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

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

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

Yigui Ou ^, and Xin Zhou

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.

Keywords: Nonsmooth convex optimization, memoryless BFGS method, scaled conjugate gradient method, nonmonotone technique, Moreau-Yosida regularization, convergence analysis.

Mathematics Subject Classification: Primary: 90C25, 90C30; Secondary: 65K05.

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. Google Scholar
[2]	A. Auslender, Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study, 30 (1987), 102-126. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. Barzilai and J. M. Borwein, Two-point stepsize gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. Google Scholar
[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. Google Scholar
[7]	J. F. Bonnans, J. C. Gilbert, C. Lemarechal and C. Sagastizabal, A family of variable-metric proximal methods, Mathematical Programming, 68 (1995), 15-47. doi: 10.1007/BF01585756. Google Scholar
[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. Google Scholar
[9]	X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization, Mathematical Programming, 85 (1999), 313-334. doi: 10.1007/s101070050059. Google Scholar
[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. Google Scholar
[11]	M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175. doi: 10.1007/BF02591883. Google Scholar
[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. Google Scholar
[13]	M. Haarala, K. 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. Google Scholar
[14]	M. Haarala, K. 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. Google Scholar
[15]	W. W. Hager and H. C. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58. Google Scholar
[16]	J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7. Google Scholar
[17]	C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization, I: Theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385. Google Scholar
[18]	Q. Li, Conjugate gradient type methods for the nondifferentiable convex minimization, Optimization Letters, 7 (2013), 533-545. Google Scholar
[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. Google Scholar
[20]	S. Lu, Z. 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. Google Scholar
[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.Google Scholar
[22]	R. Mifflin, A quasi-second-order proximal bundle algorithm, Mathematical Programming, 73 (1996), 51-72. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	N. Sagara and M. Fukushima, A trust region method for nonsmooth convex optimization, Journal of Industrial and Management Optimization, 1 (2005), 171-180. Google Scholar
[27]	D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM Journal on Numerical Analysis, 15 (1978), 1247-1257. Google Scholar
[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. Google Scholar
[29]	W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. Google Scholar
[30]	G. L. Yuan, Z. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	G. L. Yuan, Z. 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. Google Scholar
[34]	G. L. Yuan, Z. 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. Google Scholar
[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. Google Scholar
[36]	L. Zhang, W. 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. Google Scholar

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. Google Scholar
[2]	A. Auslender, Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study, 30 (1987), 102-126. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. Barzilai and J. M. Borwein, Two-point stepsize gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. Google Scholar
[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. Google Scholar
[7]	J. F. Bonnans, J. C. Gilbert, C. Lemarechal and C. Sagastizabal, A family of variable-metric proximal methods, Mathematical Programming, 68 (1995), 15-47. doi: 10.1007/BF01585756. Google Scholar
[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. Google Scholar
[9]	X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization, Mathematical Programming, 85 (1999), 313-334. doi: 10.1007/s101070050059. Google Scholar
[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. Google Scholar
[11]	M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175. doi: 10.1007/BF02591883. Google Scholar
[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. Google Scholar
[13]	M. Haarala, K. 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. Google Scholar
[14]	M. Haarala, K. 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. Google Scholar
[15]	W. W. Hager and H. C. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58. Google Scholar
[16]	J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7. Google Scholar
[17]	C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization, I: Theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385. Google Scholar
[18]	Q. Li, Conjugate gradient type methods for the nondifferentiable convex minimization, Optimization Letters, 7 (2013), 533-545. Google Scholar
[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. Google Scholar
[20]	S. Lu, Z. 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. Google Scholar
[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.Google Scholar
[22]	R. Mifflin, A quasi-second-order proximal bundle algorithm, Mathematical Programming, 73 (1996), 51-72. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	N. Sagara and M. Fukushima, A trust region method for nonsmooth convex optimization, Journal of Industrial and Management Optimization, 1 (2005), 171-180. Google Scholar
[27]	D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM Journal on Numerical Analysis, 15 (1978), 1247-1257. Google Scholar
[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. Google Scholar
[29]	W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. Google Scholar
[30]	G. L. Yuan, Z. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	G. L. Yuan, Z. 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. Google Scholar
[34]	G. L. Yuan, Z. 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. Google Scholar
[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. Google Scholar
[36]	L. Zhang, W. 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. Google Scholar

Figure 1. Performance profiles based on iterations (left) and function evaluations (right)

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Figure 2. Performance profiles based on iterations (left) and function evaluations (right)

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Table 1. Testing functions for small scale problems

No.	Functions	$n$	$x_0$	${{f}_{ops}}\left( x \right)$
1	Rosenbrock	2	(-1.2; 1)	0
2	Crescent	2	(-1.5; 2)	0
3	CB2	2	(1; -0.1)	1.9522245
4	CB3	2	(2; 2)	2.0
5	DEM	2	(1; 1)	-3
6	QL	2	(-1; 5)	7.2
7	LQ	2	(-0.5; -0.5)	-1.4142136
8	Mifflin 1	2	(0.8; 0.6)	-1.0
9	Mifflin 2	2	(-1; -1)	-1.0
10	Wolfe	2	(3; 2)	-8
11	Rosen-Suzuki	4	(0; 0; 0; 0)	-44
12	Shor	5	(0; 0; 0; 0; 1)	22.600162

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No.	Functions	$n$	$x_0$	${{f}_{ops}}\left( x \right)$
1	Rosenbrock	2	(-1.2; 1)	0
2	Crescent	2	(-1.5; 2)	0
3	CB2	2	(1; -0.1)	1.9522245
4	CB3	2	(2; 2)	2.0
5	DEM	2	(1; 1)	-3
6	QL	2	(-1; 5)	7.2
7	LQ	2	(-0.5; -0.5)	-1.4142136
8	Mifflin 1	2	(0.8; 0.6)	-1.0
9	Mifflin 2	2	(-1; -1)	-1.0
10	Wolfe	2	(3; 2)	-8
11	Rosen-Suzuki	4	(0; 0; 0; 0)	-44
12	Shor	5	(0; 0; 0; 0; 1)	22.600162

Table 2. Numerical results for five algorithms

No.	Algorithn 3.1	LWTR	YWBB	SFTR	RFBFGS
1	3/4/	6/13/	54/56/	48/49/	4/5/
	2.6178e-9	0.2976e-7	3.4484e-7	7.1545e-4	6.2072e-10
2	3/4/	3/7/	14/16/	31/32/	35/36/
	3.3026e-6	6.5430e-4	2.7450e-5	1.6000e-3	3.0590e-7
3	4/5/	5/12/	13/15/	54/55/	5/6/
	1.9522	1.9522	1.9522	1.9573	1.9522
4	2/3/	6/13/	4/8/	55/56	2/3/
	2.0000	2.0000	2.0000	2.0100	2.0076
5	3/4/	8/16/	4/7/	5/6/	3/4
	-3.0000	-3.0000	-3.0000	-3.0000	-3.0000
6	11/12/	4/9/	22/25/	48/49/	10/11/
	7.2000	7.2000	7.2000	7.2003	7.2000
7	3/4/	5/10/	6/7/	3/4/	2/3/
	-1.4142	-1.4142	-1.4142	-1.4118	-1.4033
8	34/35/	15/31/	3/6/	59/60/	57/58/
	-1.0000	-1.0000	-0.9938	-1.0000	-1.0000
9	2/3/	8/16/	12/13/	4/5/	2/3/
	-1.0000	-1.0000	-0.9999	-0.9997	-0.9813
10	3/4/	12/24/	9/12/	43/46/	4/5/
	-8.0000	-8.0000	-8.0000	-8.0000	-8.0000
11	49/106/	20/40/	8/9/	60/61/	25/31/
	-43.9999	-44.0000	-43.9493	-39.9924	-43.9982
12	14/16/	14/28/	9/10/	71/72/	66/152/
	22.6019	22.6002	22.6004	22.6892	22.6017

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No.	Algorithn 3.1	LWTR	YWBB	SFTR	RFBFGS
1	3/4/	6/13/	54/56/	48/49/	4/5/
	2.6178e-9	0.2976e-7	3.4484e-7	7.1545e-4	6.2072e-10
2	3/4/	3/7/	14/16/	31/32/	35/36/
	3.3026e-6	6.5430e-4	2.7450e-5	1.6000e-3	3.0590e-7
3	4/5/	5/12/	13/15/	54/55/	5/6/
	1.9522	1.9522	1.9522	1.9573	1.9522
4	2/3/	6/13/	4/8/	55/56	2/3/
	2.0000	2.0000	2.0000	2.0100	2.0076
5	3/4/	8/16/	4/7/	5/6/	3/4
	-3.0000	-3.0000	-3.0000	-3.0000	-3.0000
6	11/12/	4/9/	22/25/	48/49/	10/11/
	7.2000	7.2000	7.2000	7.2003	7.2000
7	3/4/	5/10/	6/7/	3/4/	2/3/
	-1.4142	-1.4142	-1.4142	-1.4118	-1.4033
8	34/35/	15/31/	3/6/	59/60/	57/58/
	-1.0000	-1.0000	-0.9938	-1.0000	-1.0000
9	2/3/	8/16/	12/13/	4/5/	2/3/
	-1.0000	-1.0000	-0.9999	-0.9997	-0.9813
10	3/4/	12/24/	9/12/	43/46/	4/5/
	-8.0000	-8.0000	-8.0000	-8.0000	-8.0000
11	49/106/	20/40/	8/9/	60/61/	25/31/
	-43.9999	-44.0000	-43.9493	-39.9924	-43.9982
12	14/16/	14/28/	9/10/	71/72/	66/152/
	22.6019	22.6002	22.6004	22.6892	22.6017

Table 3. Testing functions for large scale problems

No.	Functions	Initial points $x_0$
1	Generalization of MAXQ	$(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2	Generalization of MXHILB	$(1, 1, \cdots, 1)$
3	Chained LQ	$(-0.5, -0.5, \cdots, -0.5)$
4	Number of active faces	$(1, 1, \cdots, 1)$
5	Nonsmooth generalization of Brown 2	$(1, 0, 1, 0, \cdots)$
6	Chained Mifflin 2	$(-1, -1, \cdots, -1)$
7	Chained Crescent Ⅰ	$(-1.5, 2, -1.5, 2, \cdots)$
8	Chained Crescent Ⅱ	$(1, 0, 1, 0 \cdots)$

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No.	Functions	Initial points $x_0$
1	Generalization of MAXQ	$(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2	Generalization of MXHILB	$(1, 1, \cdots, 1)$
3	Chained LQ	$(-0.5, -0.5, \cdots, -0.5)$
4	Number of active faces	$(1, 1, \cdots, 1)$
5	Nonsmooth generalization of Brown 2	$(1, 0, 1, 0, \cdots)$
6	Chained Mifflin 2	$(-1, -1, \cdots, -1)$
7	Chained Crescent Ⅰ	$(-1.5, 2, -1.5, 2, \cdots)$
8	Chained Crescent Ⅱ	$(1, 0, 1, 0 \cdots)$

Table 4. results for Algorithms 3.1 and CG-YWL

No.	n	Algorithn 3.1	CG-YWL
1	1000	186/1601/2.6568e-10	225/4710/6.9354e-8
	5000	242/2725/1.2183e-10	250/5235/6.8798e-8
	10000	253/2997/3.4045e-10	261/5466/6.6528e-8
2	1000	94/1301/4.0582e-9	91/1482/8.2738e-9
	5000	119/1499/2.6731e-9	111/1938/9.7206e-9
	10000	129/1901/2.1905e-9	120/2127/5.8524e-9
3	1000	37/110/2.3278e-9	37/114/7.2687e-9
	5000	39/116/5.9987e-9	39/120/9.0932e-9
	10000	40/121/1.6943e-9	40/123/9.0941e-9
4	1000	71/891/3.6735e-11	77/1026/6.8037e-9
	5000	82/937/7.9864e-11	90/1281/7.8405e-9
	10000	86/1208/5.7163e-11	96/1401/9.9366e-9
5	1000	35/114/1.5021e-11	38/117/7.2687e-9
	5000	39/120/5.2352e-11	40/123/9.0932e-9
	10000	41/126/2.1136e-11	41/125/1.8188e-8
6	1000	38/116/-5.3979e+3	37/114/-2.4975e+4
	5000	41/123/-3.2153e+4	39/120/-1.2498e+5
	10000	43/129/-2.0127e+4	40/123/-2.4998e+5
7	1000	34/101/7.0463e-11	37/114/5.4897e-9
	5000	36/113/3.8145e-11	39/120/6.8294e-9
	10000	39/120/4.3654e-11	40/123/6.8253e-9
8	1000	37/112/6.0424e-11	39/120/6.8185e-9
	5000	39/121/1.8205e-11	41/126/8.5258e-9
	10000	42/125/2.6473e-11	42/129/8.5262e-9

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No.	n	Algorithn 3.1	CG-YWL
1	1000	186/1601/2.6568e-10	225/4710/6.9354e-8
	5000	242/2725/1.2183e-10	250/5235/6.8798e-8
	10000	253/2997/3.4045e-10	261/5466/6.6528e-8
2	1000	94/1301/4.0582e-9	91/1482/8.2738e-9
	5000	119/1499/2.6731e-9	111/1938/9.7206e-9
	10000	129/1901/2.1905e-9	120/2127/5.8524e-9
3	1000	37/110/2.3278e-9	37/114/7.2687e-9
	5000	39/116/5.9987e-9	39/120/9.0932e-9
	10000	40/121/1.6943e-9	40/123/9.0941e-9
4	1000	71/891/3.6735e-11	77/1026/6.8037e-9
	5000	82/937/7.9864e-11	90/1281/7.8405e-9
	10000	86/1208/5.7163e-11	96/1401/9.9366e-9
5	1000	35/114/1.5021e-11	38/117/7.2687e-9
	5000	39/120/5.2352e-11	40/123/9.0932e-9
	10000	41/126/2.1136e-11	41/125/1.8188e-8
6	1000	38/116/-5.3979e+3	37/114/-2.4975e+4
	5000	41/123/-3.2153e+4	39/120/-1.2498e+5
	10000	43/129/-2.0127e+4	40/123/-2.4998e+5
7	1000	34/101/7.0463e-11	37/114/5.4897e-9
	5000	36/113/3.8145e-11	39/120/6.8294e-9
	10000	39/120/4.3654e-11	40/123/6.8253e-9
8	1000	37/112/6.0424e-11	39/120/6.8185e-9
	5000	39/121/1.8205e-11	41/126/8.5258e-9
	10000	42/125/2.6473e-11	42/129/8.5262e-9

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2018 Impact Factor: 1.025

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2018 Impact Factor: 1.025

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2018 Impact Factor: 1.025

American Institute of Mathematical Sciences