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January  2017, 13(1): 283-295. doi: 10.3934/jimo.2016017

Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations

Jinkui Liu 1,2,, and  Shengjie Li 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China
2. 

School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, 404100, China

* Corresponding author: Jinkui Liu

Received  April 2015 Published  March 2016

Fund Project: supported by the National Natural Science Foundation of China (Grant number: 11571055), the fund of Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant number:KJ1501003) and Chongqing Three Gorges University(Grant number:14ZD-14).

In this paper, we consider a multivariate spectral DY-type projection method for solving nonlinear monotone equations with convex constraints. The search direction of the proposed method combines those of the multivariate spectral gradient method and DY conjugate gradient method. With no need for the derivative information, the proposed method is very suitable to solve large-scale nonsmooth monotone equations. Under appropriate conditions, we prove the global convergence and R-linear convergence rate of the proposed method. The preliminary numerical results also indicate that the proposed method is robust and effective.

Keywords: Nonlinear monotone equations, multivariate spectral gradient method, DY conjugate gradient method, global convergence.
Mathematics Subject Classification: Primary: 49M37, 65H10; Secondary: 65K05.
Citation: Jinkui Liu, Shengjie Li. Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2017, 13 (1) : 283-295. doi: 10.3934/jimo.2016017
References:
[1]

J. M. Barizilai and M. Borwein, Two point step size gradient methods, IMA Journal on Numerical Analysis, 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[2]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejèer-monotone methods in Hilbert spaces, Mathematical Methods and Operations Research, 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.  Google Scholar

[3]

S. Bellavia and B. Morini, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM Journal on Scientific Computing, 23 (2001), 940-960.  doi: 10.1137/S1064827599363976.  Google Scholar

[4]

Y. H. Dai and Y. X. Yuan, A nonlinear conjugate gradient with a strong global convergence property, SIAM Journal on Optimization, 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[5]

J. E. Dennis and J. J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Mathematics of Computation, 28 (1974), 549-560.  doi: 10.1090/S0025-5718-1974-0343581-1.  Google Scholar

[6]

J. E. Dennis and J. J. Moré, Quasi-Newton method, motivation and theory, SIAM Review, 19 (1997), 46-89.  doi: 10.1137/1019005.  Google Scholar

[7]

S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optimization Methods and Software, 5 (1995), 319-345.  doi: 10.1080/10556789508805619.  Google Scholar

[8]

L. Han, G. H. Yu and L. T. Guan, Multivariate spectral gradient method for unconstrained optimization, Applied Mathematics and Computation, 201 (2008), 621-630.  doi: 10.1016/j.amc.2007.12.054.  Google Scholar

[9]

A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distances for constrained optmization, Optimization, 41 (1997), 257-278.  doi: 10.1080/02331939708844339.  Google Scholar

[10]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18 (2003), 583-599.  doi: 10.1080/10556780310001610493.  Google Scholar

[11]

W. La Cruz, J. M. Mart$\acute{i}$nez and M. Raydan, Spectral residual method without gradient minformation for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[12]

Q. N. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA Journal on Numerical Analysis, 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.  Google Scholar

[13]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Applied Mathematics and Computation, 22 (1987), 333-361.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[14]

K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Transactions on Mathematical Software, 16 (1990), 143-151.  doi: 10.1145/78928.78930.  Google Scholar

[15]

L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming, 58 (1999), 353-367.  doi: 10.1007/BF01581275.  Google Scholar

[16]

M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Applied Optimization, 22, Kluwer Acad. Publ., Dordrecht, 1999, 355–369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar

[17]

C. W. Wang, Y. J. Wang and C. L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Mathematical Methods and Operations Research, 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[18]

A. J. Wood and B. F. Wollenberg, Power Generations Operations and Control, Wiley, New York, 1996. Google Scholar

[19]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, Computing, 15 (2001), 239-249.  doi: 10.1007/978-3-7091-6217-0_18.  Google Scholar

[20]

N. Yamashita and M. Fukushima, Modified Newton methods for sovling a semismooth reformulation of monotone complementary problems, Mathematical Programming, 76 (1997), 469-491.  doi: 10.1007/BF02614394.  Google Scholar

[21]

Z. S. Yu, J. Sun and Y. Qin, A multivariate spectral projected gradient method for bound constrained optimization, Journal of Computational and Applied Mathematics, 235 (2011), 2263-2269.  doi: 10.1016/j.cam.2010.10.023.  Google Scholar

[22]

G. H. Yu, S. Z. Niu and J. H. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, Journal of Industrial and Management Optimization, 9 (2013), 117-129.  doi: 10.3934/jimo.2013.9.117.  Google Scholar

[23]

L. Zhang and W. J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, Journal of Computational and Applied Mathematics, 196 (2006), 478-484.  doi: 10.1016/j.cam.2005.10.002.  Google Scholar

show all references

References:
[1]

J. M. Barizilai and M. Borwein, Two point step size gradient methods, IMA Journal on Numerical Analysis, 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[2]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejèer-monotone methods in Hilbert spaces, Mathematical Methods and Operations Research, 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.  Google Scholar

[3]

S. Bellavia and B. Morini, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM Journal on Scientific Computing, 23 (2001), 940-960.  doi: 10.1137/S1064827599363976.  Google Scholar

[4]

Y. H. Dai and Y. X. Yuan, A nonlinear conjugate gradient with a strong global convergence property, SIAM Journal on Optimization, 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[5]

J. E. Dennis and J. J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Mathematics of Computation, 28 (1974), 549-560.  doi: 10.1090/S0025-5718-1974-0343581-1.  Google Scholar

[6]

J. E. Dennis and J. J. Moré, Quasi-Newton method, motivation and theory, SIAM Review, 19 (1997), 46-89.  doi: 10.1137/1019005.  Google Scholar

[7]

S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optimization Methods and Software, 5 (1995), 319-345.  doi: 10.1080/10556789508805619.  Google Scholar

[8]

L. Han, G. H. Yu and L. T. Guan, Multivariate spectral gradient method for unconstrained optimization, Applied Mathematics and Computation, 201 (2008), 621-630.  doi: 10.1016/j.amc.2007.12.054.  Google Scholar

[9]

A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distances for constrained optmization, Optimization, 41 (1997), 257-278.  doi: 10.1080/02331939708844339.  Google Scholar

[10]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18 (2003), 583-599.  doi: 10.1080/10556780310001610493.  Google Scholar

[11]

W. La Cruz, J. M. Mart$\acute{i}$nez and M. Raydan, Spectral residual method without gradient minformation for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[12]

Q. N. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA Journal on Numerical Analysis, 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.  Google Scholar

[13]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Applied Mathematics and Computation, 22 (1987), 333-361.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[14]

K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Transactions on Mathematical Software, 16 (1990), 143-151.  doi: 10.1145/78928.78930.  Google Scholar

[15]

L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming, 58 (1999), 353-367.  doi: 10.1007/BF01581275.  Google Scholar

[16]

M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Applied Optimization, 22, Kluwer Acad. Publ., Dordrecht, 1999, 355–369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar

[17]

C. W. Wang, Y. J. Wang and C. L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Mathematical Methods and Operations Research, 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[18]

A. J. Wood and B. F. Wollenberg, Power Generations Operations and Control, Wiley, New York, 1996. Google Scholar

[19]

N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, Computing, 15 (2001), 239-249.  doi: 10.1007/978-3-7091-6217-0_18.  Google Scholar

[20]

N. Yamashita and M. Fukushima, Modified Newton methods for sovling a semismooth reformulation of monotone complementary problems, Mathematical Programming, 76 (1997), 469-491.  doi: 10.1007/BF02614394.  Google Scholar

[21]

Z. S. Yu, J. Sun and Y. Qin, A multivariate spectral projected gradient method for bound constrained optimization, Journal of Computational and Applied Mathematics, 235 (2011), 2263-2269.  doi: 10.1016/j.cam.2010.10.023.  Google Scholar

[22]

G. H. Yu, S. Z. Niu and J. H. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, Journal of Industrial and Management Optimization, 9 (2013), 117-129.  doi: 10.3934/jimo.2013.9.117.  Google Scholar

[23]

L. Zhang and W. J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, Journal of Computational and Applied Mathematics, 196 (2006), 478-484.  doi: 10.1016/j.cam.2005.10.002.  Google Scholar

Table 1.  The results of Problem 1 with given initial points
Dim Initial points MSGP method Algorithm 2.1
1000 X1 7/22/0.06/1.31135e-007 13/40/0.05/3.38669e-006
X2 2/7/0.05/0.00000e+000 4/13/0.03/0.00000e+000
X3 11/34/0.08/2.19964e-007 3/40/0.05/3.36081e-006
X4 3/10/0.05/0.00000e+000 6/19/0.05/0.00000e+000
X5 11/34/0.09/8.93938e-008 8/25/0.06/6.13420e-006
3000 X1 2/7/0.05/0.00000e+000 13/40/0.06/5.81725e-006
X2 11/34/0.25/4.90071e-006 4/13/0.03/0.00000e+000
X3 3/10/0.06/0.00000e+000 13/40/0.08/5.80007e-006
X4 11/34/0.30/2.69613e-006 6/19/0.05/0.00000e+000
X5 7/22/0.31/1.31003e-007 10/31/0.42/2.65195e-006
5000 X1 2/7/0.05/0.00000e+000 13/40/0.08/7.49753e-006
X2 11/34/0.53/7.57273e-006 4/13/0.05/ 0.00000e+000
X3 3/10/0.13/0.00000e+000 13/40/0.09/7.48340e-006
X4 12/37/0.78/8.40744e-008 6/19/0.05/0.00000e+000
X5 7/22/0.64/1.30968e-007 10/31/1.05/3.78182e-006
10000 X1 2/7/0.06/0.00000e+000 14/43/0.14/2.89200e-006
X2 11/34/1.77/6.56131e-006 4/13/0.06/0.00000e+000
X3 3/10/0.36/0.00000e+000 14/43/0.16/2.88949e-006
X4 13/40/3.00/6.90437e-008 6/19/0.08/0.00000e+000
X5 7/22/2.23/1.30940e-007 10/31/3.20/5.11269e-006
12000 X1 2/7/0.06/0.00000e+000 14/43/0.17/3.16733e-006
X2 11/34/2.48/6.34881e-006 4/13/0.06/0.00000e+000
X3 3/10/0.47/0.00000e+000 14/43/0.16/3.16500e-006
X4 13/40/4.53/6.04693e-008 6/19/0.09/0.00000e+000
X5 7/22/3.13/1.30935e-007 10/31/4.67/5.34936e-006
Table Options

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Dim Initial points MSGP method Algorithm 2.1
1000 X1 7/22/0.06/1.31135e-007 13/40/0.05/3.38669e-006
X2 2/7/0.05/0.00000e+000 4/13/0.03/0.00000e+000
X3 11/34/0.08/2.19964e-007 3/40/0.05/3.36081e-006
X4 3/10/0.05/0.00000e+000 6/19/0.05/0.00000e+000
X5 11/34/0.09/8.93938e-008 8/25/0.06/6.13420e-006
3000 X1 2/7/0.05/0.00000e+000 13/40/0.06/5.81725e-006
X2 11/34/0.25/4.90071e-006 4/13/0.03/0.00000e+000
X3 3/10/0.06/0.00000e+000 13/40/0.08/5.80007e-006
X4 11/34/0.30/2.69613e-006 6/19/0.05/0.00000e+000
X5 7/22/0.31/1.31003e-007 10/31/0.42/2.65195e-006
5000 X1 2/7/0.05/0.00000e+000 13/40/0.08/7.49753e-006
X2 11/34/0.53/7.57273e-006 4/13/0.05/ 0.00000e+000
X3 3/10/0.13/0.00000e+000 13/40/0.09/7.48340e-006
X4 12/37/0.78/8.40744e-008 6/19/0.05/0.00000e+000
X5 7/22/0.64/1.30968e-007 10/31/1.05/3.78182e-006
10000 X1 2/7/0.06/0.00000e+000 14/43/0.14/2.89200e-006
X2 11/34/1.77/6.56131e-006 4/13/0.06/0.00000e+000
X3 3/10/0.36/0.00000e+000 14/43/0.16/2.88949e-006
X4 13/40/3.00/6.90437e-008 6/19/0.08/0.00000e+000
X5 7/22/2.23/1.30940e-007 10/31/3.20/5.11269e-006
12000 X1 2/7/0.06/0.00000e+000 14/43/0.17/3.16733e-006
X2 11/34/2.48/6.34881e-006 4/13/0.06/0.00000e+000
X3 3/10/0.47/0.00000e+000 14/43/0.16/3.16500e-006
X4 13/40/4.53/6.04693e-008 6/19/0.09/0.00000e+000
X5 7/22/3.13/1.30935e-007 10/31/4.67/5.34936e-006
Table 2.  The results of Problem 2 with given initial points
Dim Initial points MSGP method Algorithm 2.1
1000 X1 290/1164/1.28/9.81195e-006 33/161/0.05/9.43667e-006
X2 285/1153/1.19/8.96144e-006 56/294/0.06/9.59066e-006
X3 86/381/0.17/9.29618e-006 37/184/0.05/8.83590e-006
X4 61/275/0.30/9.56339e-006 62/330/0.06/9.97633e-006
X5 361/1457/1.55/8.71650e-006 47/246/0.23/8.93449e-006
3000 X1 61/281/0.98/9.44368e-006 44/219/0.09/8.60313e-006
X2 347/1390/9.69/9.33905e-006 51/289/0.11/9.89992e-006
X3 110/467/2.80/8.61958e-006 38/189/0.08/7.07861e-006
X4 65/295/0.89/9.73034e-006 47/275/0.11/6.69019e-006
X5 361/1457/10.36/8.71650e-006 47/246/1.31/8.93449e-006
5000 X1 73/324/3.75/9.96592e-006 57/291/0.19/9.69435e-006
X2 305/1224/22.45/9.99741e-006 61/326/0.19/9.23331e-006
X3 99/420/6.53/9.97198e-006 44/220/0.14/8.90390e-006
X4 64/285/4.08/8.54432e-006 54/292/0.19/9.52366e-006
X5 361/1457/26.92/8.71650e-006 47/246/3.28/8.93449e-006
10000 X1 63/280/13.44/8.48262e-006 43/212/0.25/6.39422e-006
X2 367/1471/101.08/9.69111e-006 48/249/0.52/9.16252e-006
X3 79/353/18.42/8.41347e-006 65/331/0.63/8.97609e-006
X4 120/498/16.27/8.51405e-006 64/357/0.41/9.82673e-006
X5 361/1457/101.25/8.71650e-006 47/246/12.08/8.93449e-006
12000 X1 67/301/15.72/9.73091e-006 43/211/0.30/6.68736e-006
X2 390/1562/154.20/8.01930e-006 66/467/0.52/8.76219e-006
X3 82/362/25.55/9.73575e-006 56/292/0.39/9.27446e-006
X4 134/548/49.52/9.58265e-006 57/311/0.44/8.42381e-006
X5 361/1457/144.38/8.71650e-006 47/246/17.19/8.93449e-006
Table Options

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Dim Initial points MSGP method Algorithm 2.1
1000 X1 290/1164/1.28/9.81195e-006 33/161/0.05/9.43667e-006
X2 285/1153/1.19/8.96144e-006 56/294/0.06/9.59066e-006
X3 86/381/0.17/9.29618e-006 37/184/0.05/8.83590e-006
X4 61/275/0.30/9.56339e-006 62/330/0.06/9.97633e-006
X5 361/1457/1.55/8.71650e-006 47/246/0.23/8.93449e-006
3000 X1 61/281/0.98/9.44368e-006 44/219/0.09/8.60313e-006
X2 347/1390/9.69/9.33905e-006 51/289/0.11/9.89992e-006
X3 110/467/2.80/8.61958e-006 38/189/0.08/7.07861e-006
X4 65/295/0.89/9.73034e-006 47/275/0.11/6.69019e-006
X5 361/1457/10.36/8.71650e-006 47/246/1.31/8.93449e-006
5000 X1 73/324/3.75/9.96592e-006 57/291/0.19/9.69435e-006
X2 305/1224/22.45/9.99741e-006 61/326/0.19/9.23331e-006
X3 99/420/6.53/9.97198e-006 44/220/0.14/8.90390e-006
X4 64/285/4.08/8.54432e-006 54/292/0.19/9.52366e-006
X5 361/1457/26.92/8.71650e-006 47/246/3.28/8.93449e-006
10000 X1 63/280/13.44/8.48262e-006 43/212/0.25/6.39422e-006
X2 367/1471/101.08/9.69111e-006 48/249/0.52/9.16252e-006
X3 79/353/18.42/8.41347e-006 65/331/0.63/8.97609e-006
X4 120/498/16.27/8.51405e-006 64/357/0.41/9.82673e-006
X5 361/1457/101.25/8.71650e-006 47/246/12.08/8.93449e-006
12000 X1 67/301/15.72/9.73091e-006 43/211/0.30/6.68736e-006
X2 390/1562/154.20/8.01930e-006 66/467/0.52/8.76219e-006
X3 82/362/25.55/9.73575e-006 56/292/0.39/9.27446e-006
X4 134/548/49.52/9.58265e-006 57/311/0.44/8.42381e-006
X5 361/1457/144.38/8.71650e-006 47/246/17.19/8.93449e-006
Table 3.  The results of Problem 3 with given initial points
Dim Initial points MSGP method Algorithm 2.1
1000 X1 54/226/0.09/9.68368e-006 43/264/0.06/9.77486e-006
X2 556/4532/1.11/9.59335e-006 36/230/0.05/7.01507e-006
X3 1004/9105/1.70/9.64279e-006 48/311/0.06/8.53004e-006
X4 871/7511/1.70/9.98355e-006 51/318/0.06/8.62714e-006
X5 58/264/0.30/9.42953e-006 39/257/0.16/8.20033e-006
3000 X1 53/213/0.16/7.47095e-006 53/335/0.14/7.17598e-006
X2 628/5226/4.92/9.53246e-006 36/226/0.09/6.14965e-006
X3 1237/12041/7.30/9.87899e-006 47/303/0.13/7.53500e-006
X4 1059/9647/13.73/9.81351e-006 54/386/0.16/9.61675e-006
X5 58/264/1.73/9.42953e-006 39/257/1.11/8.20033e-006
5000 X1 51/212/0.20/8.70652e-006 49/311/0.19/7.89292e-006
X2 635/5353/10.84/9.92649e-006 39/247/0.16/6.13645e-006
X3 1397/13868/16.84/9.98769e-006 77/531/0.30/7.77119e-006
X4 1093/10060/36.78/9.97031e-006 84/687/0.36/5.17010e-006
X5 58/264/4.27/9.42953e-006 39/257/2.75/8.20033e-006
10000 X1 61/243/0.45/8.97371e-006 56/365/0.41/8.54468e-006
X2 206/1273/16.94/9.54721e-006 42/275/0.30/5.75051e-006
X3 1739/18415/57.48/9.96405e-006 61/418/0.45/6.13008e-006
X4 1102/10030/135.05/9.77795e-006 4/125/0.69/0.00000e+000
X5 58/264/15.72/9.42953e-006 39/257/10.11/8.20033e-006
12000 X1 65/267/0.58/9.69254e-006 52/337/0.45/8.80176e-006
X2 655/5624/48.14/9.97926e-006 42/276/0.36/8.48727e-006
X3 1439/14609/47.84/9.95483e-006 58/396/0.52/9.34846e-006
X4 1037/9446/181.69/9.44940e-006 4/71/103.92/0.00000e+000
X5 58/264/22.38/9.42953e-006 39/257/14.28/8.20033e-006
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Dim Initial points MSGP method Algorithm 2.1
1000 X1 54/226/0.09/9.68368e-006 43/264/0.06/9.77486e-006
X2 556/4532/1.11/9.59335e-006 36/230/0.05/7.01507e-006
X3 1004/9105/1.70/9.64279e-006 48/311/0.06/8.53004e-006
X4 871/7511/1.70/9.98355e-006 51/318/0.06/8.62714e-006
X5 58/264/0.30/9.42953e-006 39/257/0.16/8.20033e-006
3000 X1 53/213/0.16/7.47095e-006 53/335/0.14/7.17598e-006
X2 628/5226/4.92/9.53246e-006 36/226/0.09/6.14965e-006
X3 1237/12041/7.30/9.87899e-006 47/303/0.13/7.53500e-006
X4 1059/9647/13.73/9.81351e-006 54/386/0.16/9.61675e-006
X5 58/264/1.73/9.42953e-006 39/257/1.11/8.20033e-006
5000 X1 51/212/0.20/8.70652e-006 49/311/0.19/7.89292e-006
X2 635/5353/10.84/9.92649e-006 39/247/0.16/6.13645e-006
X3 1397/13868/16.84/9.98769e-006 77/531/0.30/7.77119e-006
X4 1093/10060/36.78/9.97031e-006 84/687/0.36/5.17010e-006
X5 58/264/4.27/9.42953e-006 39/257/2.75/8.20033e-006
10000 X1 61/243/0.45/8.97371e-006 56/365/0.41/8.54468e-006
X2 206/1273/16.94/9.54721e-006 42/275/0.30/5.75051e-006
X3 1739/18415/57.48/9.96405e-006 61/418/0.45/6.13008e-006
X4 1102/10030/135.05/9.77795e-006 4/125/0.69/0.00000e+000
X5 58/264/15.72/9.42953e-006 39/257/10.11/8.20033e-006
12000 X1 65/267/0.58/9.69254e-006 52/337/0.45/8.80176e-006
X2 655/5624/48.14/9.97926e-006 42/276/0.36/8.48727e-006
X3 1439/14609/47.84/9.95483e-006 58/396/0.52/9.34846e-006
X4 1037/9446/181.69/9.44940e-006 4/71/103.92/0.00000e+000
X5 58/264/22.38/9.42953e-006 39/257/14.28/8.20033e-006
Table 4.  The results of Problem 4 with given initial points
Dim Initial points MSGP method Algorithm 2.1
1000 X1 194/1015/0.59/9.66433e-006 30/184/0.05/5.70062e-006
X2 117/619/0.20/9.09560e-006 58/375/0.08/8.92285e-006
X3 157/833/0.24/9.74765e-006 75/484/0.08/6.43351e-006
X4 159/838/0.25/9.55038e-006 70/453/0.06/6.92855e-006
X5 156/800/0.52/9.96648e-006 55/354/0.06/8.13127e-006
3000 X1 174/908/3.66/9.87344e-006 94/610/0.19/9.67519e-006
X2 164/839/0.89/9.10283e-006 43/272/0.09/4.67179e-006
X3 168/886/0.98/9.48568e-006 75/486/0.16/8.24368e-006
X4 213/1147/1.39/9.27139e-006 61/394/0.13/5.88525e-006
X5 183/990/3.84/9.99259e-006 62/396/0.13/8.46852e-006
5000 X1 174/915/6.94/9.56987e-006 42/267/0.13/9.61170e-006
X2 163/840/1.94/9.18817e-006 63/404/0.19/9.53629e-006
X3 186/1014/2.59/9.83818e-006 67/431/0.20/9.62859e-006
X4 211/1127/2.70/9.61085e-006 62/402/0.19/6.30592e-006
X5 170/881/9.19/9.70906e-006 41/261/0.14/7.03502e-006
10000 X1 181/969/30.69/9.49929e-006 29/178/0.17/9.26476e-006
X2 161/808/7.27/8.99994e-006 31/193/0.19/8.57493e-006
X3 182/983/7.89/9.62444e-006 78/504/0.94/8.99403e-006
X4 206/1069/8.33/9.78939e-006 40/250/0.23/9.71324e-006
X5 182/972/30.75/9.75983e-006 68/439/0.38/8.26113e-006
12000 X1 190/1007/59.33/9.43919e-006 52/331/0.34/8.48201e-006
X2 177/920/9.55/9.60849e-006 68/467/0.45/6.51248e-006
X3 180/960/11.08/9.56932e-006 85/549/0.55/9.43656e-006
X4 200/1059/11.67/9.87597e-006 28/171/0.19/8.94950e-006
X5 174/936/34.61/9.79661e-006 70/453/0.47/7.91811e-006
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Dim Initial points MSGP method Algorithm 2.1
1000 X1 194/1015/0.59/9.66433e-006 30/184/0.05/5.70062e-006
X2 117/619/0.20/9.09560e-006 58/375/0.08/8.92285e-006
X3 157/833/0.24/9.74765e-006 75/484/0.08/6.43351e-006
X4 159/838/0.25/9.55038e-006 70/453/0.06/6.92855e-006
X5 156/800/0.52/9.96648e-006 55/354/0.06/8.13127e-006
3000 X1 174/908/3.66/9.87344e-006 94/610/0.19/9.67519e-006
X2 164/839/0.89/9.10283e-006 43/272/0.09/4.67179e-006
X3 168/886/0.98/9.48568e-006 75/486/0.16/8.24368e-006
X4 213/1147/1.39/9.27139e-006 61/394/0.13/5.88525e-006
X5 183/990/3.84/9.99259e-006 62/396/0.13/8.46852e-006
5000 X1 174/915/6.94/9.56987e-006 42/267/0.13/9.61170e-006
X2 163/840/1.94/9.18817e-006 63/404/0.19/9.53629e-006
X3 186/1014/2.59/9.83818e-006 67/431/0.20/9.62859e-006
X4 211/1127/2.70/9.61085e-006 62/402/0.19/6.30592e-006
X5 170/881/9.19/9.70906e-006 41/261/0.14/7.03502e-006
10000 X1 181/969/30.69/9.49929e-006 29/178/0.17/9.26476e-006
X2 161/808/7.27/8.99994e-006 31/193/0.19/8.57493e-006
X3 182/983/7.89/9.62444e-006 78/504/0.94/8.99403e-006
X4 206/1069/8.33/9.78939e-006 40/250/0.23/9.71324e-006
X5 182/972/30.75/9.75983e-006 68/439/0.38/8.26113e-006
12000 X1 190/1007/59.33/9.43919e-006 52/331/0.34/8.48201e-006
X2 177/920/9.55/9.60849e-006 68/467/0.45/6.51248e-006
X3 180/960/11.08/9.56932e-006 85/549/0.55/9.43656e-006
X4 200/1059/11.67/9.87597e-006 28/171/0.19/8.94950e-006
X5 174/936/34.61/9.79661e-006 70/453/0.47/7.91811e-006
Table 5.  The results with initial points randomly generated from (0, 1)
Dim MSGP method Algorithm 2.1
Problem 1 1000 10/31/0.08/1.29174e-007 6/19/0.03/0.00000e+000
10/31/0.05/3.60715e-006 6/19/0.00/0.00000e+000
10/31/0.05/1.31904e-007 6/19/0.02/0.00000e+000
3000 11/34/0.30/4.53840e-006 6/19/0.03/0.00000e+000
11/34/0.27/6.94297e-006 6/19/0.02/0.00000e+000
11/34/0.25/9.30472e-006 6/19/0.02/0.00000e+000
5000 11/34/0.72/9.94443e-006 6/19/0.05/0.00000e+000
12/37/0.75/1.10770e-006 6/19/0.03/0.00000e+000
12/37/0.72/9.50443e-008 6/19/0.03/0.00000e+000
1000013/40/3.03/8.74543e-0086/19/0.08/0.00000e+000
12/37/2.67/6.56124e-006 6/19/0.06/0.00000e+000
12/37/2.64/4.66009e-006 6/19/0.05/0.00000e+000
12000 13/40/4.13/1.19951e-007 6/19/0.09/0.00000e+000
14/43/4.67/1.13842e-007 6/19/0.06/0.00000e+000
13/40/4.13/1.98039e-007 6/19/0.06/0.00000e+000
Problem 2 1000 105/506/0.33/9.97726e-006 84/487/0.09/7.86183e-006
104/482/0.28/9.49717e-006 81/465/0.06/8.31377e-006
>113/554/0.31/9.22890e-006 83/449/0.06/9.84927e-006
3000 126/650/1.88/9.29974e-006 91/515/0.22/9.63881e-006
139/677/2.05/8.98283e-006 94/521/0.19/8.70734e-006
133/639/1.94/9.51118e-006 98/532/0.22/7.45756e-006
5000 143/727/5.03/8.99648e-006 97/565/0.39/9.72677e-006
134/692/5.02/7.94479e-006 100/567/0.38/8.99827e-006
141/695/5.00/8.93255e-006 102/650/0.41/9.83925e-006
10000 154/834/20.17/9.55153e-006 93/545/0.70/7.86166e-006
152/796/20.00/8.59073e-006 122/745/0.92/8.76303e-006
154/794/20.20/9.43874e-006 127/863/0.98/9.20702e-006
12000 162/855/30.14/9.10526e-006 122/748/1.17/9.67756e-006
163/854/30.45/9.97329e-006 114/805/1.16/8.33603e-006
158/828/29.94/9.08543e-006 112/751/1.08/9.98871e-006
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Dim MSGP method Algorithm 2.1
Problem 1 1000 10/31/0.08/1.29174e-007 6/19/0.03/0.00000e+000
10/31/0.05/3.60715e-006 6/19/0.00/0.00000e+000
10/31/0.05/1.31904e-007 6/19/0.02/0.00000e+000
3000 11/34/0.30/4.53840e-006 6/19/0.03/0.00000e+000
11/34/0.27/6.94297e-006 6/19/0.02/0.00000e+000
11/34/0.25/9.30472e-006 6/19/0.02/0.00000e+000
5000 11/34/0.72/9.94443e-006 6/19/0.05/0.00000e+000
12/37/0.75/1.10770e-006 6/19/0.03/0.00000e+000
12/37/0.72/9.50443e-008 6/19/0.03/0.00000e+000
1000013/40/3.03/8.74543e-0086/19/0.08/0.00000e+000
12/37/2.67/6.56124e-006 6/19/0.06/0.00000e+000
12/37/2.64/4.66009e-006 6/19/0.05/0.00000e+000
12000 13/40/4.13/1.19951e-007 6/19/0.09/0.00000e+000
14/43/4.67/1.13842e-007 6/19/0.06/0.00000e+000
13/40/4.13/1.98039e-007 6/19/0.06/0.00000e+000
Problem 2 1000 105/506/0.33/9.97726e-006 84/487/0.09/7.86183e-006
104/482/0.28/9.49717e-006 81/465/0.06/8.31377e-006
>113/554/0.31/9.22890e-006 83/449/0.06/9.84927e-006
3000 126/650/1.88/9.29974e-006 91/515/0.22/9.63881e-006
139/677/2.05/8.98283e-006 94/521/0.19/8.70734e-006
133/639/1.94/9.51118e-006 98/532/0.22/7.45756e-006
5000 143/727/5.03/8.99648e-006 97/565/0.39/9.72677e-006
134/692/5.02/7.94479e-006 100/567/0.38/8.99827e-006
141/695/5.00/8.93255e-006 102/650/0.41/9.83925e-006
10000 154/834/20.17/9.55153e-006 93/545/0.70/7.86166e-006
152/796/20.00/8.59073e-006 122/745/0.92/8.76303e-006
154/794/20.20/9.43874e-006 127/863/0.98/9.20702e-006
12000 162/855/30.14/9.10526e-006 122/748/1.17/9.67756e-006
163/854/30.45/9.97329e-006 114/805/1.16/8.33603e-006
158/828/29.94/9.08543e-006 112/751/1.08/9.98871e-006
Table 6.  The results with initial points randomly generated from (0, 1)
Dim MSGP method Algorithm 2.1
Problem 3 1000 248/1765/0.52/8.14403e-006 66/443/0.09/5.73224e-006
233/1659/0.44/8.60251e-006 73/505/0.08/8.37567e-006
265/1839/0.53/8.07376e-006 61/410/0.06/6.30228e-006
3000 291/2170/2.63/6.57364e-006 77/535/0.23/9.31672e-006
284/2163/2.41/9.63384e-006 88/612/0.23/9.82561e-006
287/2196/2.55/9.97633e-006 85/583/0.23/7.58095e-006
5000 293/2317/6.02/9.41072e-006 106/758/0.58/8.60162e-006
300/2320/6.19/9.59322e-006 89/635/0.44/8.10417e-006
286/2259/5.86/7.50859e-006 108/784/0.56/6.29653e-006
10000 262/2147/17.13/9.37057e-006 181/1544/2.89/7.61750e-006
296/2454/18.55/8.82342e-006 94/645/1.03/9.49197e-006
277/2187/18.78/8.03849e-006 65/435/0.72/5.90105e-006
12000 378/3238/32.92/6.12712e-006 82/601/1.02/9.99908e-006
301/2446/28.58/9.40122e-006 139/1096/2.63/6.13477e-006
300/2497/26.70/9.76692e-006 148/1137/2.44/8.01365e-006
Problem 4 1000 255/1646/0.55/9.88522e-006 148/969/0.14/8.52252e-006
315/2190/0.64/8.58459e-006 167/1132/0.14/7.14294e-006
263/1779/0.51/9.78341e-006 152/998/0.11/6.83836e-006
3000 266/1864/2.59/9.68351e-006 204/1466/0.52/8.15543e-006
269/1792/2.56/9.50426e-006 176/1151/0.33/7.26404e-006
303/2094/2.95/7.70559e-006 180/1257/0.42/7.24683e-006
5000 322/2299/7.41/6.83957e-006 208/1469/0.91/5.80054e-006
265/1790/5.94/9.84065e-006 181/1202/0.64/9.43111e-006
260/2023/5.14/9.80395e-006 195/1315/0.77/6.31066e-006
10000 271/2024/20.59/8.91775e-006 217/1519/2.16/6.92039e-006
283/1997/21.91/9.98757e-006 216/1535/2.09/9.27616e-006
272/2002/19.44/8.12148e-006 198/1299/1.64/7.45635e-006
12000 218/1786/13.67/5.52117e-006 235/1728/3.33/7.46955e-006
309/2230/32.14/9.85930e-006 231/1779/3.78/8.69710e-006
248/1926/24.22/8.80377e-006 209/1463/2.95/8.14360e-006
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Dim MSGP method Algorithm 2.1
Problem 3 1000 248/1765/0.52/8.14403e-006 66/443/0.09/5.73224e-006
233/1659/0.44/8.60251e-006 73/505/0.08/8.37567e-006
265/1839/0.53/8.07376e-006 61/410/0.06/6.30228e-006
3000 291/2170/2.63/6.57364e-006 77/535/0.23/9.31672e-006
284/2163/2.41/9.63384e-006 88/612/0.23/9.82561e-006
287/2196/2.55/9.97633e-006 85/583/0.23/7.58095e-006
5000 293/2317/6.02/9.41072e-006 106/758/0.58/8.60162e-006
300/2320/6.19/9.59322e-006 89/635/0.44/8.10417e-006
286/2259/5.86/7.50859e-006 108/784/0.56/6.29653e-006
10000 262/2147/17.13/9.37057e-006 181/1544/2.89/7.61750e-006
296/2454/18.55/8.82342e-006 94/645/1.03/9.49197e-006
277/2187/18.78/8.03849e-006 65/435/0.72/5.90105e-006
12000 378/3238/32.92/6.12712e-006 82/601/1.02/9.99908e-006
301/2446/28.58/9.40122e-006 139/1096/2.63/6.13477e-006
300/2497/26.70/9.76692e-006 148/1137/2.44/8.01365e-006
Problem 4 1000 255/1646/0.55/9.88522e-006 148/969/0.14/8.52252e-006
315/2190/0.64/8.58459e-006 167/1132/0.14/7.14294e-006
263/1779/0.51/9.78341e-006 152/998/0.11/6.83836e-006
3000 266/1864/2.59/9.68351e-006 204/1466/0.52/8.15543e-006
269/1792/2.56/9.50426e-006 176/1151/0.33/7.26404e-006
303/2094/2.95/7.70559e-006 180/1257/0.42/7.24683e-006
5000 322/2299/7.41/6.83957e-006 208/1469/0.91/5.80054e-006
265/1790/5.94/9.84065e-006 181/1202/0.64/9.43111e-006
260/2023/5.14/9.80395e-006 195/1315/0.77/6.31066e-006
10000 271/2024/20.59/8.91775e-006 217/1519/2.16/6.92039e-006
283/1997/21.91/9.98757e-006 216/1535/2.09/9.27616e-006
272/2002/19.44/8.12148e-006 198/1299/1.64/7.45635e-006
12000 218/1786/13.67/5.52117e-006 235/1728/3.33/7.46955e-006
309/2230/32.14/9.85930e-006 231/1779/3.78/8.69710e-006
248/1926/24.22/8.80377e-006 209/1463/2.95/8.14360e-006
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