一个带重启步的改进PRP型谱共轭梯度法

Abstract

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

CLC Number:

O221

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.

Trendmd

Figures/Tables 8

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

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

References 29

1	Hestenes M R , Stiefel E . Method of conjugate gradient for solving linear equations. Journal of Research of National Bureau of Standards, 1952, 49: 409- 436 doi: 10.6028/jres.049.044
2	Fletcher R , Reeves C . Function minimization by conjugate gradients. Computer Journal, 1964, 7 (2): 149- 154 doi: 10.1093/comjnl/7.2.149
3	Polak E , Ribiére G . Note surla convergence de directions conjugées. Revue Francaise Informat Recherche Operationelle 3e Anneé, 1969, 16 (3): 35- 43
4	Polyak B T . The conjugate gradient method in extreme problems. USSR Computational Mathematics and Mathematical Physics, 1969, 9: 94- 112 doi: 10.1016/0041-5553(69)90035-4
5	Dai Y H , Yuan Y X . A nonlinear conjugate gradient method with a strong global convergence property. SIAM Journal on Optimization, 1999, 10 (1): 177- 182 doi: 10.1137/S1052623497318992
6	Fletcher R. Unconstrained Optimization. Practical Methods of Optimization: Vol 1. New York: Wiley, 1987
7	Liu Y , Storey C . Efficient generalized conjugate gradient algorithms, part 1:theory. Journal of Optimization Theory and Applications, 1991, 69 (1): 129- 137 doi: 10.1007/BF00940464
8	Gilbert J C , Nocedal J . Global convergence properties of conjugate gradient methods for optimization. SIAM Journal on Optimization, 1992, 2 (1): 21- 42 doi: 10.1137/0802003
9	Powell M J D. Nonconvex Minimization Calculations and the Conjugate Gradient Method//Griffiths D F, et al. Numerical Analysis. Berlin: Springer, 1984: 122-141
10	Powell M J D . Convergence properties of algorithms for nonlinear optimization. SIAM Review, 1986, 28 (4): 487- 500 doi: 10.1137/1028154
11	Jiang X Z , Jian J B , Song D , et al. An improved Polak-Ribiére-Polyak conjugate gradient method with an efficient restart direction. Computational and Applied Mathematics, 2021, 40 (5): 1- 24
12	Mtagulwa P , Kaelo P . An efficient modified PRP-FR hybrid conjugate gradient method for solving unconstrained optimization problems. Applied Numerical Mathematics, 2019, 145: 111- 120 doi: 10.1016/j.apnum.2019.06.003
13	Aminifard Z , Babaie-Kafaki S . A modified descent Polak-Ribiŕe-Polyak conjugate gradient method with global convergence property for nonconvex functions. Calcolo, 2019, 56 (2): 1- 11
14	Liu J K , Feng Y M , Zou L M . A spectral conjugate gradient method for solving large-scale unconstrained optimization. Computers Mathematics with Applications, 2019, 77 (3): 731- 739 doi: 10.1016/j.camwa.2018.10.002
15	Dong X L . A modified nonlinear Polak-Ribiére-Polyak conjugate gradient method with sufficient descent property. Calcolo, 2020, 57 (3): 1- 14
16	Dai Z F , Tian B S . Global convergence of some modified PRP nonlinear conjugate gradient methods. Optimization Letters, 2011, 5 (4): 615- 630 doi: 10.1007/s11590-010-0224-8
17	Li M . A three term Polak-Ribiére-Polyak conjugate gradient method close to the memoryless bfgs quasi-newton method. Journal of Industrial Management Optimization, 2020, 16 (1): 245- 260 doi: 10.3934/jimo.2018149
18	Li X L , Shi J J , Dong X L , et al. A new conjugate gradient method based on Quasi-Newton equation for unconstrained optimization. Journal of Computational and Applied Mathematics, 2019, 350: 372- 379 doi: 10.1016/j.cam.2018.10.035
19	Dai Y H , Yuan Y X . A class of globally convergent conjugate gradient methods. Science in China Series A: Mathematics, 2003, 46 (2): 251- 261 doi: 10.1360/03ys9027
20	Barzilai J , Borwein J M . Two-point step size gradient methods. IMA Journal of Numerical Analysis, 1988, 8 (1): 141- 148 doi: 10.1093/imanum/8.1.141
21	Luengo F, Raydan M, Glunt W, et al. Preconditioned spectral gradient method for unconstrained optimization problems[R]. Technical Report 96-08, Escuela de Computación, Facultad de Ciencias, Universidad Central de Venezuela, 47002 Caracas 1041-A, Venezuela, 1996
22	Birgin E G , Martínez J M . A spectral conjugate gradient method for unconstrained optimization. Applied Mathematics and Optimization, 2001, 43 (2): 117- 128 doi: 10.1007/s00245-001-0003-0
23	Kou C X , Dai Y H . A modified self-scaling memoryless Broyden-Fletcher-Goldfard-Shanno method for unconstrained optimization. Journal of Optimization Theory and Applications, 2015, 165 (1): 209- 224 doi: 10.1007/s10957-014-0528-4
24	Dai Y H , Kou C X . A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search. SIAM Journal on Optimization, 2013, 23 (1): 296- 320 doi: 10.1137/100813026
25	Hager W W , Zhang H . A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM Journal on Optimization, 2005, 16 (1): 170- 192 doi: 10.1137/030601880
26	Gould N I M , Orban D , Toint P L . CUTEr and SifDec: A constrained and unconstrained testing environment, revisited. ACM Transactions on Mathematical Software (TOMS), 2003, 29 (4): 373- 394 doi: 10.1145/962437.962439
27	Moré J J , Garbow B S , Hillstrom K E . Testing unconstrained optimization software. ACM Transactions on Mathematical Software, 1981, 7 (1): 17- 41 doi: 10.1145/355934.355936
28	Andrei N . An unconstrained optimization test functions collection. Advanced Modeling and Optimization, 2008, 10 (1): 147- 161
29	Dolan E D , Moré J J . Benchmarking optimization software with performance profiles. Mathematical Programming, 2002, 91 (2): 201- 213 doi: 10.1007/s101070100263

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An Improved PRP Type Spectral Conjugate Gradient Method with Restart Steps

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