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

Basim A. Hassan

doi:10.3934/naco.2019049

Article Contents

2020, Volume 10, Issue 2: 227-235. Doi: 10.3934/naco.2019049

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

Basim A. Hassan^,

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

^* Corresponding author: Basim A. Hassan
^* Corresponding author: Basim A. Hassan

Received: February 2019

Revised: July 2019

Published: April 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. 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]

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Table 2. 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

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Table 3. 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%

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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.
[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.
[3]	R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, ChiChester, New York, 1987.
[4]	A. R. M. Issam, A new limited memory Quasi-Newton method for unconstrained optimization, J. KSIAM, 7 (2003), 7-14.
[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.
[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.
[7]	J. Nocedal and J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, New York, USA. doi: 10.1007/b98874.
[8]	M. J. D. Powell, Algorithms for nonlinear constraints that use Lagrange functions, Math. Programming, 14 (1978), 224-248. doi: 10.1007/BF01588967.
[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.
[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.
[11]	P. Wolfe, Convergence conditions for ascent methods, (Ⅱ): Some corrections, SIAM Review, 13 (1971), 185-188. doi: 10.1137/1013035.
[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.
[13]	Y. Yuan and W. Sun, Theory and Methods of Optimization, Science Press of China, 1999.
[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.
[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.
[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.
[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.
[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.