Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization

Shummin Nakayama; Yasushi Narushima; Hiroshi Yabe

doi:10.3934/jimo.2018122

Article Contents

2019, Volume 15, Issue 4: 1773-1793. Doi: 10.3934/jimo.2018122

This issue Previous Article Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency Next Article Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints

Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization

1.
Department of Applied Mathematics, Graduate School of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
2.
Faculty of International Social Sciences, Yokohama National University, 79-4 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
3.
Department of Applied Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

* Corresponding author: Shummin Nakayama
* Corresponding author: Shummin Nakayama

Received: June 30, 2017

Revised: January 31, 2018

Early access: August 2018

Published: July 2019

Abstract Full Text(HTML) Figure(4) / Table(5) Related Papers Cited by

Abstract

Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Recently, memoryless quasi-Newton methods based on several kinds of updating formulas were proposed. Since the methods closely related to the conjugate gradient method, the methods are promising. In this paper, we propose a memoryless quasi-Newton method based on the Broyden family with the spectral-scaling secant condition. We focus on the convex and preconvex classes of the Broyden family, and we show that the proposed method satisfies the sufficient descent condition and converges globally. Finally, some numerical experiments are given.

Keywords:

Mathematics Subject Classification: Primary: 90C30, 90C06; Secondary: 65K05.

Citation:

Full Text(HTML)

Figure 1. Performance profiles of methods in Table 2

Download: Full-size image PowerPoint slide

Figure 2. Performance profiles of methods in Table 3

Download: Full-size image PowerPoint slide

Figure 3. Performance profiles of methods in Table 4

Download: Full-size image PowerPoint slide

Figure 4. Performance profiles of methods in Table 5

Download: Full-size image PowerPoint slide

Table 1. Test problems (names and dimensions) by CUTEr library

name	n	name	n	name	n	name	n
AKIVA	2	DIXMAANC	3000	HEART8LS	8	PENALTY1	1000
ALLINITU	4	DIXMAAND	3000	HELIX	3	PENALTY2	200
ARGLINA	200	DIXMAANE	3000	HIELOW	3	PENALTY3	200
ARGLINB	200	DIXMAANF	3000	HILBERTA	2	POWELLSG	5000
ARWHEAD	5000	DIXMAANG	3000	HILBERTB	10	POWER	10000
BARD	3	DIXMAANH	3000	HIMMELBB	2	QUARTC	5000
BDQRTIC	5000	DIXMAANI	3000	HIMMELBF	4	ROSENBR	2
BEALE	2	DIXMAANJ	3000	HIMMELBG	2	S308	2
BIGGS6	6	DIXMAANK	3000	HIMMELBH	2	SCHMVETT	5000
BOX3	3	DIXMAANL	3000	HUMPS	2	SENSORS	100
BOX	10000	DIXON3DQ	10000	JENSMP	2	SINEVAL	2
BRKMCC	2	DJTL	2	KOWOSB	4	SINQUAD	5000
BROWNAL	200	DQDRTIC	5000	LIARWHD	5000	SISSER	2
BROWNBS	2	DQRTIC	5000	LOGHAIRY	2	SNAIL	2
BROWNDEN	4	EDENSCH	2000	MANCINO	100	SPARSINE	5000
BROYDN7D	5000	EG2	1000	MARATOSB	2	SPARSQUR	10000
BRYBND	5000	ENGVAL1	5000	MEXHAT	2	SPMSRTLS	4999
CHAINWOO	4000	ENGVAL2	3	MOREBV	5000	SROSENBR	5000
CHNROSNB	50	ERRINROS	50	MSQRTALS	1024	STRATEC	10
CLIFF	2	EXPFIT	2	MSQRTBLS	1024	TESTQUAD	5000
COSINE	10000	EXTROSNB	1000	NONCVXU2	5000	TOINTGOR	50
CRAGGLVY	5000	FLETCBV2	5000	NONDIA	5000	TOINTGSS	5000
CUBE	2	FLETCHCR	1000	NONDQUAR	5000	TOINTPSP	50
CURLY10	10000	FMINSRF2	5625	OSBORNEA	5	TOINTQOR	50
CURLY20	10000	FMINSURF	5625	OSBORNEB	11	TQUARTIC	5000
CURLY30	10000	FREUROTH	5000	OSCIPATH	10	TRIDIA	5000
DECONVU	63	GENHUMPS	5000	PALMER1C	8	VARDIM	200
DENSCHNA	2	GENROSE	500	PALMER1D	7	VAREIGVL	50
DENSCHNB	2	GROWTHLS	3	PALMER2C	8	VIBRBEAM	8
DENSCHNC	2	GULF	3	PALMER3C	8	WATSON	12
DENSCHND	3	HAIRY	2	PALMER4C	8	WOODS	4000
DENSCHNE	3	HATFLDD	3	PALMER5C	6	YFITU	3
DENSCHNF	2	HATFLDE	3	PALMER6C	8	ZANGWIL2	2
DIXMAANA	3000	HATFLDFL	3	PALMER7C	8
DIXMAANB	3000	HEART6LS	6	PALMER8C	8

| Show Table

DownLoad: CSV

Table 2. Tested methods (Methods 1-9)

Method number	$ \theta_{k-1}$	Class	Global convergence

1 (DFP)	0	convex	not established
2	0.25	convex	established
3	0.5	convex	established
4	0.75	convex	established
5 (BFGS)	1	convex	established
6	1.25	preconvex	established
7	1.5	preconvex	established
8	1.75	preconvex	established
9	2	preconvex	not established

| Show Table

DownLoad: CSV

Table 3. Tested methods (Methods 5 and 10-17)

Method number	$ \theta_{k-1}$	Class	Global convergence

(BFGS)	1	convex	established
10	0.8	convex	established
11	0.85	convex	established
12	0.9	convex	established
13	0.95	convex	established
14	1.05	preconvex	established
15	1.1	preconvex	established
16	1.15	preconvex	established
17	1.2	preconvex	established

| Show Table

DownLoad: CSV

Table 4. Tested methods (Methods 5 and 18-21)

Method number	$\hat\theta_{k-1}$	Class	Global convergence

5 (BFGS)	1	convex	established
18 (Hoshino)	$\dfrac{s_{k-1}^Ty_{k-1}}{(s_{k-1} + \hat\gamma_{k-1}^{-1}y_{k-1})^T y_{k-1}}$	convex	established
19	$1 + \left\|\dfrac{d_{k-1}^Tg_k}{d_{k-1}^Tg_{k-1}}\right\|$	preconvex	established
20	$1 + \dfrac{ d_{k-1}^Tg_k }{\\|d_{k-1}\\| \\|g_{k}\\|}$	unknown	established
21	$1 + \dfrac{ \|d_{k-1}^Tg_k\| }{\\|d_{k-1}\\| \\|g_{k}\\|}$	preconvex	established

| Show Table

DownLoad: CSV

Table 5. Tested methods (Methods 5, 19 and 21-25)

Method number	$\hat\gamma_{k-1}$	$\hat\theta_{k-1}$	Global convergence
5 (BFGS)	(54)	1	established
19	(54)	$1 + \left\|\dfrac{d_{k-1}^Tg_k}{d_{k-1}^Tg_{k-1}}\right\| $	established
21	(54)	$1 + \dfrac{ \|d_{k-1}^Tg_k\| }{\\|d_{k-1}\\| \\|g_{k}\\|}$	established
22 (BFGS)	(56)	1	not established
23	(56)	$1 + \left\|\dfrac{d_{k-1}^Tg_k}{d_{k-1}^Tg_{k-1}}\right\|$	not established
24	(56)	$1 + \dfrac{ \|d_{k-1}^Tg_k\| }{\\|d_{k-1}\\| \\|g_{k}\\|}$	not established
25 (CG_DESCENT) [11,12,14]			established

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	M. Al-Baali, Y. Narushima and H. Yabe, A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Computational Optimization and Applications, 60 (2015), 89-110. doi: 10.1007/s10589-014-9662-z.
[2]	I. Bongart, A. R. Conn, N. I. M. Gould and P. L. Toint, CUTE: constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.
[3]	Z. Chen and W. Cheng, Spectral-scaling quasi-Newton methods with updates from the one parameter of the Broyden family, Journal of Computational and Applied Mathematics, 248 (2013), 88-98. doi: 10.1016/j.cam.2013.01.012.
[4]	W. Y. Cheng and D. H. Li, Spectral scaling BFGS method, Journal of Optimization Theory and Applications, 146 (2010), 305-319. doi: 10.1007/s10957-010-9652-y.
[5]	Y. H. Dai and C. X. Kou, A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search, SIAM Journal on Optimization, 23 (2013), 296-320. doi: 10.1137/100813026.
[6]	Y. H. Dai and L. Z. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Applied Mathematics and Optimization, 43 (2001), 87-101. doi: 10.1007/s002450010019.
[7]	E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213. doi: 10.1007/s101070100263.
[8]	J. A. Ford and I. A. Moghrabi, Multi-step quasi-Newton methods for optimization, Journal of Computational and Applied Mathematics, 50 (1994), 305-323. doi: 10.1016/0377-0427(94)90309-3.
[9]	J. C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42. doi: 10.1137/0802003.
[10]	N.I.M Gould, D. Orban and P.L. Toint, CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 373-394. doi: 10.1145/962437.962439.
[11]	W. W. Hager, Hager's web page: https://people.clas.ufl.edu/hager/.
[12]	W. W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM Journal on Optimization, 16 (2005), 170-192. doi: 10.1137/030601880.
[13]	W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient method, Pacific Journal of Optimization, 2 (2006), 35-58.
[14]	W. W. Hager and H. Zhang, Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent, ACM Transactions on Mathematical Software, 32 (2006), 113-137. doi: 10.1145/1132973.1132979.
[15]	M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436. doi: 10.6028/jres.049.044.
[16]	S. Hoshino, A formulation of variable metric methods, IMA Journal of Applied Mathematics, 10 (1972), 394-403. doi: 10.1093/imamat/10.3.394.
[17]	C. X. Kou and Y. H. Dai, A modified self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno method for unconstrained optimization, Journal of Optimization Theory and Applications, 165 (2015), 209-224. doi: 10.1007/s10957-014-0528-4.
[18]	D. H. Li and M. Fukushima, A modified BFGS method and its global convergence in nonconvex minimization, Journal of Computational and Applied Mathematics, 129 (2001), 15-35. doi: 10.1016/S0377-0427(00)00540-9.
[19]	F. Modarres, M. A. Hassan and W. J. Leong, Memoryless modified symmetric rank-one method for large-scale unconstrained optimization, American Journal of Applied Sciences, 6 (2009), 2054-2059.
[20]	A.U. Moyi and W.J. Leong, A sufficient descent three-term conjugate gradient method via symmetric rank-one update for large-scale optimization, Optimization, 65 (2016), 121-143. doi: 10.1080/02331934.2014.994625.
[21]	S. Nakayama, Y. Narushima and H. Yabe, A memoryless symmetric rank-one method with sufficient descent property for unconstrained optimization, Journal of the Operations Research Society of Japan, 61 (2018), 53-70. doi: 10.15807/jorsj.61.53.
[22]	Y. Narushima and H. Yabe, A survey of sufficient descent conjugate gradient methods for unconstrained optimization, SUT Journal of Mathematics, 50 (2014), 167-203.
[23]	Y. Narushima, H. Yabe and J. A. Ford, A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM Journal on Optimization, 21 (2011), 212-230. doi: 10.1137/080743573.
[24]	J. Nocedal and S. J. Wright, Numerical Optimization, 2^nd edition, Springer, 2006.
[25]	S. S. Oren, Self-scaling variable metric (SSVM) algorithms, Part Ⅱ: Implementation and experiments, Management Science, 20 (1974), 863-874. doi: 10.1287/mnsc.20.5.863.
[26]	S. S. Oren and D. G. Luenberger, Self-scaling variable metric (SSVM) algorithms, Part Ⅰ: Criteria and sufficient conditions for scaling a class of algorithms, Management Science, 20 (1974), 845-862. doi: 10.1287/mnsc.20.5.845.
[27]	D. F. Shanno, Conjugate gradient methods with inexact searches, Mathematics of Operations Research, 3 (1978), 244-256. doi: 10.1287/moor.3.3.244.
[28]	K. Sugiki, Y. Narushima and H. Yabe, Globally convergent three-term conjugate gradient methods that use secant condition and generate descent search directions for unconstrained optimization, Journal of Optimization Theory and Applications, 153 (2012), 733-757. doi: 10.1007/s10957-011-9960-x.
[29]	L. Sun, An approach to scaling symmetric rank-one update, Pacific Journal of Optimization, 2 (2006), 105-118.
[30]	W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, 2006.
[31]	Z. Wei, G. Li and L. Qi, New quasi-Newton methods for unconstrained optimization problems, Applied Mathematics and Computation, 175 (2006), 1156-1188. doi: 10.1016/j.amc.2005.08.027.
[32]	J. Z. Zhang, N. Y. Deng and L. H. Chen, New quasi-Newton equation and related methods for unconstrained optimization, Journal of Optimization Theory and Applications, 102 (1999), 147-167. doi: 10.1023/A:1021898630001.
[33]	L. Zhang, W. Zhou and D. H. Li, A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640. doi: 10.1093/imanum/drl016.
[34]	Y. Zhang and R. P. Tewarson, Quasi-Newton algorithms with updates from the preconvex part of Broyden's family, IMA Journal of Numerical Analysis, 8 (1988), 487-509. doi: 10.1093/imanum/8.4.487.