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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