一个带重启步的改进PRP型谱共轭梯度法

数学物理学报 ›› 2022, Vol. 42 ›› Issue (1): 216-227.

一个带重启步的改进PRP型谱共轭梯度法

江羡珍(),廖伟(),简金宝*(),毋晓迪()

广西民族大学数学与物理学院, 应用数学与人工智能研究中心 & 广西混杂计算与集成电路设计分析重点实验室南宁 530006

收稿日期:2021-01-18 出版日期:2022-02-26 发布日期:2022-02-23
通讯作者: 简金宝 E-mail:yl2811280@163.com;1436134351@qq.com;jianjb@gxu.edu.cn;1710703068@qq.com
作者简介:江羡珍, E-mail: yl2811280@163.com|廖伟, E-mail: 1436134351@qq.com|毋晓迪, E-mail: 1710703068@qq.com
基金资助:
广西自然科学基金(2020GXNSFDA238017);广西自然科学基金(2018GXNSFAA281099);广西民族大学科研基金(2018KJQD02);广西民族大学研究生创新项目(gxun-chxp201909)

An Improved PRP Type Spectral Conjugate Gradient Method with Restart Steps

Xianzhen Jiang(),Wei Liao(),Jinbao Jian*(),Xiaodi Wu()

Center for Applied Mathematics and Artificial Intelligence & Guangxi Key Laboratory of Hybrid Compution and IC Design Analysis, College of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006

Received:2021-01-18 Online:2022-02-26 Published:2022-02-23
Contact: Jinbao Jian E-mail:yl2811280@163.com;1436134351@qq.com;jianjb@gxu.edu.cn;1710703068@qq.com
Supported by:
the NSF of Guangxi Province(2020GXNSFDA238017);the NSF of Guangxi Province(2018GXNSFAA281099);the Research Project of Guangxi University for Nationalities(2018KJQD02);the Innovation Project of Guangxi University for Nationalities Graduate Education(gxun-chxp201909)

摘要/Abstract

摘要：

Polak-Ribière-Polak（PRP）方法是经典共轭梯度法中数值表现较好的方法之一.结合Wolfe非精确线搜索准则对PRP公式进行改进，从而产生新的共轭参数，并基于新共轭参数设计新的谱参数，引入重启条件并构造新的重启方向，进而建立一个带重启步的谱共轭梯度算法.在常规假设及强Wolfe非精确线搜索步长准则下，算法具有充分下降性和全局收敛性.最后，对算法进行中大规模数值实验并与当前公认数值效果较好的同类方法进行比较，结果表明新算法是很有效的.

关键词: 无约束优化, 谱共轭梯度法, 重启方向, 强Wolfe线搜索

Abstract:

The Polak-Ribière-Polak algorithm is considered one of the most efficient methods among classical conjugate gradient methods (CGMs). To generate new conjugate parameter, an improved PRP formula is proposed by combining the strong Wolfe line search condition. Furthermore, a new spectral parameter and a new restart direction are designed, and thus a new spectral conjugate gradient method with restart steps is established. Using the strong Wolfe line search condition to yield the step length, the sufficient descent property and global convergence of the new algorithm are obtained under the general assumptions. Finally, for the proposed algorithm, a medium-large scale numerical experiments is performed, and compared with some existing efficient CGMs, the numerical results show that the proposed algorithm is very promising.

Key words: Unconstrained optimization, Spectral conjugate gradient method, Restart direction, Strong Wolfe line search

中图分类号:

O221

江羡珍,廖伟,简金宝,毋晓迪. 一个带重启步的改进PRP型谱共轭梯度法[J]. 数学物理学报, 2022, 42(1): 216-227.

Xianzhen Jiang,Wei Liao,Jinbao Jian,Xiaodi Wu. An Improved PRP Type Spectral Conjugate Gradient Method with Restart Steps[J]. Acta mathematica scientia,Series A, 2022, 42(1): 216-227.

图/表 8

图 1

图 2

图 3

图 4

表 1

数值试验报告"

Problems	JLJW+	KD	DK	HZ	SPF2
Name/n	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|
bard 3	1430/4.93/7.6e-07	1565/5.19/8.0e-07	F/F/3.2e-05	620/2.00/4.0e-07	235/0.64/6.6e-07
beale 2	631/0.96/3.6e-07	326/0.52/7.7e-07	86/0.11/1.6e-07	251/0.37/6.2e-07	142/0.20/4.3e-07
box 3	150/0.26/2.1e-07	330/0.57/6.0e-07	134/0.26/1.4e-08	475/0.89/8.9e-07	91/0.13/5.2e-07
cosine 300	19/0.03/5.2e-07	26/0.09/3.3e-07	F/F/1.2e-04	32/0.06/4.9e-08	F/F/3.0e+02
cosine 1500	1560/11.26/1.5e-07	F/F/1.4e+03	F/F/2.8e+03	F/F/1.2e+02	F/F/1.8e+03
cosine 4500	F/F/2.9e+02	F/F/5.8e-04	F/F/1.2e+04	477/11.19/2.9e-07	F/F/2.1e+03
dixmaana 3000	18/0.99/3.4e-07	17/0.90/3.1e-07	23/1.22/6.0e-07	26/1.75/1.3e-07	24/1.38/1.9e-07
dixmaana 12000	18/3.14/6.8e-07	17/2.93/6.2e-07	25/4.16/1.1e-07	21/3.70/1.6e-07	24/4.05/3.7e-07
dixmaanb 3000	11/0.39/7.4e-07	12/0.50/3.3e-07	34/1.76/1.7e-07	49/3.92/9.6e-08	14/0.61/1.5e-07
dixmaanb 12000	11/1.17/4.4e-07	12/1.51/5.8e-07	47/10.10/9.0e-08	40/10.30/6.5e-07	13/1.55/2.7e-07
dixmaanc 3000	24/1.48/3.8e-08	25/1.54/5.5e-07	29/1.62/2.4e-07	33/2.20/1.9e-07	35/2.18/8.0e-07
dixmaanc 12000	16/2.37/1.0e-06	25/4.38/4.5e-07	27/4.39/8.2e-07	73/18.86/5.6e-07	67/15.50/9.6e-07
dixmaand 3000	25/1.28/1.3e-07	25/1.32/3.6e-07	29/1.56/3.3e-07	54/4.45/2.3e-07	45/2.73/6.7e-07
dixmaand 12000	22/3.41/5.5e-07	23/3.67/6.5e-07	44/8.42/7.7e-07	29/5.73/7.2e-07	50/10.73/2.7e-07
dixmaane 6000	1308/199.45/9.0e-07	1229/192.04/9.3e-07	1941/331.01/8.5e-07	1716/274.31/9.3e-07	564/82.65/7.7e-07
dixmaanf 4500	775/94.63/8.3e-07	1258/161.65/7.9e-07	396/53.77/8.5e-07	1165/155.96/9.0e-07	470/59.77/7.1e-07
dixmaanf 9000	1080/237.42/9.3e-07	1078/246.31/8.8e-07	1396/340.53/9.4e-07	F/F/1.8e-05	383/80.92/4.1e-07
dixmaang 7500	889/168.72/8.8e-07	1154/220.67/5.8e-07	1239/250.95/8.9e-07	1682/327.37/9.0e-07	614/112.05/9.5e-07
dixmaanh 4500	1417/179.40/9.2e-07	1324/169.60/9.8e-07	1006/137.94/7.2e-07	953/120.53/9.9e-07	704/81.78/9.8e-07
dixmaani 120	1930/9.09/8.7e-07	1595/7.68/6.7e-07	F/F/1.2e-03	F/F/8.4e-07	731/3.29/4.2e-07
dixmaanj 2700	1193/94.28/8.5e-07	1382/108.65/6.2e-07	1320/110.36/9.5e-07	F/F/1.0e-05	1153/85.27/6.4e-07
dixmaank 3000	1142/98.31/6.5e-07	1158/100.76/7.8e-07	F/F/4.0e-05	F/F/2.7e-05	1799/154.65/6.6e-07
dixmaanl 300	1554/14.02/9.8e-07	1424/12.64/8.9e-07	F/F/4.6e-04	F/F/8.0e-05	826/7.12/7.4e-07
dixon3dq 50	1199/1.50/8.8e-07	1613/2.28/6.0e-07	F/F/5.9e-05	1836/2.41/5.3e-07	629/0.82/6.0e-07
dixon3dq 88	1934/2.80/8.3e-07	1476/2.09/6.7e-07	F/F/1.0e-02	F/F/4.9e-04	1227/1.65/5.2e-07
dqdrtic 60000	496/35.92/7.8e-07	426/31.92/2.8e-07	197/13.41/4.1e-07	917/67.92/7.1e-07	312/21.78/6.2e-07
dqrtic 100	22/0.06/5.7e-07	23/0.04/6.4e-07	34/0.04/7.7e-07	28/0.03/1.8e-07	60/0.12/1.8e-07
dqrtic 450	34/0.13/2.8e-07	47/0.19/3.4e-07	33/0.15/7.4e-07	31/0.13/5.0e-07	60/0.36/5.7e-07
edensch 10000	46/4.59/9.5e-07	54/6.25/6.2e-07	F/F/8.7e-06	F/F/3.8e-06	F/F/3.3e-06
edensch 50000	55/24.21/1.9e-07	65/29.17/8.8e-07	F/F/1.0e-04	F/F/1.8e-05	F/F/1.2e-04
edensch 100000	126/93.46/9.7e-07	F/F/4.8e-05	F/F/2.3e-05	120/139.38/5.3e-07	F/F/2.0e-04
eg2 30	F/F/2.9e-06	F/F/3.0e-01	313/0.43/8.0e-07	F/F/3.2e-03	F/F/1.1e-05
eg2 80	F/F/2.5e-04	F/F/5.8e-06	F/F/4.0e-01	F/F/2.8e-02	F/F/1.9e-05
engval1 6	87/0.11/1.9e-07	95/0.14/2.4e-08	F/F/4.7e-01	F/F/8.8e-01	F/F/1.8e-01
fletchcr 10000	145/3.81/1.3e-07	F/F/1.1e-03	F/F/3.1e-04	F/F/9.5e-04	F/F/6.9e-04
fletchcr 300000	211/118.78/3.9e-07	F/F/9.2e-03	F/F/6.2e-03	F/F/9.5e-04	F/F/2.6e-02
freuroth 20	1700/2.52/9.8e-07	476/0.66/9.2e-07	F/F/1.5e-05	455/0.60/6.8e-07	F/F/6.8e-05
freuroth 36	F/F/2.2e-06	F/F/6.3e-06	F/F/8.2e-06	F/F/6.5e-05	F/F/7.5e-05
genrose 2000	574/2.28/2.8e-07	399/1.94/4.1e-07	395/1.77/6.1e-07	1259/5.68/9.6e-07	692/2.85/4.4e-07
genrose 47000	270/25.44/6.7e-07	336/31.54/1.3e-07	474/42.83/7.1e-07	854/76.03/9.0e-07	571/48.69/8.7e-07
gulf 3	2/0.00/0.0e+00	2/0.00/0.0e+00	2/0.00/0.0e+00	2/0.00/0.0e+00	2/0.00/0.0e+00
helix 3	479/1.02/1.6e-07	282/0.62/4.2e-07	316/0.60/7.0e-07	731/1.56/6.7e-07	F/F/3.5e+03
himmelbg 1000	3/0.01/1.6e-28	3/0.00/1.3e-28	3/0.00/1.6e-28	3/0.00/1.2e-28	3/0.00/1.6e-28
himmelbg 10000	3/0.01/5.2e-28	3/0.01/4.2e-28	3/0.01/5.0e-28	3/0.01/3.8e-28	3/0.01/5.0e-28
himmelbg 100000	3/0.10/1.6e-27	3/0.11/1.3e-27	3/0.12/1.6e-27	3/0.11/1.2e-27	3/0.12/1.6e-27
kowosb 4	774/1.40/7.0e-07	817/1.55/4.7e-07	1574/3.07/8.7e-07	F/F/5.2e-04	239/0.43/7.5e-07
liarwhd 500	487/1.06/1.9e-07	750/1.63/9.4e-07	1168/2.52/8.5e-07	F/F/2.9e-03	268/0.49/1.2e-07
liarwhd 1000	534/1.43/2.3e-07	F/F/1.3e-04	F/F/1.6e-02	1267/3.35/9.8e-07	294/0.75/4.7e-07
liarwhd 10000	F/F/5.6e-02	F/F/5.8e+00	F/F/3.2e+03	F/F/1.5e+03	409/8.96/8.9e-07
nondquar 4	F/F/1.2e-04	F/F/5.0e-05	F/F/4.9e-03	395/0.66/2.2e-07	87/0.12/6.5e-07
penalty1 1000	15/0.82/1.2e-07	15/0.82/1.2e-07	15/0.86/1.2e-07	15/0.87/1.2e-07	15/0.89/1.2e-07
penalty1 10000	9/28.41/9.6e-07	9/28.32/9.6e-07	9/28.02/9.6e-07	9/28.38/9.6e-07	9/28.46/9.6e-07
quartc 100	22/0.04/5.7e-07	23/0.04/6.4e-07	34/0.05/7.7e-07	28/0.05/1.8e-07	60/0.13/1.8e-07
quartc 450	34/0.14/2.8e-07	47/0.20/3.4e-07	33/0.14/7.4e-07	31/0.12/5.0e-07	60/0.31/5.7e-07

表 1

表 2

数值试验报告(续)"

Problems	JLJW+	KD	DK	HZ	SPF2
Name/n	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|	Itr/Tcpu/\|\|g_*\|\|
tridia 200	1491/2.44/3.2e-07	1380/2.41/9.8e-07	1699/2.90/9.3e-07	F/F/5.9e-06	593/0.94/5.5e-07
tridia 400	1905/3.90/5.7e-07	F/F/1.1e-03	F/F/8.8e-03	F/F/3.8e-05	733/1.29/6.9e-07
sinquad 3	406/0.57/4.7e-07	272/0.34/8.9e-07	F/F/3.1e-04	501/0.63/3.0e-07	366/0.45/9.2e-07
vardim 8	12/0.01/4.0e-08	12/0.01/4.0e-08	12/0.01/4.0e-08	12/0.01/4.0e-08	12/0.02/4.0e-08
watson 3	107/0.26/2.6e-07	147/0.36/6.3e-07	51/0.12/2.1e-07	121/0.32/5.8e-07	108/0.31/9.8e-07
woods 10000	F/F/1.3e-02	792/20.82/9.5e-07	883/22.24/6.2e-07	922/22.70/8.8e-07	309/7.19/9.9e-07
bdexp 1000	3/0.01/5.9e-107	3/0.00/7.1e-108	3/0.00/4.4e-107	3/0.00/2.5e-108	3/0.00/4.5e-107
bdexp 10000	3/0.01/1.2e-109	3/0.01/9.4e-110	3/0.01/1.1e-109	3/0.01/8.4e-110	3/0.01/1.1e-109
bdexp 100000	3/0.12/1.7e-109	3/0.13/1.7e-109	3/0.15/1.7e-109	3/0.15/1.7e-109	3/0.12/1.7e-109
exdenschnf 1000	25/0.05/4.6e-07	68/0.18/2.4e-07	65/0.17/5.2e-07	91/0.35/3.8e-07	96/0.28/9.7e-07
exdenschnf 10000	26/0.56/3.1e-07	68/2.43/7.6e-07	73/2.21/2.5e-07	85/2.85/2.6e-07	843/33.85/8.8e-07
exdenschnf 100000	27/4.41/1.1e-07	69/15.80/6.0e-07	73/14.86/7.8e-07	59/12.45/2.2e-07	114/25.09/2.1e-07
mccormak 2	21/0.05/9.3e-08	19/0.01/8.1e-07	63/0.07/9.0e-08	39/0.06/1.2e-07	F/F/2.9e+04
exdenschnb 1000	22/0.03/8.2e-08	27/0.05/8.0e-07	74/0.19/4.5e-07	32/0.05/3.5e-07	102/0.27/4.8e-07
exdenschnb 10000	22/0.23/2.6e-07	28/0.40/7.1e-07	82/1.38/3.2e-07	25/0.29/1.0e-06	153/2.80/8.0e-07
exdenschnb 100000	22/1.58/8.2e-07	30/2.59/2.6e-07	88/9.13/2.7e-07	29/2.19/4.5e-07	91/9.42/1.5e-07
genquartic 5000	24/0.20/4.7e-07	31/0.32/9.0e-08	108/1.30/2.6e-07	33/0.22/8.6e-07	115/1.37/2.6e-07
genquartic 30000	35/1.33/1.9e-07	38/1.37/2.0e-07	96/4.61/5.6e-07	36/1.43/7.0e-08	157/8.53/2.4e-07
genquartic 100000	27/2.41/7.1e-08	37/3.79/5.2e-07	98/13.02/3.0e-07	81/11.20/8.8e-07	182/26.69/4.3e-07
biggsb1 100	1240/1.82/6.9e-07	1381/1.97/9.8e-07	F/F/2.9e-05	F/F/7.2e-06	621/0.82/9.4e-07
sine 4000	64/1.65/7.7e-09	180/4.19/1.3e-08	F/F/8.0e-02	91/2.17/3.0e-07	F/F/1.1e+03
sine 20000	94/7.82/1.7e-09	181/14.39/6.0e-13	F/F/9.7e-03	81/7.52/9.5e-07	F/F/7.6e-04
sine 40000	87/13.53/2.3e-08	394/67.01/7.9e-07	F/F/5.0e-01	80/13.64/3.2e-07	F/F/2.1e+03
fletcbv3 8	43/0.04/5.3e-07	51/0.05/4.7e-07	47/0.05/7.7e-07	30/0.03/7.4e-07	52/0.05/3.9e-07
nonscomp 2000	176/0.65/1.3e-07	F/F/3.0e-03	100/0.37/4.2e-07	F/F/4.6e-03	F/F/5.0e-05
nonscomp 10000	275/6.17/6.5e-07	F/F/6.2e-02	F/F/3.8e-02	413/9.78/4.7e-07	F/F/1.2e-04
power1 55	1699/2.30/9.5e-07	F/F/5.1e-06	F/F/1.4e-04	F/F/3.6e-04	737/0.84/9.5e-07
raydan1 2000	857/2.94/6.9e-07	1118/3.86/6.5e-07	F/F/1.4e-04	1335/4.78/7.8e-07	F/F/9.2e-05
raydan1 3000	1252/8.69/7.5e-07	F/F/2.5e-04	F/F/2.6e-04	F/F/3.8e-04	F/F/2.3e-04
raydan2 500	15/0.02/2.2e-07	18/0.02/1.2e-08	24/0.04/2.6e-08	21/0.03/8.5e-08	14/0.02/6.6e-08
raydan2 5000	18/0.21/1.6e-07	21/0.28/7.2e-08	46/0.71/6.6e-07	38/0.56/2.7e-07	12/0.10/3.6e-07
raydan2 50000	18/1.44/8.9e-07	19/1.65/5.7e-07	21/1.89/9.8e-07	22/1.90/9.3e-07	48/5.15/5.0e-07
diagonal1 60	164/0.23/7.3e-07	88/0.09/9.4e-07	84/0.08/8.2e-07	F/F/2.4e-06	179/0.28/8.7e-07
diagonal1 100	242/0.34/1.7e-07	F/F/2.6e-05	F/F/3.3e-06	348/0.57/9.7e-07	F/F/5.5e-06
diagonal2 2000	1118/6.78/7.2e-07	1049/6.59/9.3e-07	334/1.97/6.9e-07	1988/12.48/7.6e-07	444/2.32/3.9e-07
diagonal3 150	F/F/2.3e-05	194/0.40/7.9e-07	F/F/8.4e-06	F/F/6.0e-06	F/F/4.3e-05
bv 2000	13/5.24/3.3e-07	14/5.79/3.1e-07	22/9.65/2.3e-07	9/3.23/4.6e-07	123/53.66/5.8e-07
bv 20000	1/0.14/1.2e-08	1/0.00/1.2e-08	1/0.00/1.2e-08	1/0.00/1.2e-08	1/0.00/1.2e-08
ie 100	15/1.33/2.8e-07	13/1.12/6.2e-07	20/1.94/5.0e-07	14/1.34/7.5e-07	38/4.96/6.6e-07
ie 200	15/4.47/3.9e-07	13/3.56/7.2e-07	18/5.35/3.1e-07	17/5.68/9.3e-08	40/17.36/3.2e-07
singx 200	348/2.28/7.4e-07	1218/7.82/6.3e-07	F/F/3.5e-03	F/F/7.0e-06	339/1.98/9.9e-07
singx 1500	1105/318.57/6.4e-07	1940/559.01/9.8e-07	F/F/6.3e-03	F/F/1.8e-05	292/75.74/9.0e-07
band 3	111/0.23/1.3e-07	67/0.12/6.3e-07	38/0.06/4.2e-07	77/0.11/6.9e-07	46/0.05/8.9e-08
gauss 3	59/0.16/5.6e-07	47/0.10/6.4e-07	15/0.02/1.7e-07	14/0.03/8.1e-07	30/0.08/7.4e-07
jensam 2	201/0.35/7.7e-07	119/0.19/6.7e-07	153/0.24/5.3e-07	211/0.39/7.0e-07	173/0.29/9.7e-07
lin 100	2/0.01/2.0e-14	2/0.00/2.0e-14	2/0.00/2.0e-14	2/0.00/2.0e-14	2/0.00/2.0e-14
lin 1000	12/83.01/1.3e-07	12/83.63/1.3e-07	12/82.60/1.3e-07	12/81.73/1.3e-07	12/82.49/1.3e-07
osb2 11	1717/6.76/8.4e-07	F/F/2.5e-03	F/F/8.2e-04	F/F/6.4e-04	F/F/4.9e-03
pen1 55	626/1.47/6.2e-07	448/1.02/9.2e-07	1475/3.47/9.5e-07	839/1.99/4.7e-07	106/0.18/1.5e-08
pen2 100	F/F/4.1e-05	274/2.25/6.2e-07	F/F/1.1e-05	430/3.57/9.0e-07	F/F/1.6e-05
rosex 40	F/F/7.3e-04	1077/2.00/5.3e-07	1496/2.58/9.4e-07	828/1.45/5.5e-07	141/0.18/7.1e-07
sing 4	545/0.97/4.7e-07	380/0.60/6.7e-07	F/F/5.0e-04	F/F/5.5e-06	254/0.50/3.4e-08
trid 500	F/F/7.2e-05	899/55.29/1.0e-06	F/F/1.5e-02	410/24.09/8.6e-07	F/F/1.4e-04
trid 1000	500/76.37/7.1e-07	211/29.37/3.9e-07	F/F/4.5e-04	754/111.54/8.3e-07	391/55.48/9.2e-07

表 2

图 5

图 6

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