# American Institute of Mathematical Sciences

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

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

 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.

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
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
 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
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
 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
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
 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
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
 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
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 10000 13/40/3.03/8.74543e-008 6/19/0.08/0.00000e+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
 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 10000 13/40/3.03/8.74543e-008 6/19/0.08/0.00000e+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
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
 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

2018 Impact Factor: 1.025