April 2018, 14(2): 785-801. doi: 10.3934/jimo.2017075

A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization

Department of Applied Mathematics, Hainan University, Haikou 570228, China

* Corresponding author: Yigui Ou

Received  January 2017 Revised  March 2017 Published  September 2017

Fund Project: The work is supported by NNSF of China (No. 11261015) and NSF of Hainan Province (No. 2016CXTD004; No. 111001; No. 117011)

This paper presents a nonmonotone scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving nonsmooth convex optimization problems, which combines the idea of scaled memoryless BFGS preconditioned conjugate gradient method with the nonmonotone technique and the Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under mild conditions, the global convergence of the proposed method is established. Preliminary numerical results and related comparisons show that the proposed method can be applied to solve large scale nonsmooth convex optimization problems.

Citation: 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
References:
[1]

N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Computational Optimization and Applications, 38 (2007), 401-416. doi: 10.1007/s10589-007-9055-7.

[2]

A. Auslender, Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study, 30 (1987), 102-126.

[3]

S. Babaie-Kafaki, A modified scaled memoryless BFGS preconditioned conjugate gradient method for unconstrained optimization, 4OR, A Quarterly Journal of Operations Research, 11 (2013), 361-374. doi: 10.1007/s10288-013-0233-4.

[4]

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

[5]

J. Barzilai and J. M. Borwein, Two-point stepsize gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.

[6]

E. Birgin and J. M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Applied Mathematics and Optimization, 43 (2001), 117-128. doi: 10.1007/s00245-001-0003-0.

[7]

J. F. BonnansJ. C. GilbertC. Lemarechal and C. Sagastizabal, A family of variable-metric proximal methods, Mathematical Programming, 68 (1995), 15-47. doi: 10.1007/BF01585756.

[8]

J. V. Burke and M. Qian, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Mathematical Programming, 88 (2000), 157-181. doi: 10.1007/PL00011373.

[9]

X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization, Mathematical Programming, 85 (1999), 313-334. doi: 10.1007/s101070050059.

[10]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, Serial A, 91 (2002), 201-213. doi: 10.1007/s101070100263.

[11]

M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175. doi: 10.1007/BF02591883.

[12]

M. Fukushima and L. Q. Qi, A globally and superlinearly convergent algorithm for nonsmooth convex minimization, SIAM Journal on Optimization, 6 (1996), 1106-1120. doi: 10.1137/S1052623494278839.

[13]

M. HaaralaK. Miettinen and M. M. Mäkelä, New limited memory bundle method for large-scale nonsmooth optimization, Optimization Methods and Software, 19 (2004), 673-692. doi: 10.1080/10556780410001689225.

[14]

M. HaaralaK. Miettinen and M. M. Mäkelä, Globally convergent limited memory bundle method for large-scale nonsmooth optimization, Mathematical Programming, 109 (2007), 181-205. doi: 10.1007/s10107-006-0728-2.

[15]

W. W. Hager and H. C. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.

[16]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.

[17]

C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization, I: Theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385.

[18]

Q. Li, Conjugate gradient type methods for the nondifferentiable convex minimization, Optimization Letters, 7 (2013), 533-545.

[19]

D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optimization Methods and Software, 13 (2000), 181-201.

[20]

S. LuZ. X. Wei and L. Li, A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization, Computational Optimization and Applications, 51 (2012), 551-573.

[21]

L. Luk$\check{s}$an and J. Vl$\check{c}$ek, Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization, Technical Report No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000.

[22]

R. Mifflin, A quasi-second-order proximal bundle algorithm, Mathematical Programming, 73 (1996), 51-72.

[23]

Y. G. Ou and H. C. Lin, An ODE-like nonmonotone method for nonsmooth convex optimization, Journal of Applied Mathematics and Computing, 52 (2016), 265-285.

[24]

L. Q. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18 (1993), 227-244. doi: 10.1287/moor.18.1.227.

[25]

A. I. Rauf and M. Fukushima, A globally convergent BFGS method for nonsmooth convex optimization, Journal of Optimization Theory and Applications, 104 (2000), 539-558.

[26]

N. Sagara and M. Fukushima, A trust region method for nonsmooth convex optimization, Journal of Industrial and Management Optimization, 1 (2005), 171-180.

[27]

D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM Journal on Numerical Analysis, 15 (1978), 1247-1257.

[28]

J. Shen, L. P. Pang and D. Li, An approximate quasi-Newton bundle-type method for nonsmooth optimization, Abstract and Applied Analysis, 2013, Art. ID 697474, 7 pp. doi: 10.1155/2013/697474.

[29]

W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006.

[30]

G. L. YuanZ. H. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large scale nonsmooth minimizations and nonlinear equations, Journal of Optimization Theory and Applications, 168 (2016), 129-152.

[31]

G. L. Yuan, Z. Sheng and W. J. Liu, The modified HZ conjugate gradient algorithm for large scale nonsmooth optimization Plos One, 11(2016), e0164289, 15pp. doi: 10.1371/journal.pone.0164289.

[32]

G. L. Yuan and Z. X. Wei, The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems, Mathematical Modelling and Analysis, 17 (2012), 203-216.

[33]

G. L. YuanZ. X. Wei and G. Y. Li, A modified Polak-Ribiére-Polyak conjugate gradient algorithm for nonsmooth convex programs, Journal of Computational and Applied mathematics, 255 (2014), 86-96.

[34]

G. L. YuanZ. X. Wei and Z. X. Wang, Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minization, Computational Optimization and Applications, 54 (2013), 45-64.

[35]

H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM Jpournal on Optimization, 14 (2004), 1043-1056. doi: 10.1137/S1052623403428208.

[36]

L. ZhangW. J. Zhou and D. H. Li, A descent modified Polak-Ribiére-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640. doi: 10.1093/imanum/drl016.

show all references

References:
[1]

N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Computational Optimization and Applications, 38 (2007), 401-416. doi: 10.1007/s10589-007-9055-7.

[2]

A. Auslender, Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study, 30 (1987), 102-126.

[3]

S. Babaie-Kafaki, A modified scaled memoryless BFGS preconditioned conjugate gradient method for unconstrained optimization, 4OR, A Quarterly Journal of Operations Research, 11 (2013), 361-374. doi: 10.1007/s10288-013-0233-4.

[4]

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

[5]

J. Barzilai and J. M. Borwein, Two-point stepsize gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.

[6]

E. Birgin and J. M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Applied Mathematics and Optimization, 43 (2001), 117-128. doi: 10.1007/s00245-001-0003-0.

[7]

J. F. BonnansJ. C. GilbertC. Lemarechal and C. Sagastizabal, A family of variable-metric proximal methods, Mathematical Programming, 68 (1995), 15-47. doi: 10.1007/BF01585756.

[8]

J. V. Burke and M. Qian, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Mathematical Programming, 88 (2000), 157-181. doi: 10.1007/PL00011373.

[9]

X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization, Mathematical Programming, 85 (1999), 313-334. doi: 10.1007/s101070050059.

[10]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, Serial A, 91 (2002), 201-213. doi: 10.1007/s101070100263.

[11]

M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175. doi: 10.1007/BF02591883.

[12]

M. Fukushima and L. Q. Qi, A globally and superlinearly convergent algorithm for nonsmooth convex minimization, SIAM Journal on Optimization, 6 (1996), 1106-1120. doi: 10.1137/S1052623494278839.

[13]

M. HaaralaK. Miettinen and M. M. Mäkelä, New limited memory bundle method for large-scale nonsmooth optimization, Optimization Methods and Software, 19 (2004), 673-692. doi: 10.1080/10556780410001689225.

[14]

M. HaaralaK. Miettinen and M. M. Mäkelä, Globally convergent limited memory bundle method for large-scale nonsmooth optimization, Mathematical Programming, 109 (2007), 181-205. doi: 10.1007/s10107-006-0728-2.

[15]

W. W. Hager and H. C. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.

[16]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.

[17]

C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization, I: Theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385.

[18]

Q. Li, Conjugate gradient type methods for the nondifferentiable convex minimization, Optimization Letters, 7 (2013), 533-545.

[19]

D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optimization Methods and Software, 13 (2000), 181-201.

[20]

S. LuZ. X. Wei and L. Li, A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization, Computational Optimization and Applications, 51 (2012), 551-573.

[21]

L. Luk$\check{s}$an and J. Vl$\check{c}$ek, Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization, Technical Report No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000.

[22]

R. Mifflin, A quasi-second-order proximal bundle algorithm, Mathematical Programming, 73 (1996), 51-72.

[23]

Y. G. Ou and H. C. Lin, An ODE-like nonmonotone method for nonsmooth convex optimization, Journal of Applied Mathematics and Computing, 52 (2016), 265-285.

[24]

L. Q. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18 (1993), 227-244. doi: 10.1287/moor.18.1.227.

[25]

A. I. Rauf and M. Fukushima, A globally convergent BFGS method for nonsmooth convex optimization, Journal of Optimization Theory and Applications, 104 (2000), 539-558.

[26]

N. Sagara and M. Fukushima, A trust region method for nonsmooth convex optimization, Journal of Industrial and Management Optimization, 1 (2005), 171-180.

[27]

D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM Journal on Numerical Analysis, 15 (1978), 1247-1257.

[28]

J. Shen, L. P. Pang and D. Li, An approximate quasi-Newton bundle-type method for nonsmooth optimization, Abstract and Applied Analysis, 2013, Art. ID 697474, 7 pp. doi: 10.1155/2013/697474.

[29]

W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006.

[30]

G. L. YuanZ. H. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large scale nonsmooth minimizations and nonlinear equations, Journal of Optimization Theory and Applications, 168 (2016), 129-152.

[31]

G. L. Yuan, Z. Sheng and W. J. Liu, The modified HZ conjugate gradient algorithm for large scale nonsmooth optimization Plos One, 11(2016), e0164289, 15pp. doi: 10.1371/journal.pone.0164289.

[32]

G. L. Yuan and Z. X. Wei, The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems, Mathematical Modelling and Analysis, 17 (2012), 203-216.

[33]

G. L. YuanZ. X. Wei and G. Y. Li, A modified Polak-Ribiére-Polyak conjugate gradient algorithm for nonsmooth convex programs, Journal of Computational and Applied mathematics, 255 (2014), 86-96.

[34]

G. L. YuanZ. X. Wei and Z. X. Wang, Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minization, Computational Optimization and Applications, 54 (2013), 45-64.

[35]

H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM Jpournal on Optimization, 14 (2004), 1043-1056. doi: 10.1137/S1052623403428208.

[36]

L. ZhangW. J. Zhou and D. H. Li, A descent modified Polak-Ribiére-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640. doi: 10.1093/imanum/drl016.

Figure 1.  Performance profiles based on iterations (left) and function evaluations (right)
Figure 2.  Performance profiles based on iterations (left) and function evaluations (right)
Table 1.  Testing functions for small scale problems
No.Functions$n$$x_0$${{f}_{ops}}\left( x \right)$
1Rosenbrock2(-1.2; 1)0
2Crescent2(-1.5; 2)0
3CB22(1; -0.1)1.9522245
4CB32(2; 2)2.0
5DEM2(1; 1)-3
6QL2(-1; 5)7.2
7LQ2(-0.5; -0.5)-1.4142136
8Mifflin 12(0.8; 0.6)-1.0
9Mifflin 22(-1; -1)-1.0
10Wolfe2(3; 2)-8
11Rosen-Suzuki4(0; 0; 0; 0)-44
12Shor5(0; 0; 0; 0; 1)22.600162
No.Functions$n$$x_0$${{f}_{ops}}\left( x \right)$
1Rosenbrock2(-1.2; 1)0
2Crescent2(-1.5; 2)0
3CB22(1; -0.1)1.9522245
4CB32(2; 2)2.0
5DEM2(1; 1)-3
6QL2(-1; 5)7.2
7LQ2(-0.5; -0.5)-1.4142136
8Mifflin 12(0.8; 0.6)-1.0
9Mifflin 22(-1; -1)-1.0
10Wolfe2(3; 2)-8
11Rosen-Suzuki4(0; 0; 0; 0)-44
12Shor5(0; 0; 0; 0; 1)22.600162
Table 2.  Numerical results for five algorithms
No.Algorithn 3.1LWTRYWBBSFTRRFBFGS
13/4/6/13/54/56/48/49/4/5/
2.6178e-90.2976e-73.4484e-77.1545e-46.2072e-10
23/4/3/7/14/16/31/32/35/36/
3.3026e-66.5430e-42.7450e-51.6000e-33.0590e-7
34/5/5/12/13/15/54/55/5/6/
1.95221.95221.95221.95731.9522
42/3/6/13/4/8/55/562/3/
2.00002.00002.00002.01002.0076
53/4/8/16/4/7/5/6/3/4
-3.0000-3.0000-3.0000-3.0000-3.0000
611/12/4/9/22/25/48/49/10/11/
7.20007.20007.20007.20037.2000
73/4/5/10/6/7/3/4/2/3/
-1.4142-1.4142-1.4142-1.4118-1.4033
834/35/15/31/3/6/59/60/57/58/
-1.0000-1.0000-0.9938-1.0000-1.0000
92/3/8/16/12/13/4/5/2/3/
-1.0000-1.0000-0.9999-0.9997-0.9813
103/4/12/24/9/12/43/46/4/5/
-8.0000-8.0000-8.0000-8.0000-8.0000
1149/106/20/40/8/9/60/61/25/31/
-43.9999-44.0000-43.9493-39.9924-43.9982
1214/16/14/28/9/10/71/72/66/152/
22.601922.600222.600422.689222.6017
No.Algorithn 3.1LWTRYWBBSFTRRFBFGS
13/4/6/13/54/56/48/49/4/5/
2.6178e-90.2976e-73.4484e-77.1545e-46.2072e-10
23/4/3/7/14/16/31/32/35/36/
3.3026e-66.5430e-42.7450e-51.6000e-33.0590e-7
34/5/5/12/13/15/54/55/5/6/
1.95221.95221.95221.95731.9522
42/3/6/13/4/8/55/562/3/
2.00002.00002.00002.01002.0076
53/4/8/16/4/7/5/6/3/4
-3.0000-3.0000-3.0000-3.0000-3.0000
611/12/4/9/22/25/48/49/10/11/
7.20007.20007.20007.20037.2000
73/4/5/10/6/7/3/4/2/3/
-1.4142-1.4142-1.4142-1.4118-1.4033
834/35/15/31/3/6/59/60/57/58/
-1.0000-1.0000-0.9938-1.0000-1.0000
92/3/8/16/12/13/4/5/2/3/
-1.0000-1.0000-0.9999-0.9997-0.9813
103/4/12/24/9/12/43/46/4/5/
-8.0000-8.0000-8.0000-8.0000-8.0000
1149/106/20/40/8/9/60/61/25/31/
-43.9999-44.0000-43.9493-39.9924-43.9982
1214/16/14/28/9/10/71/72/66/152/
22.601922.600222.600422.689222.6017
Table 3.  Testing functions for large scale problems
No.FunctionsInitial points $x_0$
1Generalization of MAXQ $(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2Generalization of MXHILB $(1, 1, \cdots, 1)$
3Chained LQ $(-0.5, -0.5, \cdots, -0.5)$
4Number of active faces $(1, 1, \cdots, 1)$
5Nonsmooth generalization of Brown 2 $(1, 0, 1, 0, \cdots)$
6Chained Mifflin 2 $(-1, -1, \cdots, -1)$
7Chained Crescent Ⅰ $(-1.5, 2, -1.5, 2, \cdots)$
8Chained Crescent Ⅱ $(1, 0, 1, 0 \cdots)$
No.FunctionsInitial points $x_0$
1Generalization of MAXQ $(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2Generalization of MXHILB $(1, 1, \cdots, 1)$
3Chained LQ $(-0.5, -0.5, \cdots, -0.5)$
4Number of active faces $(1, 1, \cdots, 1)$
5Nonsmooth generalization of Brown 2 $(1, 0, 1, 0, \cdots)$
6Chained Mifflin 2 $(-1, -1, \cdots, -1)$
7Chained Crescent Ⅰ $(-1.5, 2, -1.5, 2, \cdots)$
8Chained Crescent Ⅱ $(1, 0, 1, 0 \cdots)$
Table 4.  results for Algorithms 3.1 and CG-YWL
No.nAlgorithn 3.1CG-YWL
11000186/1601/2.6568e-10225/4710/6.9354e-8
5000242/2725/1.2183e-10250/5235/6.8798e-8
10000253/2997/3.4045e-10261/5466/6.6528e-8
2100094/1301/4.0582e-991/1482/8.2738e-9
5000119/1499/2.6731e-9111/1938/9.7206e-9
10000129/1901/2.1905e-9120/2127/5.8524e-9
3100037/110/2.3278e-937/114/7.2687e-9
500039/116/5.9987e-939/120/9.0932e-9
1000040/121/1.6943e-940/123/9.0941e-9
4100071/891/3.6735e-1177/1026/6.8037e-9
500082/937/7.9864e-1190/1281/7.8405e-9
1000086/1208/5.7163e-1196/1401/9.9366e-9
5100035/114/1.5021e-1138/117/7.2687e-9
500039/120/5.2352e-1140/123/9.0932e-9
1000041/126/2.1136e-1141/125/1.8188e-8
6100038/116/-5.3979e+337/114/-2.4975e+4
500041/123/-3.2153e+439/120/-1.2498e+5
1000043/129/-2.0127e+440/123/-2.4998e+5
7100034/101/7.0463e-1137/114/5.4897e-9
500036/113/3.8145e-1139/120/6.8294e-9
1000039/120/4.3654e-1140/123/6.8253e-9
8100037/112/6.0424e-1139/120/6.8185e-9
500039/121/1.8205e-1141/126/8.5258e-9
1000042/125/2.6473e-1142/129/8.5262e-9
No.nAlgorithn 3.1CG-YWL
11000186/1601/2.6568e-10225/4710/6.9354e-8
5000242/2725/1.2183e-10250/5235/6.8798e-8
10000253/2997/3.4045e-10261/5466/6.6528e-8
2100094/1301/4.0582e-991/1482/8.2738e-9
5000119/1499/2.6731e-9111/1938/9.7206e-9
10000129/1901/2.1905e-9120/2127/5.8524e-9
3100037/110/2.3278e-937/114/7.2687e-9
500039/116/5.9987e-939/120/9.0932e-9
1000040/121/1.6943e-940/123/9.0941e-9
4100071/891/3.6735e-1177/1026/6.8037e-9
500082/937/7.9864e-1190/1281/7.8405e-9
1000086/1208/5.7163e-1196/1401/9.9366e-9
5100035/114/1.5021e-1138/117/7.2687e-9
500039/120/5.2352e-1140/123/9.0932e-9
1000041/126/2.1136e-1141/125/1.8188e-8
6100038/116/-5.3979e+337/114/-2.4975e+4
500041/123/-3.2153e+439/120/-1.2498e+5
1000043/129/-2.0127e+440/123/-2.4998e+5
7100034/101/7.0463e-1137/114/5.4897e-9
500036/113/3.8145e-1139/120/6.8294e-9
1000039/120/4.3654e-1140/123/6.8253e-9
8100037/112/6.0424e-1139/120/6.8185e-9
500039/121/1.8205e-1141/126/8.5258e-9
1000042/125/2.6473e-1142/129/8.5262e-9
[1]

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

[2]

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

[3]

Yigui Ou, Yuanwen Liu. A memory gradient method based on the nonmonotone technique. Journal of Industrial & Management Optimization, 2017, 13 (2) : 857-872. doi: 10.3934/jimo.2016050

[4]

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