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A linear-quadratic control problem of uncertain discrete-time switched systems
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
J. E. Dennis and J. J. Moré,
Quasi-Newton method, motivation and theory, SIAM Review, 19 (1997), 46-89.
doi: 10.1137/1019005. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
L. Qi and J. Sun,
A nonsmooth version of Newton's method, Mathematical Programming, 58 (1999), 353-367.
doi: 10.1007/BF01581275. |
[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. |
[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. |
[18] |
A. J. Wood and B. F. Wollenberg,
Power Generations Operations and Control, Wiley, New York, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
J. E. Dennis and J. J. Moré,
Quasi-Newton method, motivation and theory, SIAM Review, 19 (1997), 46-89.
doi: 10.1137/1019005. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
L. Qi and J. Sun,
A nonsmooth version of Newton's method, Mathematical Programming, 58 (1999), 353-367.
doi: 10.1007/BF01581275. |
[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. |
[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. |
[18] |
A. J. Wood and B. F. Wollenberg,
Power Generations Operations and Control, Wiley, New York, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
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 |
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 |
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 |
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 |
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 |
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 |
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