doi: 10.3934/jimo.2018149

A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method

Department of Mathematics and Computational Science, Huaihua University, Huaihua, Hunan 418008, China

* Corresponding author: Min Li

Received  September 2017 Revised  May 2018 Published  September 2018

Fund Project: This work is supported by the NSF (11401242) of China and Scientific Research Fund of Hunan Provincial Education Department(14B139)

In this paper, we develop a three-term Polak-Ribière-Polyak conjugate gradient method, in which the search direction is close to the direction in the memoryless BFGS quasi-Newton method. The new scheme reduces to the standard Polak-Ribière-Polyak method when an exact line search is used. For any line search, the method satisfies the sufficient descent condition $g_{k}^{T}d_{k}≤ -{(1-\frac{1}{4}(1+\overline{t})^2})\|g_k\|^2$, where $\overline{t}∈[0,1)$ is a constant. The global convergence results of the new algorithm are established with suitable line search methods. Numerical results show that the proposed method is efficient for the unconstrained problems in the CUTEr library.

Citation: Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018149
References:
[1]

N. Andrei, Open problems in nonlinear conjugate gradient algorithms for unconstrained optimization, Bull. Malays. Math. Sci. Soc., 34 (2011), 319-330.

[2]

S. Babaie-Kafaki and G. Reza, A descent family of Dai-Liao conjugate gradient methods, Optim. Method. Softw., 21 (2013), 1-9. doi: 10.1080/10556788.2013.833199.

[3]

I. BongartzA. ConnN. Gould and P. Toint, CUTE: constrained and unconstrained testing environments, ACM Trans. Math. Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.

[4]

Y. Dai and C. Kou, A nonlinear conjugate gradient algorithm with an optimal property and an improved wolfe line search, SIAM J. Optim, 23 (2013), 296-320. doi: 10.1137/100813026.

[5]

Y. Dai and L. Liao, New conjugate conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101. doi: 10.1007/s002450010019.

[6]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (2000), 177-182. doi: 10.1137/S1052623497318992.

[7]

Z. Dai and B. Tian, Global convergence of some modified PRP nonlinear conjugate gradient methods, Optim. Lett., 5 (2011), 615-630. doi: 10.1007/s11590-010-0224-8.

[8]

E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213. doi: 10.1007/s101070100263.

[9]

R. Fletcher, Practical Method of Optimization, vol. 1: Unconstrained Optimization, John Wiley & Sons, New York, 1987.

[10]

R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.

[11]

J. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM. J. Optim., 2 (1992), 21-42. doi: 10.1137/0802003.

[12]

L. Grippo and S. Lucidi, A globally convergent version of the Polak-Ribière-Polyak conjugate gradient method, Math. Program., 78 (1979), 375-391. doi: 10.1007/BF02614362.

[13]

W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192. doi: 10.1137/030601880.

[14]

W. Hager and H. Zhang, Algorithm 851: CG_ DESCENT, a conjugate gradient method with guaranteed descent, ACM Trans. Math. Software, 32 (2006), 113-137. doi: 10.1145/1132973.1132979.

[15]

W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.

[16]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49 (1952), 409-436. doi: 10.6028/jres.049.044.

[17]

G. LiC. Tang and Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math., 202 (2007), 523-539. doi: 10.1016/j.cam.2006.03.005.

[18]

M. LiJ. Liu and H. Feng, The global convergence of a descent PRP conjugate gradient method, Comput. Appl. Math., 31 (2012), 59-83.

[19]

D. Liu and J. Nocedal, On the limited memory BFGS method for large-scale optimization, Math. Program., 45 (1989), 503-528. doi: 10.1007/BF01589116.

[20]

Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, part 1: Theory, J. Optim. Theory Appl., 69 (1991), 177-182. doi: 10.1007/BF00940464.

[21]

J. Nocedal, Updating quasi-Newton matrices with limited storage, Math. Comput., 35 (1980), 773-782. doi: 10.1090/S0025-5718-1980-0572855-7.

[22]

J. M. Perry, A class of conjugate gradient algorithms with a two-step variable-metric memory, Discussion Paper 269, Center for Mathematical Studies in Economics and Management Sciences, Northwestern University, Evanston, Illinois, 1977. doi: 10.1287/opre.26.6.1073.

[23]

B. Polak and G. Ribière, Note sur la convergence de directions conjuguées, Rev. Francaise Informat. Recherche Opertionelle, 3e Année, 16 (1969), 35-43.

[24]

B. Polyak, The conjugate gradient method in extreme problems, USSR Comp. Math. Math. Phys., 9 (1969), 94-112.

[25]

M. Powell, Nonvonvex minimization calculations and the conjugate gradient method, in: Lecture Notes in Mathematics, vol. 1066, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099521.

[26]

D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM J. Numer. Anal., 15 (1978), 1247-1257. doi: 10.1137/0715085.

[27]

D. Shanno, Conjugate gradient methods with inexact searches, Math. Oper. Res., 3 (1978), 244-256. doi: 10.1287/moor.3.3.244.

[28]

H. Yabe and M. Takano, Global convergence properties of nonlinear conjugate gradient methods with modified secant condition, Comput. Optim. Appl., 28 (2004), 203-225. doi: 10.1023/B:COAP.0000026885.81997.88.

[29]

G. Yu and L. Guan, Modified PRP methods with sufficient desent property and their convergence properties, Acta Scientiarum Naturalium Universitatis Sunyatseni(Chinese), 45 (2006), 11-14.

[30]

G. YuanZ. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optimz. Theory App., 168 (2016), 129-152. doi: 10.1007/s10957-015-0781-1.

[31]

G. YuanZ. ShengB. WangW. Hu and C. Li, The global convergence of a modified BFGS method for nonconvex functions, J. Comput. Appl. Math., 327 (2018), 274-294. doi: 10.1016/j.cam.2017.05.030.

[32]

G. YuanZ. Wei and G. Li, A modified Polak-Ribiéere-Polyak conjugate gradient algorithm with nonmonotone line search for nonsmooth convex minimization, J. Comput. Appl. Math., 255 (2014), 86-96. doi: 10.1007/s12190-015-0912-8.

[33]

G. Yuan, Z. Wei and X. Lu, Global convergence of the BFGS method and the PRP method for general functions under a modified weak Wolfe-Powell line search, Appl. Math. Model., 47 (2017), 811-825 doi: 10.1016/j.apm.2017.02.008.

[34]

G. Yuan, Modified nonlinear conjugate gradient methods with sufficient descent property for largescale optimization problems, Optim. Lett., 3 (2009), 11-21. doi: 10.1007/s11590-008-0086-5.

[35]

J. ZhangN. Deng and L. Chen, New quasi-newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl., 102 (1999), 147-167. doi: 10.1023/A:1021898630001.

[36]

L. Zhang, New versions of the Hestenes-Stiefel nonlinear conjugate gradient method based on the secant condition for optimization, Comp. Appl. Math., 28 (2009), 1-23. doi: 10.1590/S0101-82052009000100006.

[37]

L. ZhangW. Zhou and D. Li, A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal., 26 (2006), 629-640. doi: 10.1093/imanum/drl016.

[38]

L. ZhangW. Zhou and D. Li, Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search, Numer. Math., 104 (2006), 561-572. doi: 10.1007/s00211-006-0028-z.

[39]

G. Zoutendijk, Nonlinear programming, computational methods, in Integer and Nonlinear Programming (ed. J. Abadie), North-Holland, Amsterdam, 1970, 37-86.

show all references

References:
[1]

N. Andrei, Open problems in nonlinear conjugate gradient algorithms for unconstrained optimization, Bull. Malays. Math. Sci. Soc., 34 (2011), 319-330.

[2]

S. Babaie-Kafaki and G. Reza, A descent family of Dai-Liao conjugate gradient methods, Optim. Method. Softw., 21 (2013), 1-9. doi: 10.1080/10556788.2013.833199.

[3]

I. BongartzA. ConnN. Gould and P. Toint, CUTE: constrained and unconstrained testing environments, ACM Trans. Math. Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.

[4]

Y. Dai and C. Kou, A nonlinear conjugate gradient algorithm with an optimal property and an improved wolfe line search, SIAM J. Optim, 23 (2013), 296-320. doi: 10.1137/100813026.

[5]

Y. Dai and L. Liao, New conjugate conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101. doi: 10.1007/s002450010019.

[6]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (2000), 177-182. doi: 10.1137/S1052623497318992.

[7]

Z. Dai and B. Tian, Global convergence of some modified PRP nonlinear conjugate gradient methods, Optim. Lett., 5 (2011), 615-630. doi: 10.1007/s11590-010-0224-8.

[8]

E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213. doi: 10.1007/s101070100263.

[9]

R. Fletcher, Practical Method of Optimization, vol. 1: Unconstrained Optimization, John Wiley & Sons, New York, 1987.

[10]

R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.

[11]

J. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM. J. Optim., 2 (1992), 21-42. doi: 10.1137/0802003.

[12]

L. Grippo and S. Lucidi, A globally convergent version of the Polak-Ribière-Polyak conjugate gradient method, Math. Program., 78 (1979), 375-391. doi: 10.1007/BF02614362.

[13]

W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192. doi: 10.1137/030601880.

[14]

W. Hager and H. Zhang, Algorithm 851: CG_ DESCENT, a conjugate gradient method with guaranteed descent, ACM Trans. Math. Software, 32 (2006), 113-137. doi: 10.1145/1132973.1132979.

[15]

W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.

[16]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49 (1952), 409-436. doi: 10.6028/jres.049.044.

[17]

G. LiC. Tang and Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math., 202 (2007), 523-539. doi: 10.1016/j.cam.2006.03.005.

[18]

M. LiJ. Liu and H. Feng, The global convergence of a descent PRP conjugate gradient method, Comput. Appl. Math., 31 (2012), 59-83.

[19]

D. Liu and J. Nocedal, On the limited memory BFGS method for large-scale optimization, Math. Program., 45 (1989), 503-528. doi: 10.1007/BF01589116.

[20]

Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, part 1: Theory, J. Optim. Theory Appl., 69 (1991), 177-182. doi: 10.1007/BF00940464.

[21]

J. Nocedal, Updating quasi-Newton matrices with limited storage, Math. Comput., 35 (1980), 773-782. doi: 10.1090/S0025-5718-1980-0572855-7.

[22]

J. M. Perry, A class of conjugate gradient algorithms with a two-step variable-metric memory, Discussion Paper 269, Center for Mathematical Studies in Economics and Management Sciences, Northwestern University, Evanston, Illinois, 1977. doi: 10.1287/opre.26.6.1073.

[23]

B. Polak and G. Ribière, Note sur la convergence de directions conjuguées, Rev. Francaise Informat. Recherche Opertionelle, 3e Année, 16 (1969), 35-43.

[24]

B. Polyak, The conjugate gradient method in extreme problems, USSR Comp. Math. Math. Phys., 9 (1969), 94-112.

[25]

M. Powell, Nonvonvex minimization calculations and the conjugate gradient method, in: Lecture Notes in Mathematics, vol. 1066, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099521.

[26]

D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM J. Numer. Anal., 15 (1978), 1247-1257. doi: 10.1137/0715085.

[27]

D. Shanno, Conjugate gradient methods with inexact searches, Math. Oper. Res., 3 (1978), 244-256. doi: 10.1287/moor.3.3.244.

[28]

H. Yabe and M. Takano, Global convergence properties of nonlinear conjugate gradient methods with modified secant condition, Comput. Optim. Appl., 28 (2004), 203-225. doi: 10.1023/B:COAP.0000026885.81997.88.

[29]

G. Yu and L. Guan, Modified PRP methods with sufficient desent property and their convergence properties, Acta Scientiarum Naturalium Universitatis Sunyatseni(Chinese), 45 (2006), 11-14.

[30]

G. YuanZ. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optimz. Theory App., 168 (2016), 129-152. doi: 10.1007/s10957-015-0781-1.

[31]

G. YuanZ. ShengB. WangW. Hu and C. Li, The global convergence of a modified BFGS method for nonconvex functions, J. Comput. Appl. Math., 327 (2018), 274-294. doi: 10.1016/j.cam.2017.05.030.

[32]

G. YuanZ. Wei and G. Li, A modified Polak-Ribiéere-Polyak conjugate gradient algorithm with nonmonotone line search for nonsmooth convex minimization, J. Comput. Appl. Math., 255 (2014), 86-96. doi: 10.1007/s12190-015-0912-8.

[33]

G. Yuan, Z. Wei and X. Lu, Global convergence of the BFGS method and the PRP method for general functions under a modified weak Wolfe-Powell line search, Appl. Math. Model., 47 (2017), 811-825 doi: 10.1016/j.apm.2017.02.008.

[34]

G. Yuan, Modified nonlinear conjugate gradient methods with sufficient descent property for largescale optimization problems, Optim. Lett., 3 (2009), 11-21. doi: 10.1007/s11590-008-0086-5.

[35]

J. ZhangN. Deng and L. Chen, New quasi-newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl., 102 (1999), 147-167. doi: 10.1023/A:1021898630001.

[36]

L. Zhang, New versions of the Hestenes-Stiefel nonlinear conjugate gradient method based on the secant condition for optimization, Comp. Appl. Math., 28 (2009), 1-23. doi: 10.1590/S0101-82052009000100006.

[37]

L. ZhangW. Zhou and D. Li, A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal., 26 (2006), 629-640. doi: 10.1093/imanum/drl016.

[38]

L. ZhangW. Zhou and D. Li, Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search, Numer. Math., 104 (2006), 561-572. doi: 10.1007/s00211-006-0028-z.

[39]

G. Zoutendijk, Nonlinear programming, computational methods, in Integer and Nonlinear Programming (ed. J. Abadie), North-Holland, Amsterdam, 1970, 37-86.

Figure 1.  Performance profiles relative to the CPU time (left) and the total number of iterations (right)
Figure 2.  Performance profiles relative to the total numbers of function evaluations (left) and gradient evaluations (right)
Table 1.  The problems and their dimensions
No. Prob Dim No. Prob Dim No. Prob Dim
No. Prob Dim No. Prob Dim No. Prob Dim
1 ARGLINA 100 52 DIXMAANK 1500 103 NONDQUAR 1000
2 ARGLINA 200 53 DIXMAANL 300 104 NONMSQRT 100
3 ARGLINB 100 54 DIXMAANL 1500 105 OSCIPATH 100
4 ARGLINB 200 55 DIXON3DQ 100 106 OSCIPATH 500
5 ARGLINC 50 56 DIXON3DQ 1000 107 PENALTY1 50
6 ARGLINC 200 57 DQDRTIC 1000 108 PENALTY1 100
7 ARWHEAD 100 58 DQDRTIC 5000 109 PENALTY2 100
8 ARWHEAD 1000 59 DQRTIC 500 110 PENALTY2 200
9 BDQRTIC 100 60 DQRTIC 1000 111 PENALTY3 50
10 BDQRTIC 500 61 EDENSCH 2000 112 PENALTY3 100
11 BDQRTIC 1000 62 EG2 1000 113 POWELLSG 100
12 BOX 100 63 ENGVAL1 1000 114 POWELLSG 10000
13 BOX 1000 64 ENGVAL1 5000 115 POWER 5000
14 BROWNAL 200 65 ERRINROS 50 116 POWER 10000
15 BROYDN7D 100 66 EXTROSNB 100 117 QUARTC 100
16 BROYDN7D 10000 67 EXTROSNB 1000 118 QUARTC 10000
17 BRYBND 100 68 FLETCBV2 5000 119 SCHMVETT 100
18 BRYBND 500 69 FLETCBV2 10000 120 SCHMVETT 10000
19 CHAINWOO 1000 70 FLETCHBV 100 121 SCOSINE 100
20 COSINE 1000 71 FLETCHCR 1000 122 SCURLY10 100
21 COSINE 10000 72 FMINSRF2 5625 123 SCURLY20 100
22 CRAGGLVY 1000 73 FMINSRF2 10000 124 SCURLY30 100
23 CRAGGLVY 5000 74 FMINSURF 121 125 SENSORS 100
24 CURLY10 100 75 FMINSURF 10000 126 SINQUAD 500
25 CURLY10 1000 76 FREUROTH 100 127 SINQUAD 10000
26 CURLY20 100 77 FREUROTH 5000 128 SPARSINE 50
27 CURLY20 1000 78 GENHUMPS 1000 129 SPARSINE 1000
28 CURLY30 100 79 GENHUMPS 5000 130 SPARSQUR 5000
29 CURLY30 1000 80 GENROSE 100 131 SPARSQUR 10000
30 DECONVU 61 81 GENROSE 500 132 SPMSRTLS 4999
31 DIXMAANA 3000 82 HILBERTA 10 133 SPMSRTLS 10000
32 DIXMAANA 9000 83 HILBERTB 50 134 SROSENBR 100
33 DIXMAANB 300 84 HYDC20LS 99 135 SROSENBR 5000
34 DIXMAANB 9000 85 LIARWHD 100 136 SROSENBR 10000
35 DIXMAANC 90 86 LIARWHD 10000 137 TESTQUAD 1000
36 DIXMAANC 9000 87 MANCINO 50 138 TESTQUAD 5000
37 DIXMAAND 300 88 MANCINO 100 139 TOINTGOR 50
38 DIXMAAND 1500 89 MODBEALE 200 140 TOINTGSS 100
39 DIXMAANE 90 90 MODBEALE 2000 141 TOINTGSS 1000
40 DIXMAANE 1500 91 MOREBV 50 142 TOINTPSP 50
41 DIXMAANF 1500 92 MOREBV 500 143 TOINTQOR 50
42 DIXMAANF 9000 93 MSQRTALS 100 144 TQUARTIC 500
43 DIXMAANG 90 94 MSQRTALS 529 145 TQUARTIC 5000
44 DIXMAANG 300 95 MSQRTBLS 100 146 TRIDIA 100
45 DIXMAANH 3000 96 MSQRTBLS 1024 147 TRIDIA 1000
46 DIXMAANH 9000 97 NONCVXU2 1000 148 VARDIM 50
47 DIXMAANI 300 98 NONCVXU2 10000 149 VARDIM 100
48 DIXMAANI 1500 99 NONCVXUN 100 150 VAREIGVL 1000
49 DIXMAANJ 300 100 NONDIA 5000 151 VAREIGVL 5000
50 DIXMAANJ 1500 101 NONDIA 10000 152 WOODS 4000
51 DIXMAANK 300 102 NONDQUAR 500 153 WOODS 10000
No. Prob Dim No. Prob Dim No. Prob Dim
No. Prob Dim No. Prob Dim No. Prob Dim
1 ARGLINA 100 52 DIXMAANK 1500 103 NONDQUAR 1000
2 ARGLINA 200 53 DIXMAANL 300 104 NONMSQRT 100
3 ARGLINB 100 54 DIXMAANL 1500 105 OSCIPATH 100
4 ARGLINB 200 55 DIXON3DQ 100 106 OSCIPATH 500
5 ARGLINC 50 56 DIXON3DQ 1000 107 PENALTY1 50
6 ARGLINC 200 57 DQDRTIC 1000 108 PENALTY1 100
7 ARWHEAD 100 58 DQDRTIC 5000 109 PENALTY2 100
8 ARWHEAD 1000 59 DQRTIC 500 110 PENALTY2 200
9 BDQRTIC 100 60 DQRTIC 1000 111 PENALTY3 50
10 BDQRTIC 500 61 EDENSCH 2000 112 PENALTY3 100
11 BDQRTIC 1000 62 EG2 1000 113 POWELLSG 100
12 BOX 100 63 ENGVAL1 1000 114 POWELLSG 10000
13 BOX 1000 64 ENGVAL1 5000 115 POWER 5000
14 BROWNAL 200 65 ERRINROS 50 116 POWER 10000
15 BROYDN7D 100 66 EXTROSNB 100 117 QUARTC 100
16 BROYDN7D 10000 67 EXTROSNB 1000 118 QUARTC 10000
17 BRYBND 100 68 FLETCBV2 5000 119 SCHMVETT 100
18 BRYBND 500 69 FLETCBV2 10000 120 SCHMVETT 10000
19 CHAINWOO 1000 70 FLETCHBV 100 121 SCOSINE 100
20 COSINE 1000 71 FLETCHCR 1000 122 SCURLY10 100
21 COSINE 10000 72 FMINSRF2 5625 123 SCURLY20 100
22 CRAGGLVY 1000 73 FMINSRF2 10000 124 SCURLY30 100
23 CRAGGLVY 5000 74 FMINSURF 121 125 SENSORS 100
24 CURLY10 100 75 FMINSURF 10000 126 SINQUAD 500
25 CURLY10 1000 76 FREUROTH 100 127 SINQUAD 10000
26 CURLY20 100 77 FREUROTH 5000 128 SPARSINE 50
27 CURLY20 1000 78 GENHUMPS 1000 129 SPARSINE 1000
28 CURLY30 100 79 GENHUMPS 5000 130 SPARSQUR 5000
29 CURLY30 1000 80 GENROSE 100 131 SPARSQUR 10000
30 DECONVU 61 81 GENROSE 500 132 SPMSRTLS 4999
31 DIXMAANA 3000 82 HILBERTA 10 133 SPMSRTLS 10000
32 DIXMAANA 9000 83 HILBERTB 50 134 SROSENBR 100
33 DIXMAANB 300 84 HYDC20LS 99 135 SROSENBR 5000
34 DIXMAANB 9000 85 LIARWHD 100 136 SROSENBR 10000
35 DIXMAANC 90 86 LIARWHD 10000 137 TESTQUAD 1000
36 DIXMAANC 9000 87 MANCINO 50 138 TESTQUAD 5000
37 DIXMAAND 300 88 MANCINO 100 139 TOINTGOR 50
38 DIXMAAND 1500 89 MODBEALE 200 140 TOINTGSS 100
39 DIXMAANE 90 90 MODBEALE 2000 141 TOINTGSS 1000
40 DIXMAANE 1500 91 MOREBV 50 142 TOINTPSP 50
41 DIXMAANF 1500 92 MOREBV 500 143 TOINTQOR 50
42 DIXMAANF 9000 93 MSQRTALS 100 144 TQUARTIC 500
43 DIXMAANG 90 94 MSQRTALS 529 145 TQUARTIC 5000
44 DIXMAANG 300 95 MSQRTBLS 100 146 TRIDIA 100
45 DIXMAANH 3000 96 MSQRTBLS 1024 147 TRIDIA 1000
46 DIXMAANH 9000 97 NONCVXU2 1000 148 VARDIM 50
47 DIXMAANI 300 98 NONCVXU2 10000 149 VARDIM 100
48 DIXMAANI 1500 99 NONCVXUN 100 150 VAREIGVL 1000
49 DIXMAANJ 300 100 NONDIA 5000 151 VAREIGVL 5000
50 DIXMAANJ 1500 101 NONDIA 10000 152 WOODS 4000
51 DIXMAANK 300 102 NONDQUAR 500 153 WOODS 10000
Table 2.  The numerical results
CG_DESCENT method MPRP method NPRP+ method
No. Iter/Nf/Ng/Time Iter/Nf/Ng/Time Iter/Nf/Ng/Time
1 1/3/2/0.001 1/3/2/0.001 1/3/2/0
2 1/3/2/0.002 1/3/2/0.002 1/3/2/0.001
3 4/8/7/0.001 5/10/10/0.001 5/10/10/0.000999
4 9/16/20/0.005999 7/13/15/0.004999 7/13/15/0.004999
5 3/7/5/0 3/7/5/0 3/7/5/0
6 8/14/17/0.003999 5/11/11/0.003 5/11/11/0.004
7 9/21/15/0 12/27/19/0.000999 9/21/15/0
8 10/24/16/0.002 9/21/14/0.002 8/21/16/0.002
9 131/255/180/0.003999 126/245/164/0.001999 101/195/173/0.003
10 375/765/503/0.027996 510/890/740/0.039994 490/700/1025/0.046993
11 531/1100/718/0.078988 405/861/589/0.06499 238/533/540/0.052992
12 15/31/23/0.001 11/24/16/0.001 12/25/17/0.001
13 13/34/28/0.002999 36/60/77/0.007999 18/43/33/0.004999
14 4/9/6/0.001 15/31/21/0.003999 18/37/26/0.004
15 82/157/91/0.006999 90/171/101/0.005 79/152/87/0.004
16 2909/5808/2925/15.238 2819/5621/2840/15.294 2794/5573/2811/15.206
17 113/227/114/0.002 112/225/114/0.002 106/213/107/0.001999
18 37/75/38/0.003 27/55/28/0.003999 29/60/31/0.002999
19 448/834/531/0.061991 436/770/577/0.062991 4517/8797/4945/0.61991
20 12/28/24/0.003999 10/25/21/0.003999 11/27/22/0.003
21 12/32/28/0.038994 11/27/25/0.033995 10/25/21/0.029996
22 103/185/128/0.041994 122/230/159/0.050992 108/194/133/0.043993
23 110/200/139/0.22697 132/252/184/0.30595 99/179/131/0.24196
24 991/1777/1387/0.010999 1018/1825/1492/0.009999 936/1718/1286/0.008999
25 8654/13686/12760/0.84087 9755/14820/15245/0.97685 8911/13925/13444/0.94286
26 875/1612/1299/0.015998 852/1590/1249/0.016997 894/1658/1277/0.014998
27 9816/15450/15254/1.3928 10806/16627/17405/1.5678 9776/15443/15149/1.4718
28 986/1857/1447/0.018997 978/1828/1484/0.017997 989/1833/1419/0.016998
29 9832/15701/15229/1.7937 9824/15661/15379/1.8267 10778/16930/17516/2.2757
30 337/676/339/0.005999 385/773/390/0.006999 395/793/400/0.007998
31 9/19/10/0.003999 7/15/8/0.004999 7/15/8/0.004
32 9/19/10/0.008999 7/15/8/0.008999 7/15/8/0.006999
33 9/19/10/0.001 8/17/9/0.001 8/17/9/0.000999
34 9/19/10/0.008999 8/17/9/0.007999 8/17/9/0.007998
35 10/21/11/0.001 9/19/10/0.001 9/19/10/0.000999
36 10/21/11/0.009999 9/19/10/0.008998 9/19/10/0.008999
37 12/25/13/0.000999 11/23/12/0.001 11/23/12/0.001
38 12/25/13/0.001999 11/23/12/0.001999 11/23/12/0.001999
39 48/97/49/0.000999 49/99/50/0.001 48/97/49/0.000999
40 167/335/168/0.028996 168/337/169/0.024996 169/339/170/0.023996
41 133/267/134/0.019997 129/259/130/0.018997 127/255/128/0.018997
42 269/539/270/0.24496 265/531/266/0.24396 263/527/264/0.25996
43 54/109/55/0.000999 52/105/53/0.001 52/105/53/0.000999
44 83/167/84/0.002999 81/163/82/0.003 79/159/80/0.002
45 167/335/168/0.057992 169/339/170/0.058991 164/329/165/0.054992
46 263/527/264/0.23896 266/533/267/0.24796 256/513/257/0.23396
47 1046/2093/1047/0.030995 1057/2115/1058/0.032995 938/1877/939/0.027996
48 2926/5853/2927/0.43093 2914/5829/2915/0.43593 2918/5837/2919/0.42993
49 635/1271/636/0.018997 607/1215/608/0.018997 599/1199/600/0.018997
50 1467/2935/1468/0.20897 1478/2957/1479/0.21497 1413/2827/1414/0.21397
51 606/1213/607/0.019997 602/1205/603/0.022997 481/963/482/0.013998
52 1434/2869/1435/0.21897 1413/2827/1414/0.20497 1387/2775/1388/0.28396
53 596/1193/597/0.020997 604/1209/605/0.018997 388/777/389/0.016998
54 1374/2749/1375/0.20497 1424/2849/1425/0.20697 1356/2713/1357/0.20697
55 200/401/202/0.001999 200/401/202/0.001999 200/401/202/0.001999
56 1000/2001/1002/0.048992 1000/2001/1002/0.06199 1000/2001/1002/0.050992
57 7/15/8/0.002 7/15/8/0.002 6/13/7/0.001999
58 7/15/8/0.004 7/15/8/0.004999 7/15/8/0.003999
59 28/57/29/0.001 27/55/28/0.001 27/55/28/0.001
60 29/59/30/0.000999 29/59/30/0.002 29/59/30/0.002
61 32/59/40/0.010999 32/60/45/0.009998 31/56/39/0.011998
62 4/9/6/0.001 4/9/6/0.001 4/9/6/0.001
63 26/48/33/0.003 26/49/33/0.003 22/42/28/0.002999
64 27/50/40/0.016997 25/44/34/0.014998 23/42/30/0.013998
65 1151/2285/1599/0.008998 1150/2270/1564/0.006998 1708/3414/2234/0.013998
66 5481/11373/6023/0.041993 5510/11246/5845/0.041994 5906/12296/6550/0.045993
67 6354/13073/6816/0.47393 7741/15623/7929/0.54592 7866/16031/8277/0.61791
68 0/1/1/0.001999 0/1/1/0.001999 0/1/1/0.001
69 0/1/1/0.002999 0/1/1/0.003 0/1/1/0.002
70 F/F/F/F F/F/F/F F/F/F/F
71 6604/13682/7147/0.6859 4310/8650/4348/0.43493 6879/14253/7386/0.74789
72 368/739/371/0.25896 305/611/306/0.20897 395/792/397/0.28096
73 434/869/435/0.54392 375/752/377/0.53992 460/922/462/0.60891
74 88/178/90/0.001999 92/186/94/0.001999 80/162/82/0.002
75 603/1209/606/0.85987 454/910/456/0.6459 655/1311/656/0.96185
76 52/103/80/0.000999 107/205/145/0.003 36/72/65/0.001
77 53/107/80/0.047993 49/97/74/0.044993 33/68/56/0.032995
78 2720/5482/2769/1.2288 1895/4067/2224/0.93486 2964/5978/3028/1.3378
79 6653/13399/6765/14.238 5844/12355/6649/12.715 7015/14081/7077/14.59
80 297/631/344/0.005999 297/622/337/0.003 319/677/367/0.005999
81 1257/2553/1321/0.070989 1094/2231/1154/0.054992 1183/2404/1237/0.06899
82 7/15/10/0.001 7/15/10/0 7/15/10/0.001
83 5/11/6/0.001 5/11/6/0.001 5/11/6/0.001
84 F/F/F/F F/F/F/F F/F/F/F
85 18/37/19/0.000999 19/40/23/0.001 19/39/23/0
86 23/54/38/0.031995 23/50/32/0.028996 24/50/30/0.026996
87 9/19/10/0.013998 9/19/10/0.011998 10/21/11/0.012998
88 11/23/12/0.057991 10/21/11/0.052991 10/21/11/0.052992
89 334/684/430/0.025996 453/910/511/0.027996 249/501/306/0.015998
90 1085/2106/1395/0.75888 F/F/F/F 561/1137/584/0.34195
91 5640/11281/5776/0.024996 7051/14103/7209/0.031995 4262/8525/4365/0.018997
92 502/1005/503/0.026996 489/979/490/0.020997 484/969/485/0.019997
93 284/577/295/0.012998 282/573/293/0.008999 282/573/293/0.008998
94 4753/9513/4762/1.8517 5342/10691/5351/1.9547 5242/10491/5251/1.9357
95 356/717/362/0.011998 355/716/363/0.017996 356/718/364/0.010999
96 2309/4624/2316/2.6026 2209/4425/2218/2.4486 2227/4460/2234/2.4826
97 1801/3346/2059/0.49393 1968/3821/2085/0.48993 1913/3704/2037/0.47193
98 9063/17431/9760/22.829 8928/17306/9480/22.166 8989/17216/9753/23.235
99 168/329/179/0.005998 150/298/154/0.004999 160/315/173/0.005999
100 10/23/16/0.006999 10/37/32/0.010998 9/31/25/0.008999
101 8/26/22/0.014997 9/22/14/0.011999 7/16/10/0.008999
102 4004/8014/4257/0.11298 3540/7087/3693/0.10298 1758/3522/1999/0.06199
103 3027/6061/3162/0.21297 2014/4032/2071/0.11298 1705/3414/1878/0.10898
104 F/F/F/F F/F/F/F F/F/F/F
105 13/26/15/0.001 13/28/15/0 13/28/15/0
106 14/28/16/0.001 15/31/18/0.002 14/29/15/0.001
107 44/121/84/0.000999 47/140/101/0.001 41/106/73/0.000999
108 44/134/98/0.001 49/126/87/0.001 44/123/87/0.000999
109 84/141/143/0.009999 97/160/158/0.009998 81/140/132/0.010999
110 200/235/367/0.044993 211/245/395/0.042994 191/224/353/0.038994
111 70/174/124/0.025997 95/214/150/0.024996 70/167/118/0.019997
112 F/F/F/F F/F/F/F 78/175/124/0.080988
113 103/214/121/0.001 102/206/112/0.000999 191/383/237/0.001999
114 224/461/272/0.11898 115/232/124/0.059991 201/403/240/0.10798
115 258/517/259/0.043993 762/1525/763/0.13298 372/745/373/0.06499
116 369/739/370/0.12398 1116/2233/1117/0.40794 473/947/474/0.17397
117 24/49/25/0.001 23/47/24/0.001 23/47/24/0.000999
118 35/71/36/0.016998 34/69/35/0.016998 34/69/35/0.015997
119 46/82/58/0.002999 44/79/55/0.003 44/78/56/0.003
120 39/65/54/0.26096 41/69/56/0.27796 41/69/56/0.27096
121 F/F/F/F F/F/F/F F/F/F/F
122 F/F/F/F F/F/F/F F/F/F/F
123 F/F/F/F F/F/F/F F/F/F/F
124 F/F/F/F F/F/F/F F/F/F/F
125 24/57/43/0.19897 32/72/57/0.25396 22/49/32/0.15398
126 44/108/92/0.010998 54/125/104/0.010998 43/115/95/0.009999
127 88/166/179/0.35795 68/166/142/0.27996 37/95/83/0.16198
128 164/331/168/0.003 185/375/192/0.003 165/333/170/0.002999
129 4499/8999/4500/1.1628 5259/10519/5260/1.3568 4737/9475/4738/1.2478
130 21/43/22/0.016998 21/43/22/0.016998 21/43/22/0.017998
131 22/45/23/0.039994 22/45/23/0.038994 22/45/23/0.038994
132 218/443/227/0.15398 206/419/215/0.14698 201/409/210/0.14298
133 225/457/234/0.33295 214/435/223/0.33495 207/421/216/0.30895
134 11/24/14/0.001 10/22/13/0.001 11/24/15/0.001
135 12/26/16/0.004998 10/22/13/0.002999 11/24/15/0.003999
136 12/26/17/0.008999 10/22/13/0.006999 11/24/15/0.007999
137 870/1741/871/0.041994 856/1713/857/0.033995 855/1711/856/0.033995
138 1641/3283/1642/0.30795 1728/3457/1729/0.33695 1682/3365/1683/0.32395
139 122/221/153/0.002999 120/221/149/0.001999 121/223/150/0.001999
140 13/26/15/0.000999 14/28/16/0.001 14/28/16/0.000999
141 6/13/7/0.003 6/13/7/0.003 6/13/7/0.002999
142 156/325/213/0.001 130/270/191/0.001 137/282/194/0.002
143 32/61/41/0.001 32/60/40/0 31/58/37/0.000999
144 25/62/42/0.002 20/76/63/0.002 18/45/33/0.002
145 22/54/39/0.015998 18/71/61/0.016997 23/72/56/0.014998
146 92/185/93/0.000999 92/185/93/0.001 91/183/92/0.000999
147 338/677/339/0.017997 339/679/340/0.016997 338/677/339/0.016996
148 21/43/22/0.001 21/43/22/0 21/43/22/0
149 25/51/26/0.001 25/51/26/0.001 25/51/26/0.001
150 93/243/150/0.026996 101/271/170/0.024996 81/214/133/0.019997
151 119/303/184/0.15498 108/291/184/0.14598 106/285/184/0.13598
152 268/554/301/0.082988 271/562/303/0.084987 157/355/225/0.057991
153 241/521/308/0.21097 208/462/286/0.18797 161/358/228/0.17597
CG_DESCENT method MPRP method NPRP+ method
No. Iter/Nf/Ng/Time Iter/Nf/Ng/Time Iter/Nf/Ng/Time
1 1/3/2/0.001 1/3/2/0.001 1/3/2/0
2 1/3/2/0.002 1/3/2/0.002 1/3/2/0.001
3 4/8/7/0.001 5/10/10/0.001 5/10/10/0.000999
4 9/16/20/0.005999 7/13/15/0.004999 7/13/15/0.004999
5 3/7/5/0 3/7/5/0 3/7/5/0
6 8/14/17/0.003999 5/11/11/0.003 5/11/11/0.004
7 9/21/15/0 12/27/19/0.000999 9/21/15/0
8 10/24/16/0.002 9/21/14/0.002 8/21/16/0.002
9 131/255/180/0.003999 126/245/164/0.001999 101/195/173/0.003
10 375/765/503/0.027996 510/890/740/0.039994 490/700/1025/0.046993
11 531/1100/718/0.078988 405/861/589/0.06499 238/533/540/0.052992
12 15/31/23/0.001 11/24/16/0.001 12/25/17/0.001
13 13/34/28/0.002999 36/60/77/0.007999 18/43/33/0.004999
14 4/9/6/0.001 15/31/21/0.003999 18/37/26/0.004
15 82/157/91/0.006999 90/171/101/0.005 79/152/87/0.004
16 2909/5808/2925/15.238 2819/5621/2840/15.294 2794/5573/2811/15.206
17 113/227/114/0.002 112/225/114/0.002 106/213/107/0.001999
18 37/75/38/0.003 27/55/28/0.003999 29/60/31/0.002999
19 448/834/531/0.061991 436/770/577/0.062991 4517/8797/4945/0.61991
20 12/28/24/0.003999 10/25/21/0.003999 11/27/22/0.003
21 12/32/28/0.038994 11/27/25/0.033995 10/25/21/0.029996
22 103/185/128/0.041994 122/230/159/0.050992 108/194/133/0.043993
23 110/200/139/0.22697 132/252/184/0.30595 99/179/131/0.24196
24 991/1777/1387/0.010999 1018/1825/1492/0.009999 936/1718/1286/0.008999
25 8654/13686/12760/0.84087 9755/14820/15245/0.97685 8911/13925/13444/0.94286
26 875/1612/1299/0.015998 852/1590/1249/0.016997 894/1658/1277/0.014998
27 9816/15450/15254/1.3928 10806/16627/17405/1.5678 9776/15443/15149/1.4718
28 986/1857/1447/0.018997 978/1828/1484/0.017997 989/1833/1419/0.016998
29 9832/15701/15229/1.7937 9824/15661/15379/1.8267 10778/16930/17516/2.2757
30 337/676/339/0.005999 385/773/390/0.006999 395/793/400/0.007998
31 9/19/10/0.003999 7/15/8/0.004999 7/15/8/0.004
32 9/19/10/0.008999 7/15/8/0.008999 7/15/8/0.006999
33 9/19/10/0.001 8/17/9/0.001 8/17/9/0.000999
34 9/19/10/0.008999 8/17/9/0.007999 8/17/9/0.007998
35 10/21/11/0.001 9/19/10/0.001 9/19/10/0.000999
36 10/21/11/0.009999 9/19/10/0.008998 9/19/10/0.008999
37 12/25/13/0.000999 11/23/12/0.001 11/23/12/0.001
38 12/25/13/0.001999 11/23/12/0.001999 11/23/12/0.001999
39 48/97/49/0.000999 49/99/50/0.001 48/97/49/0.000999
40 167/335/168/0.028996 168/337/169/0.024996 169/339/170/0.023996
41 133/267/134/0.019997 129/259/130/0.018997 127/255/128/0.018997
42 269/539/270/0.24496 265/531/266/0.24396 263/527/264/0.25996
43 54/109/55/0.000999 52/105/53/0.001 52/105/53/0.000999
44 83/167/84/0.002999 81/163/82/0.003 79/159/80/0.002
45 167/335/168/0.057992 169/339/170/0.058991 164/329/165/0.054992
46 263/527/264/0.23896 266/533/267/0.24796 256/513/257/0.23396
47 1046/2093/1047/0.030995 1057/2115/1058/0.032995 938/1877/939/0.027996
48 2926/5853/2927/0.43093 2914/5829/2915/0.43593 2918/5837/2919/0.42993
49 635/1271/636/0.018997 607/1215/608/0.018997 599/1199/600/0.018997
50 1467/2935/1468/0.20897 1478/2957/1479/0.21497 1413/2827/1414/0.21397
51 606/1213/607/0.019997 602/1205/603/0.022997 481/963/482/0.013998
52 1434/2869/1435/0.21897 1413/2827/1414/0.20497 1387/2775/1388/0.28396
53 596/1193/597/0.020997 604/1209/605/0.018997 388/777/389/0.016998
54 1374/2749/1375/0.20497 1424/2849/1425/0.20697 1356/2713/1357/0.20697
55 200/401/202/0.001999 200/401/202/0.001999 200/401/202/0.001999
56 1000/2001/1002/0.048992 1000/2001/1002/0.06199 1000/2001/1002/0.050992
57 7/15/8/0.002 7/15/8/0.002 6/13/7/0.001999
58 7/15/8/0.004 7/15/8/0.004999 7/15/8/0.003999
59 28/57/29/0.001 27/55/28/0.001 27/55/28/0.001
60 29/59/30/0.000999 29/59/30/0.002 29/59/30/0.002
61 32/59/40/0.010999 32/60/45/0.009998 31/56/39/0.011998
62 4/9/6/0.001 4/9/6/0.001 4/9/6/0.001
63 26/48/33/0.003 26/49/33/0.003 22/42/28/0.002999
64 27/50/40/0.016997 25/44/34/0.014998 23/42/30/0.013998
65 1151/2285/1599/0.008998 1150/2270/1564/0.006998 1708/3414/2234/0.013998
66 5481/11373/6023/0.041993 5510/11246/5845/0.041994 5906/12296/6550/0.045993
67 6354/13073/6816/0.47393 7741/15623/7929/0.54592 7866/16031/8277/0.61791
68 0/1/1/0.001999 0/1/1/0.001999 0/1/1/0.001
69 0/1/1/0.002999 0/1/1/0.003 0/1/1/0.002
70 F/F/F/F F/F/F/F F/F/F/F
71 6604/13682/7147/0.6859 4310/8650/4348/0.43493 6879/14253/7386/0.74789
72 368/739/371/0.25896 305/611/306/0.20897 395/792/397/0.28096
73 434/869/435/0.54392 375/752/377/0.53992 460/922/462/0.60891
74 88/178/90/0.001999 92/186/94/0.001999 80/162/82/0.002
75 603/1209/606/0.85987 454/910/456/0.6459 655/1311/656/0.96185
76 52/103/80/0.000999 107/205/145/0.003 36/72/65/0.001
77 53/107/80/0.047993 49/97/74/0.044993 33/68/56/0.032995
78 2720/5482/2769/1.2288 1895/4067/2224/0.93486 2964/5978/3028/1.3378
79 6653/13399/6765/14.238 5844/12355/6649/12.715 7015/14081/7077/14.59
80 297/631/344/0.005999 297/622/337/0.003 319/677/367/0.005999
81 1257/2553/1321/0.070989 1094/2231/1154/0.054992 1183/2404/1237/0.06899
82 7/15/10/0.001 7/15/10/0 7/15/10/0.001
83 5/11/6/0.001 5/11/6/0.001 5/11/6/0.001
84 F/F/F/F F/F/F/F F/F/F/F
85 18/37/19/0.000999 19/40/23/0.001 19/39/23/0
86 23/54/38/0.031995 23/50/32/0.028996 24/50/30/0.026996
87 9/19/10/0.013998 9/19/10/0.011998 10/21/11/0.012998
88 11/23/12/0.057991 10/21/11/0.052991 10/21/11/0.052992
89 334/684/430/0.025996 453/910/511/0.027996 249/501/306/0.015998
90 1085/2106/1395/0.75888 F/F/F/F 561/1137/584/0.34195
91 5640/11281/5776/0.024996 7051/14103/7209/0.031995 4262/8525/4365/0.018997
92 502/1005/503/0.026996 489/979/490/0.020997 484/969/485/0.019997
93 284/577/295/0.012998 282/573/293/0.008999 282/573/293/0.008998
94 4753/9513/4762/1.8517 5342/10691/5351/1.9547 5242/10491/5251/1.9357
95 356/717/362/0.011998 355/716/363/0.017996 356/718/364/0.010999
96 2309/4624/2316/2.6026 2209/4425/2218/2.4486 2227/4460/2234/2.4826
97 1801/3346/2059/0.49393 1968/3821/2085/0.48993 1913/3704/2037/0.47193
98 9063/17431/9760/22.829 8928/17306/9480/22.166 8989/17216/9753/23.235
99 168/329/179/0.005998 150/298/154/0.004999 160/315/173/0.005999
100 10/23/16/0.006999 10/37/32/0.010998 9/31/25/0.008999
101 8/26/22/0.014997 9/22/14/0.011999 7/16/10/0.008999
102 4004/8014/4257/0.11298 3540/7087/3693/0.10298 1758/3522/1999/0.06199
103 3027/6061/3162/0.21297 2014/4032/2071/0.11298 1705/3414/1878/0.10898
104 F/F/F/F F/F/F/F F/F/F/F
105 13/26/15/0.001 13/28/15/0 13/28/15/0
106 14/28/16/0.001 15/31/18/0.002 14/29/15/0.001
107 44/121/84/0.000999 47/140/101/0.001 41/106/73/0.000999
108 44/134/98/0.001 49/126/87/0.001 44/123/87/0.000999
109 84/141/143/0.009999 97/160/158/0.009998 81/140/132/0.010999
110 200/235/367/0.044993 211/245/395/0.042994 191/224/353/0.038994
111 70/174/124/0.025997 95/214/150/0.024996 70/167/118/0.019997
112 F/F/F/F F/F/F/F 78/175/124/0.080988
113 103/214/121/0.001 102/206/112/0.000999 191/383/237/0.001999
114 224/461/272/0.11898 115/232/124/0.059991 201/403/240/0.10798
115 258/517/259/0.043993 762/1525/763/0.13298 372/745/373/0.06499
116 369/739/370/0.12398 1116/2233/1117/0.40794 473/947/474/0.17397
117 24/49/25/0.001 23/47/24/0.001 23/47/24/0.000999
118 35/71/36/0.016998 34/69/35/0.016998 34/69/35/0.015997
119 46/82/58/0.002999 44/79/55/0.003 44/78/56/0.003
120 39/65/54/0.26096 41/69/56/0.27796 41/69/56/0.27096
121 F/F/F/F F/F/F/F F/F/F/F
122 F/F/F/F F/F/F/F F/F/F/F
123 F/F/F/F F/F/F/F F/F/F/F
124 F/F/F/F F/F/F/F F/F/F/F
125 24/57/43/0.19897 32/72/57/0.25396 22/49/32/0.15398
126 44/108/92/0.010998 54/125/104/0.010998 43/115/95/0.009999
127 88/166/179/0.35795 68/166/142/0.27996 37/95/83/0.16198
128 164/331/168/0.003 185/375/192/0.003 165/333/170/0.002999
129 4499/8999/4500/1.1628 5259/10519/5260/1.3568 4737/9475/4738/1.2478
130 21/43/22/0.016998 21/43/22/0.016998 21/43/22/0.017998
131 22/45/23/0.039994 22/45/23/0.038994 22/45/23/0.038994
132 218/443/227/0.15398 206/419/215/0.14698 201/409/210/0.14298
133 225/457/234/0.33295 214/435/223/0.33495 207/421/216/0.30895
134 11/24/14/0.001 10/22/13/0.001 11/24/15/0.001
135 12/26/16/0.004998 10/22/13/0.002999 11/24/15/0.003999
136 12/26/17/0.008999 10/22/13/0.006999 11/24/15/0.007999
137 870/1741/871/0.041994 856/1713/857/0.033995 855/1711/856/0.033995
138 1641/3283/1642/0.30795 1728/3457/1729/0.33695 1682/3365/1683/0.32395
139 122/221/153/0.002999 120/221/149/0.001999 121/223/150/0.001999
140 13/26/15/0.000999 14/28/16/0.001 14/28/16/0.000999
141 6/13/7/0.003 6/13/7/0.003 6/13/7/0.002999
142 156/325/213/0.001 130/270/191/0.001 137/282/194/0.002
143 32/61/41/0.001 32/60/40/0 31/58/37/0.000999
144 25/62/42/0.002 20/76/63/0.002 18/45/33/0.002
145 22/54/39/0.015998 18/71/61/0.016997 23/72/56/0.014998
146 92/185/93/0.000999 92/185/93/0.001 91/183/92/0.000999
147 338/677/339/0.017997 339/679/340/0.016997 338/677/339/0.016996
148 21/43/22/0.001 21/43/22/0 21/43/22/0
149 25/51/26/0.001 25/51/26/0.001 25/51/26/0.001
150 93/243/150/0.026996 101/271/170/0.024996 81/214/133/0.019997
151 119/303/184/0.15498 108/291/184/0.14598 106/285/184/0.13598
152 268/554/301/0.082988 271/562/303/0.084987 157/355/225/0.057991
153 241/521/308/0.21097 208/462/286/0.18797 161/358/228/0.17597
[1]

Gaohang Yu, Lutai Guan, Guoyin Li. Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. Journal of Industrial & Management Optimization, 2008, 4 (3) : 565-579. doi: 10.3934/jimo.2008.4.565

[2]

Saman Babaie–Kafaki, Reza Ghanbari. A class of descent four–term extension of the Dai–Liao conjugate gradient method based on the scaled memoryless BFGS update. Journal of Industrial & Management Optimization, 2017, 13 (2) : 649-658. doi: 10.3934/jimo.2016038

[3]

C.Y. Wang, M.X. Li. Convergence property of the Fletcher-Reeves conjugate gradient method with errors. Journal of Industrial & Management Optimization, 2005, 1 (2) : 193-200. doi: 10.3934/jimo.2005.1.193

[4]

Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe. Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (3) : 595-619. doi: 10.3934/jimo.2013.9.595

[5]

El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883

[6]

Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030

[7]

Yigui Ou, Xin Zhou. A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2018, 14 (2) : 785-801. doi: 10.3934/jimo.2017075

[8]

Guanghui Zhou, Qin Ni, Meilan Zeng. A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 595-608. doi: 10.3934/jimo.2016034

[9]

Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems & Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195

[10]

Xing Li, Chungen Shen, Lei-Hong Zhang. A projected preconditioned conjugate gradient method for the linear response eigenvalue problem. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 389-412. doi: 10.3934/naco.2018025

[11]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

[12]

Nora Merabet. Global convergence of a memory gradient method with closed-form step size formula. Conference Publications, 2007, 2007 (Special) : 721-730. doi: 10.3934/proc.2007.2007.721

[13]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507

[14]

Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505

[15]

M. S. Lee, B. S. Goh, H. G. Harno, K. H. Lim. On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 315-326. doi: 10.3934/naco.2018020

[16]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117

[17]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[18]

Liyan Qi, Xiantao Xiao, Liwei Zhang. On the global convergence of a parameter-adjusting Levenberg-Marquardt method. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 25-36. doi: 10.3934/naco.2015.5.25

[19]

Kai Wang, Lingling Xu, Deren Han. A new parallel splitting descent method for structured variational inequalities. Journal of Industrial & Management Optimization, 2014, 10 (2) : 461-476. doi: 10.3934/jimo.2014.10.461

[20]

Shishun Li, Zhengda Huang. Guaranteed descent conjugate gradient methods with modified secant condition. Journal of Industrial & Management Optimization, 2008, 4 (4) : 739-755. doi: 10.3934/jimo.2008.4.739

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (8)
  • HTML views (117)
  • Cited by (0)

Other articles
by authors

[Back to Top]