), 727-739.  doi: 10.1137/0726042.  Google Scholar [2] X. W. Fang, Q. Ni and M. L. Zeng, A modified quasi-Newton method for nonlinear equation, Journal of Computational and Applied Mathematics, 328 (2018), 44-58.  doi: 10.1016/j.cam.2017.06.024.  Google Scholar [3] R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, ChiChester, New York, 1987.  Google Scholar [4] A. R. M. Issam, A new limited memory Quasi-Newton method for unconstrained optimization, J. KSIAM, 7 (2003), 7-14.   Google Scholar [5] 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.  Google Scholar [6] J. More, B. Garbow and K. Hillstrome, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41.  doi: 10.1145/355934.355936.  Google Scholar [7] J. Nocedal and J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, New York, USA. doi: 10.1007/b98874.  Google Scholar [8] M. J. D. Powell, Algorithms for nonlinear constraints that use Lagrange functions, Math. Programming, 14 (1978), 224-248.  doi: 10.1007/BF01588967.  Google Scholar [9] Z. Wei, G. Li and L. Qi, New quasi-Newton methods for unconstrained optimization problems, Math Program. Applied Mathematics and Computation, 175 (2006), 1156-1188.  doi: 10.1016/j.amc.2005.08.027.  Google Scholar [10] Z. Wei, G. Li and L. Qi, The superlinear convergence of a modified BFGS- type method for unconstrained optimization, Comput. Optim. Appl., 29 (2004), 315-332.  doi: 10.1023/B:COAP.0000044184.25410.39.  Google Scholar [11] P. Wolfe, Convergence conditions for ascent methods, (Ⅱ): Some corrections, SIAM Review, 13 (1971), 185-188.  doi: 10.1137/1013035.  Google Scholar [12] Y. H. Xiao, Z. X. Wei and L. Zhang, A modified BFGS method without line searches for nonconvex unconstrained optimization, Advances in Theoretical and Applied Mathematics, 1 (2006), 149-162.   Google Scholar [13] Y. Yuan and W. Sun, Theory and Methods of Optimization, Science Press of China, 1999.  Google Scholar [14] G. Yuan and Z. Wei, Convergence analysis of a modified BFGS method on convex minimizations, Comp. Optim. Appl., 47 (2010), 237-255.  doi: 10.1007/s10589-008-9219-0.  Google Scholar [15] G. Yuan, Z. Wei and X. Lu, Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search, Applied Mathematical Modelling, 47 (2017), 811-825.  doi: 10.1016/j.apm.2017.02.008.  Google Scholar [16] G. Yuan, Z. Sheng, B. Wang, W. Hu and C. Li, The global convergence of a modified BFGS method for nonconvex functions, Journal of Computational and Applied Mathematics, 327 (2018), 274-294.  doi: 10.1016/j.cam.2017.05.030.  Google Scholar [17] G. Yuan, Z. Wei and Y. Wu, Modified limited memory BFGS method with nonmonotone line search for unconstrained optimization, J. Korean Math. Soc., 47 (2010), 767-788.  doi: 10.4134/JKMS.2010.47.4.767.  Google Scholar [18] J. Z. Zhang, N. Y. Deng and L. H. Chen, Quasi-Newton equation and related methods for unconstrained optimization, JOTA, 102 (1999), 147-167.  doi: 10.1023/A:1021898630001.  Google Scholar
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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