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An optimal algorithm for the obstacle neutralization problem
A memory gradient method based on the nonmonotone technique
Department of Applied Mathematics, Hainan University, Haikou 570228, China |
In this paper, we present a new nonmonotone memory gradient algorithm for unconstrained optimization problems. An attractive property of the proposed method is that the search direction always provides sufficient descent step at each iteration. This property is independent of the line search used. Under mild assumptions, the global and local convergence results of the proposed algorithm are established respectively. Numerical results are also reported to show that the proposed method is suitable to solve large-scale optimization problems and is more stable than other similar methods in practical computation.
References:
[1] |
N. Andrei,
An unconstrained optimization test functions, Advanced Modelling and Optimization, 10 (2008), 147-161.
|
[2] |
J. Barzilai and J. M. Borwein,
Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.
doi: 10.1093/imanum/8.1.141. |
[3] |
W. Y. Cheng and Q. F. Liu,
Sufficient descent nonlinear conjugate gradient methods with conjugacy condition, Numerical Algorithms, 53 (2010), 113-131.
doi: 10.1007/s11075-009-9318-8. |
[4] |
Y. H. Dai,
On the nonmonotone line search, Journal of Optimization Theory and Applications, 112 (2002), 315-330.
doi: 10.1023/A:1013653923062. |
[5] |
Y. H. Dai and Y. X. Yuan,
A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10 (1999), 177-182.
doi: 10.1137/S1052623497318992. |
[6] |
Y. H. Dai and Y. X. Yuan, Nonlinear Conjugate Gradient Methods (in chinese), Shanghai Scientific and Technical Publishers, Shanghai, 2000. Google Scholar |
[7] |
N. Y. Deng, Y. Xiao and F. J. Zhou,
Nonmonotone trust region algorithm, Journal of Optimization Theory and Applications, 76 (1993), 259-285.
doi: 10.1007/BF00939608. |
[8] |
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. |
[9] |
J. C. Gilbert and J. Nocedal,
Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42.
doi: 10.1137/0802003. |
[10] |
N. I. M. Gould, D. Orban and Ph. L. Toint,
CUTEr: a constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 353-372.
doi: 10.1145/962437.962438. |
[11] |
L. Grippo, F. Lampariello and S. Lucidi,
A nonmonotone line search technique for Newton's method, SIAM Journal on Numerical Analysis, 23 (1986), 707-716.
doi: 10.1137/0723046. |
[12] |
N. Z. Gu and J. T. Mo,
Incorporating nonmonotone strategies into the trust region method for unconstrained optimization, Computers and Mathematics with Applications, 55 (2008), 2158-2172.
doi: 10.1016/j.camwa.2007.08.038. |
[13] |
W. W. Hager and H. C. Zhang,
A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM Journal on Optimization, 16 (2005), 170-192.
doi: 10.1137/030601880. |
[14] |
W. W. Hager and H. C. Zhang,
A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.
|
[15] |
H. Iiduka and Y. Narushima,
Conjugate gradient methods using value of objective function for unconstrained optimization, Optimization Letters, 6 (2012), 941-955.
doi: 10.1007/s11590-011-0324-0. |
[16] |
Y. Ji, Y. J. Li, K. C. Zhang and X. L. Zhang,
A new nonmonotone trust region method for conic model for solving unconstrained optimization, Journal of Computational and Applied Mathematics, 233 (2010), 1746-1754.
doi: 10.1016/j.cam.2009.09.011. |
[17] |
J. T. Mo, K. C. Zhang and Z. X. Wei,
A nonmonotone trust region method for unconstrained optimization, Applied Mathematics and Computation, 171 (2005), 371-384.
doi: 10.1016/j.amc.2005.01.048. |
[18] |
Y. Narushima and H. Yabe,
Global convergence of a memory gradient method for unconstrained optimization, Computational Optimization and Applications, 35 (2006), 325-346.
doi: 10.1007/s10589-006-8719-z. |
[19] |
Y. Narushima, H. Yabe and J. A. Ford,
A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM Journal on Optimization, 21 (2011), 212-230.
doi: 10.1137/080743573. |
[20] |
J. Nocedal and S. J. Wright,
Numerical Optimization, Springer, New York, 1999.
doi: 10.1007/b98874. |
[21] |
Y. G. Ou and G. S. Wang,
A new supermemory gradient method for unconstrained optimization problems, Optimization Letters, 6 (2012), 975-992.
doi: 10.1007/s11590-011-0328-9. |
[22] |
Y. G. Ou and Q. Zhou,
A nonmonotonic trust region algorithm for a class of semi-infinite minimax programming, Applied Mathematics and Computation, 215 (2009), 474-480.
doi: 10.1016/j.amc.2009.05.009. |
[23] |
M. Raydan,
The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7 (1977), 26-33.
doi: 10.1137/S1052623494266365. |
[24] |
Z. J. Shi and J. Shen,
A new supermemory gradient method with curve search rule, Applied Mathematics and Computation, 170 (2005), 1-16.
doi: 10.1016/j.amc.2004.10.063. |
[25] |
Z. J. Shi and J. Shen,
On memory gradient method with trust region for unconstrained optimization, Numerical Algorithms, 41 (2006), 173-196.
doi: 10.1007/s11075-005-9008-0. |
[26] |
Z. J. Shi, S. Q. Wang and Z. W. Xu,
The convergence of conjugate gradient method with nonmonotone line search, Applied Mathematics and Computation, 217 (2010), 1921-1932.
doi: 10.1016/j.amc.2010.06.047. |
[27] |
M. Sun and Q. G. Bai,
A new descent memory gradient method and its global convergence, Journal of System Science and Complexity, 24 (2011), 784-794.
doi: 10.1007/s11424-011-8150-0. |
[28] |
W. Y. Sun,
Nonmonotone trust region method for solving optimization problems, Applied Mathematics and Computation, 156 (2004), 159-174.
doi: 10.1016/j.amc.2003.07.008. |
[29] |
W. Y. Sun and Y. X. Yuan,
Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. |
[30] |
W. Y. Sun and Q. Y. Zhou,
An unconstrained optimization method using nonmonotone second order Goldstein's linesearch, Science in China (Series A): Mathematics, 50 (2007), 1389-1400.
doi: 10.1007/s11425-007-0072-x. |
[31] |
Ph. L. Toint,
An assessment of nonmonotone line search techniques for unconstrained optimization, SIAM Journal on Scientific and Statistical Computing, 17 (1996), 725-739.
doi: 10.1137/S106482759427021X. |
[32] |
Z. S. Yu, W. G. Zhang and B. F. Wu,
Strong global convergence of an adaptive nonmonotone memory gradient method, Applied Mathematics and Computation, 185 (2007), 681-688.
doi: 10.1016/j.amc.2006.07.075. |
[33] |
H. C. Zhang and W. W. Hager,
A nonmonotone line search technique and its application to unconstrained optimization, SIAM Journal on Optimization, 14 (2004), 1043-1056.
doi: 10.1137/S1052623403428208. |
[34] |
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. |
[35] |
Y. Zheng and Z. P. Wan,
A new variant of the memory gradient method for unconstrained optimization, Optimization Letters, 6 (2012), 1643-1655.
doi: 10.1007/s11590-011-0355-6. |
[36] |
W. J. Zhou and L. Zhang,
Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization, Journal of Computational and Applied Mathematics, 223 (2009), 40-47.
doi: 10.1016/j.cam.2007.12.011. |
show all references
References:
[1] |
N. Andrei,
An unconstrained optimization test functions, Advanced Modelling and Optimization, 10 (2008), 147-161.
|
[2] |
J. Barzilai and J. M. Borwein,
Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.
doi: 10.1093/imanum/8.1.141. |
[3] |
W. Y. Cheng and Q. F. Liu,
Sufficient descent nonlinear conjugate gradient methods with conjugacy condition, Numerical Algorithms, 53 (2010), 113-131.
doi: 10.1007/s11075-009-9318-8. |
[4] |
Y. H. Dai,
On the nonmonotone line search, Journal of Optimization Theory and Applications, 112 (2002), 315-330.
doi: 10.1023/A:1013653923062. |
[5] |
Y. H. Dai and Y. X. Yuan,
A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10 (1999), 177-182.
doi: 10.1137/S1052623497318992. |
[6] |
Y. H. Dai and Y. X. Yuan, Nonlinear Conjugate Gradient Methods (in chinese), Shanghai Scientific and Technical Publishers, Shanghai, 2000. Google Scholar |
[7] |
N. Y. Deng, Y. Xiao and F. J. Zhou,
Nonmonotone trust region algorithm, Journal of Optimization Theory and Applications, 76 (1993), 259-285.
doi: 10.1007/BF00939608. |
[8] |
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. |
[9] |
J. C. Gilbert and J. Nocedal,
Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42.
doi: 10.1137/0802003. |
[10] |
N. I. M. Gould, D. Orban and Ph. L. Toint,
CUTEr: a constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 353-372.
doi: 10.1145/962437.962438. |
[11] |
L. Grippo, F. Lampariello and S. Lucidi,
A nonmonotone line search technique for Newton's method, SIAM Journal on Numerical Analysis, 23 (1986), 707-716.
doi: 10.1137/0723046. |
[12] |
N. Z. Gu and J. T. Mo,
Incorporating nonmonotone strategies into the trust region method for unconstrained optimization, Computers and Mathematics with Applications, 55 (2008), 2158-2172.
doi: 10.1016/j.camwa.2007.08.038. |
[13] |
W. W. Hager and H. C. Zhang,
A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM Journal on Optimization, 16 (2005), 170-192.
doi: 10.1137/030601880. |
[14] |
W. W. Hager and H. C. Zhang,
A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.
|
[15] |
H. Iiduka and Y. Narushima,
Conjugate gradient methods using value of objective function for unconstrained optimization, Optimization Letters, 6 (2012), 941-955.
doi: 10.1007/s11590-011-0324-0. |
[16] |
Y. Ji, Y. J. Li, K. C. Zhang and X. L. Zhang,
A new nonmonotone trust region method for conic model for solving unconstrained optimization, Journal of Computational and Applied Mathematics, 233 (2010), 1746-1754.
doi: 10.1016/j.cam.2009.09.011. |
[17] |
J. T. Mo, K. C. Zhang and Z. X. Wei,
A nonmonotone trust region method for unconstrained optimization, Applied Mathematics and Computation, 171 (2005), 371-384.
doi: 10.1016/j.amc.2005.01.048. |
[18] |
Y. Narushima and H. Yabe,
Global convergence of a memory gradient method for unconstrained optimization, Computational Optimization and Applications, 35 (2006), 325-346.
doi: 10.1007/s10589-006-8719-z. |
[19] |
Y. Narushima, H. Yabe and J. A. Ford,
A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM Journal on Optimization, 21 (2011), 212-230.
doi: 10.1137/080743573. |
[20] |
J. Nocedal and S. J. Wright,
Numerical Optimization, Springer, New York, 1999.
doi: 10.1007/b98874. |
[21] |
Y. G. Ou and G. S. Wang,
A new supermemory gradient method for unconstrained optimization problems, Optimization Letters, 6 (2012), 975-992.
doi: 10.1007/s11590-011-0328-9. |
[22] |
Y. G. Ou and Q. Zhou,
A nonmonotonic trust region algorithm for a class of semi-infinite minimax programming, Applied Mathematics and Computation, 215 (2009), 474-480.
doi: 10.1016/j.amc.2009.05.009. |
[23] |
M. Raydan,
The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7 (1977), 26-33.
doi: 10.1137/S1052623494266365. |
[24] |
Z. J. Shi and J. Shen,
A new supermemory gradient method with curve search rule, Applied Mathematics and Computation, 170 (2005), 1-16.
doi: 10.1016/j.amc.2004.10.063. |
[25] |
Z. J. Shi and J. Shen,
On memory gradient method with trust region for unconstrained optimization, Numerical Algorithms, 41 (2006), 173-196.
doi: 10.1007/s11075-005-9008-0. |
[26] |
Z. J. Shi, S. Q. Wang and Z. W. Xu,
The convergence of conjugate gradient method with nonmonotone line search, Applied Mathematics and Computation, 217 (2010), 1921-1932.
doi: 10.1016/j.amc.2010.06.047. |
[27] |
M. Sun and Q. G. Bai,
A new descent memory gradient method and its global convergence, Journal of System Science and Complexity, 24 (2011), 784-794.
doi: 10.1007/s11424-011-8150-0. |
[28] |
W. Y. Sun,
Nonmonotone trust region method for solving optimization problems, Applied Mathematics and Computation, 156 (2004), 159-174.
doi: 10.1016/j.amc.2003.07.008. |
[29] |
W. Y. Sun and Y. X. Yuan,
Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. |
[30] |
W. Y. Sun and Q. Y. Zhou,
An unconstrained optimization method using nonmonotone second order Goldstein's linesearch, Science in China (Series A): Mathematics, 50 (2007), 1389-1400.
doi: 10.1007/s11425-007-0072-x. |
[31] |
Ph. L. Toint,
An assessment of nonmonotone line search techniques for unconstrained optimization, SIAM Journal on Scientific and Statistical Computing, 17 (1996), 725-739.
doi: 10.1137/S106482759427021X. |
[32] |
Z. S. Yu, W. G. Zhang and B. F. Wu,
Strong global convergence of an adaptive nonmonotone memory gradient method, Applied Mathematics and Computation, 185 (2007), 681-688.
doi: 10.1016/j.amc.2006.07.075. |
[33] |
H. C. Zhang and W. W. Hager,
A nonmonotone line search technique and its application to unconstrained optimization, SIAM Journal on Optimization, 14 (2004), 1043-1056.
doi: 10.1137/S1052623403428208. |
[34] |
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. |
[35] |
Y. Zheng and Z. P. Wan,
A new variant of the memory gradient method for unconstrained optimization, Optimization Letters, 6 (2012), 1643-1655.
doi: 10.1007/s11590-011-0355-6. |
[36] |
W. J. Zhou and L. Zhang,
Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization, Journal of Computational and Applied Mathematics, 223 (2009), 40-47.
doi: 10.1016/j.cam.2007.12.011. |

Fun. | n | FR | PRP+ | DY | HZ | Algor.2.1 |
cpu(s) | cpu(s) | cpu(s) | cpu(s) | cpu(s) | ||
Ex.F.Roth | 104 | 4.888547 | 0.550768 | 2.957423 | 0.521059 | 1.823025 |
105 | 35.695368 | 6.393457 | 25.041071 | 5.640529 | 18.592210 | |
106 | 287.654246 | 48.039330 | 228.797928 | 46.116688 | 220.881878 | |
Ex.Trig. | 104 | 6.477725 | 0.879721 | 4.214142 | 1.521069 | 2.249725 |
105 | F | 12.156054 | 116.274122 | 74.643352 | 13.436983 | |
106 | F | 137.648005 | 209.220891 | 159.74406 | 140.003615 | |
Ex.Rosen. | 104 | 0.937497 | 0.334996 | 1.779752 | 0.269760 | 1.198996 |
105 | 3.666262 | 2.938029 | 19.050805 | 2.371920 | 12.510256 | |
106 | 76.992755 | 28.496619 | 154.785583 | 19.094775 | 100.920667 | |
Ex.W.Holst | 104 | 0.927501 | 0.902176 | 0.953671 | 0.641670 | 0.667595 |
105 | 11.200851 | 6.359390 | 5.016512 | 5.216413 | 5.085647 | |
106 | 108.520029 | 63.355358 | 53.428723 | 55.247560 | 52.571641 | |
Ex.Beale | 104 | 0.324522 | 0.275613 | 0.418340 | 0.157054 | 0.228633 |
105 | 2.474278 | 1.870461 | 2.407946 | 1.499047 | 2.405817 | |
106 | 23.457059 | 19.579839 | 26.324518 | 15.640053 | 24.912454 | |
Ex.Penalty | 104 | 0.275020 | 0.201060 | 0.208781 | 0.176900 | 0.203811 |
105 | 1.234828 | 2.239090 | 1.827919 | 1.724014 | 1.821343 | |
106 | 9.546747 | 17.71788 | 27.437794 | 14.134732 | 16.033043 | |
Per.Quad. | 104 | 3.02902 | 3.077180 | 3.040349 | 3.358025 | 46.029533 |
105 | 209.611665 | 103.424010 | 114.210432 | 108.280599 | F | |
Raydan 1 | 104 | 5.081648 | 3.277429 | 4.728205 | 3.713061 | 11.254018 |
105 | 97.715159 | 46.250008 | 127.696125 | 65.739349 | 141.324543 | |
Raydan 2 | 104 | 0.109849 | 0.087747 | 0.228718 | 0.057772 | 0.057323 |
105 | 0.833817 | 0.494746 | 1.978097 | 0.504858 | 0.533447 | |
106 | 8.214171 | 4.698048 | 16.906394 | 4.956234 | 5.812667 | |
Diag.1 | 104 | 5.789406 | 3.827681 | 6.911255 | 3.708083 | 3.538550 |
105 | F | 76.219241 | F | 70.737746 | 70.438309 | |
Diag.2 | 104 | 5.678352 | F | 5.285281 | F | 5.166993 |
105 | 154.534720 | F | 152.123264 | F | 145.797863 | |
Diag.3 | 104 | 6.181573 | 4.845072 | 6.228637 | 6.442592 | 5.311859 |
105 | 57.195661 | 52.810276 | 55.066374 | 48.976185 | 52.161715 | |
Hager | 104 | 1.380050 | 0.721078 | 1.358417 | 0.732179 | 1.320932 |
105 | 22.745343 | 11.942894 | 25.253252 | 11.895805 | 22.043234 | |
106 | F | 122.218850 | F | 156.087713 | 187.476220 | |
Gen.Trid.1 | 104 | 2.445174 | 2.512342 | 2.528296 | 2.258533 | 1.755517 |
105 | 25.293375 | 21.644389 | 25.202319 | 24.239381 | 20.201541 | |
106 | 234.095747 | 254.975842 | 204.126831 | 197.991582 | 145.845123 | |
Ex.Trid.1 | 104 | 6.224192 | 17.140057 | 1.533758 | 1.532657 | 13.182526 |
105 | 143.494302 | 244.260502 | 16.205910 | 14.276184 | 133.624706 | |
Ex.Three.Exp. | 104 | 0.199302 | 0.225054 | 0.198419 | 0.113357 | 0.142117 |
105 | 2.162488 | 2.405353 | 1.905207 | 1.265936 | 1.468474 | |
106 | 18.841829 | 21.440576 | 15.564795 | 10.447416 | 14.452928 | |
Gen.Trid.2 | 104 | 13.541982 | 3.192677 | 4.418622 | 2.970148 | 2.529354 |
105 | F | 30.674646 | F | 24.129818 | 23.986342 | |
Diag.4 | 104 | 0.094771 | 0.057988 | 0.114438 | 0.062960 | 0.093890 |
105 | 1.202458 | 0.388207 | 1.013161 | 0.202075 | 0.890367 | |
106 | 19.755904 | 3.749717 | 8.999546 | 1.906101 | 8.861827 | |
Diag.5 | 104 | 0.217310 | 0.089796 | 0.238359 | 0.141571 | 0.083229 |
105 | 1.373886 | 0.744259 | 2.028916 | 1.963680 | 0.693929 | |
106 | 13.342298 | 8.012713 | 18.812321 | 18.810192 | 6.833566 | |
Ex.Himm. | 104 | 0.611196 | 0.536196 | 0.141640 | 0.087034 | 0.132999 |
105 | 5.446782 | 5.385943 | 1.156565 | 0.724150 | 1.104406 | |
106 | 53.854410 | 53.065089 | 10.564224 | 7.176464 | 11.838695 | |
Gen.PSC1 | 104 | 2.934187 | 1.250311 | 11.817101 | 2.056868 | 28.300200 |
105 | 7.925998 | 15.507716 | 57.672689 | 14.110938 | 98.215964 | |
Ex.PSC1 | 104 | 0.114309 | 0.116711 | 0.137112 | 0.121277 | 0.120501 |
105 | 1.172552 | 0.942523 | 1.131255 | 1.341338 | 1.095562 | |
106 | 13.738276 | 10.067906 | 9.391618 | 12.643046 | 10.754309 | |
Ex.B.Diag.BD1 | 104 | 0.481994 | 0.093661 | 7.119658 | 0.153198 | 0.184555 |
105 | 4.780717 | 0.828624 | 63.779966 | 1.219591 | 1.886973 | |
106 | 45.244221 | 7.678318 | F | 13.094154 | 17.652279 | |
Ex.Maratos | 104 | 7.298980 | 0.215135 | 7.417174 | 0.400084 | 1.720770 |
105 | 72.871313 | 2.399795 | 69.118730 | 2.917748 | 15.965102 | |
106 | F | 24.968870 | F | 37.905524 | 188.771631 | |
Qua.Diag.Pert. | 104 | 20.021639 | 20.021639 | 23.190386 | 6.862350 | 20.0167895 |
Ex.Wood | 104 | 3.626209 | 0.449196 | 2.218795 | 0.481428 | 1.548869 |
105 | 30.297717 | 5.710406 | 25.452955 | 5.599862 | 24.178781 | |
106 | F | 83.358482 | F | 50.926588 | 80.176211 | |
Ex.Hiebert | 104 | 12.325250 | 0.423625 | 4.209053 | 0.505931 | 4.111580 |
105 | 61.420777 | 7.301559 | 44.310328 | 4.483430 | 43.492047 | |
106 | F | 98.625677 | F | 40.245693 | F | |
Ex.Qua.Pena.QP1 | 104 | 0.209738 | 0.131130 | 0.120703 | 0.213959 | 0.120383 |
105 | 0.931057 | 2.044439 | 0.928968 | 1.702844 | 1.680728 | |
106 | 19.710059 | 13.914293 | 5.142260 | 6.478819 | 13.674789 | |
Ex.Qua.Pena.QP2 | 104 | F | 0.368244 | 7.436318 | 0.329816 | 1.646625 |
105 | F | 3.211811 | 214.833356 | 3.229403 | 14.055916 | |
106 | F | 37.820369 | F | 31.182550 | 183.466287 | |
Qua.QF2 | 104 | 6.548727 | 6.315666 | 10.817725 | 6.994623 | 6.088805 |
FLETCBV3 | 1000 | 3.391656 | 3.208908 | 3.322490 | 9.722141 | 10.174606 |
FLETCHCR | 5000 | 17.258755 | 14.429972 | 16.787908 | 14.802347 | 10.449042 |
BDQRTIC | 104 | 83.808804 | F | 11.207452 | 9.377903 | 75.611094 |
TRIDIA | 104 | 20.207208 | 7.295958 | 13.794344 | 31.899388 | 85.858224 |
ARGLINB | 104 | 0.110820 | 0.107058 | 0.077671 | 0.068173 | 0.104807 |
105 | 0.722081 | 5.972367 | 0.946074 | 0.622368 | 0.790848 | |
106 | 7.334012 | 8.595424 | 6.361517 | 13.706507 | 5.689481 | |
ARWHEAD | 104 | 0.370001 | 0.077944 | 0.132404 | 0.090989 | 0.101133 |
105 | 1.579815 | 0.563831 | 1.544622 | 1.490355 | 1.243687 | |
106 | 5.837702 | 5.826076 | 4.586703 | 6.565910 | 5.671701 | |
NONBIA | 104 | 0.537552 | 0.102890 | 1.058223 | 0.103249 | 0.8355213 |
105 | 2.747465 | 1.337234 | 2.297892 | 2.992103 | 3.579316 | |
106 | 5.671371 | 22.742947 | 3.420499 | 6.936500 | 29.025324 | |
NONDQUAR | 104 | 0.442060 | 0.190346 | 0.158307 | 0.400302 | 0.649100 |
105 | 0.796334 | 2.503826 | 0.704471 | 1.828935 | 7.990690 | |
106 | 5.778999 | 20.906151 | 4.532799 | 17.272681 | 67.882148 | |
EG2 | 104 | 0.994080 | 0.313402 | 1.554111 | 1.337630 | 2.278667 |
105 | 7.160837 | 35.861218 | 3.615775 | 7.705992 | 10.859091 | |
106 | 43.235295 | 14.166499 | 11.586350 | 10.875278 | 50.416843 | |
DIXMAANA | 3*104 | 1.271498 | 0.762316 | 1.282368 | 0.736608 | 0.756586 |
3*105 | 11.612663 | 10.441716 | 11.562110 | 8.106027 | 7.118575 | |
3*106 | 114.760219 | 102.835981 | 125.107882 | 79.990697 | 75.923830 | |
DIXMAANB | 3*104 | 0.517920 | 0.385757 | 0.542675 | 0.537647 | 0.354278 |
3*105 | 4.638275 | 7.027447 | 9.318548 | 7.237380 | 4.477140 | |
3*106 | 82.225445 | 70.181132 | 91.019613 | 71.300978 | 43.957930 | |
DIXMAANC | 3*104 | 1.841306 | 0.543875 | 1.021507 | 0.939805 | 0.525989 |
3*105 | 16.686708 | 5.288750 | 9.791455 | 8.450863 | 5.193175 | |
3*106 | 218.619095 | 51.116314 | 98.113856 | 83.487215 | 50.139642 | |
DIXMAAND | 3*104 | 1.368970 | 0.791623 | 1.046740 | 0.935925 | 0.779501 |
3*105 | 12.372986 | 7.759932 | 10.334092 | 9.122740 | 7.526088 | |
3*106 | 120.560906 | 114.310394 | 101.432738 | 89.735515 | 83.011910 | |
DIXMAANE | 3*104 | 32.531045 | F | 34.405608 | 47.489975 | 296.898199 |
DIXMAANF | 3*104 | 35.843660 | F | 29.699410 | 62.706209 | 283.669710 |
DIXMAANG | 3*104 | 26.189755 | F | 36.058009 | 36.637936 | 293.464782 |
DIXMAANH | 3*104 | 47.59872 | F | 25.163339 | 33.846983 | 227.372189 |
DIXMAANI | 3*104 | 108.335661 | F | 139.441850 | 199.234093 | 271.533107 |
DIXMAANJ | 3*104 | 33.383252 | F | 51.195703 | 46.553180 | 207.800703 |
DIXMAANK | 3*104 | 29.999585 | F | 48.466471 | 48.972313 | 193.059611 |
DIXMAANL | 3*104 | 29.847873 | F | 46.188729 | 24.687342 | 240.765512 |
LIARWHD | 104 | 0.293234 | 0.167451 | 0.602854 | 0.194428 | 1.350052 |
105 | 23.934621 | 5.071408 | 4.936705 | 1.784358 | 20.476402 | |
106 | F | 17.774764 | 84.730927 | 20.451442 | 286.401251 | |
ENGVAL1 | 104 | 2.945172 | 1.918970 | 2.481500 | 1.763492 | 2.291727 |
105 | 25.852729 | 21.748152 | 16.928397 | 14.810821 | 14.884762 | |
106 | 165.309436 | 138.481046 | 194.509697 | 198.810671 | 190.808154 | |
EDENSCH | 104 | 0.847047 | 0.669810 | 0.844103 | 0.561064 | 0.506037 |
105 | 7.170456 | 6.061741 | 7.142879 | 5.470726 | 5.411347 | |
106 | 60.312014 | 53.737460 | 70.688592 | 46.010424 | 51.866055 | |
VARDIM | 104 | 0.208694 | 0.744199 | 0.206950 | 0.186933 | 0.177388 |
105 | 1.870152 | 1.437232 | 1.864668 | 2.162191 | 2.020566 | |
106 | 28.081020 | 23.333807 | 27.601484 | 22.723724 | 22.667290 | |
QUARTC | 104 | 105.093070 | 50.764779 | 4.301317 | 54.501230 | 0.196144 |
105 | F | F | 63.741145 | F | 3.083412 | |
106 | F | F | F | F | 50.399742 | |
SINQUAD | 5000 | 31.929046 | 49.651847 | 131.459433 | 29.357854 | 119.515009 |
DENSCHNB | 104 | 0.053052 | 0.039400 | 0.044617 | 0.042022 | 0.039368 |
105 | 0.331962 | 0.373278 | 0.470038 | 0.411125 | 0.381222 | |
106 | 3.100415 | 3.535380 | 5.128739 | 4.312925 | 3.585630 | |
DENSCHNF | 104 | 0.099564 | 0.084990 | 0.105827 | 0.120146 | 0.111866 |
105 | 0.904616 | 0.758068 | 0.936884 | 1.111403 | 1.071840 | |
106 | 9.215575 | 8.414838 | 9.944450 | 11.481290 | 10.755364 | |
COSINE | 104 | 0.172759 | 0.482814 | 0.222049 | 0.133286 | 0.170851 |
105 | 1.422683 | 4.501162 | 1.686364 | 1.103350 | 1.410227 | |
106 | 13.332279 | 39.732100 | 17.426555 | 10.334436 | 13.239945 |
Fun. | n | FR | PRP+ | DY | HZ | Algor.2.1 |
cpu(s) | cpu(s) | cpu(s) | cpu(s) | cpu(s) | ||
Ex.F.Roth | 104 | 4.888547 | 0.550768 | 2.957423 | 0.521059 | 1.823025 |
105 | 35.695368 | 6.393457 | 25.041071 | 5.640529 | 18.592210 | |
106 | 287.654246 | 48.039330 | 228.797928 | 46.116688 | 220.881878 | |
Ex.Trig. | 104 | 6.477725 | 0.879721 | 4.214142 | 1.521069 | 2.249725 |
105 | F | 12.156054 | 116.274122 | 74.643352 | 13.436983 | |
106 | F | 137.648005 | 209.220891 | 159.74406 | 140.003615 | |
Ex.Rosen. | 104 | 0.937497 | 0.334996 | 1.779752 | 0.269760 | 1.198996 |
105 | 3.666262 | 2.938029 | 19.050805 | 2.371920 | 12.510256 | |
106 | 76.992755 | 28.496619 | 154.785583 | 19.094775 | 100.920667 | |
Ex.W.Holst | 104 | 0.927501 | 0.902176 | 0.953671 | 0.641670 | 0.667595 |
105 | 11.200851 | 6.359390 | 5.016512 | 5.216413 | 5.085647 | |
106 | 108.520029 | 63.355358 | 53.428723 | 55.247560 | 52.571641 | |
Ex.Beale | 104 | 0.324522 | 0.275613 | 0.418340 | 0.157054 | 0.228633 |
105 | 2.474278 | 1.870461 | 2.407946 | 1.499047 | 2.405817 | |
106 | 23.457059 | 19.579839 | 26.324518 | 15.640053 | 24.912454 | |
Ex.Penalty | 104 | 0.275020 | 0.201060 | 0.208781 | 0.176900 | 0.203811 |
105 | 1.234828 | 2.239090 | 1.827919 | 1.724014 | 1.821343 | |
106 | 9.546747 | 17.71788 | 27.437794 | 14.134732 | 16.033043 | |
Per.Quad. | 104 | 3.02902 | 3.077180 | 3.040349 | 3.358025 | 46.029533 |
105 | 209.611665 | 103.424010 | 114.210432 | 108.280599 | F | |
Raydan 1 | 104 | 5.081648 | 3.277429 | 4.728205 | 3.713061 | 11.254018 |
105 | 97.715159 | 46.250008 | 127.696125 | 65.739349 | 141.324543 | |
Raydan 2 | 104 | 0.109849 | 0.087747 | 0.228718 | 0.057772 | 0.057323 |
105 | 0.833817 | 0.494746 | 1.978097 | 0.504858 | 0.533447 | |
106 | 8.214171 | 4.698048 | 16.906394 | 4.956234 | 5.812667 | |
Diag.1 | 104 | 5.789406 | 3.827681 | 6.911255 | 3.708083 | 3.538550 |
105 | F | 76.219241 | F | 70.737746 | 70.438309 | |
Diag.2 | 104 | 5.678352 | F | 5.285281 | F | 5.166993 |
105 | 154.534720 | F | 152.123264 | F | 145.797863 | |
Diag.3 | 104 | 6.181573 | 4.845072 | 6.228637 | 6.442592 | 5.311859 |
105 | 57.195661 | 52.810276 | 55.066374 | 48.976185 | 52.161715 | |
Hager | 104 | 1.380050 | 0.721078 | 1.358417 | 0.732179 | 1.320932 |
105 | 22.745343 | 11.942894 | 25.253252 | 11.895805 | 22.043234 | |
106 | F | 122.218850 | F | 156.087713 | 187.476220 | |
Gen.Trid.1 | 104 | 2.445174 | 2.512342 | 2.528296 | 2.258533 | 1.755517 |
105 | 25.293375 | 21.644389 | 25.202319 | 24.239381 | 20.201541 | |
106 | 234.095747 | 254.975842 | 204.126831 | 197.991582 | 145.845123 | |
Ex.Trid.1 | 104 | 6.224192 | 17.140057 | 1.533758 | 1.532657 | 13.182526 |
105 | 143.494302 | 244.260502 | 16.205910 | 14.276184 | 133.624706 | |
Ex.Three.Exp. | 104 | 0.199302 | 0.225054 | 0.198419 | 0.113357 | 0.142117 |
105 | 2.162488 | 2.405353 | 1.905207 | 1.265936 | 1.468474 | |
106 | 18.841829 | 21.440576 | 15.564795 | 10.447416 | 14.452928 | |
Gen.Trid.2 | 104 | 13.541982 | 3.192677 | 4.418622 | 2.970148 | 2.529354 |
105 | F | 30.674646 | F | 24.129818 | 23.986342 | |
Diag.4 | 104 | 0.094771 | 0.057988 | 0.114438 | 0.062960 | 0.093890 |
105 | 1.202458 | 0.388207 | 1.013161 | 0.202075 | 0.890367 | |
106 | 19.755904 | 3.749717 | 8.999546 | 1.906101 | 8.861827 | |
Diag.5 | 104 | 0.217310 | 0.089796 | 0.238359 | 0.141571 | 0.083229 |
105 | 1.373886 | 0.744259 | 2.028916 | 1.963680 | 0.693929 | |
106 | 13.342298 | 8.012713 | 18.812321 | 18.810192 | 6.833566 | |
Ex.Himm. | 104 | 0.611196 | 0.536196 | 0.141640 | 0.087034 | 0.132999 |
105 | 5.446782 | 5.385943 | 1.156565 | 0.724150 | 1.104406 | |
106 | 53.854410 | 53.065089 | 10.564224 | 7.176464 | 11.838695 | |
Gen.PSC1 | 104 | 2.934187 | 1.250311 | 11.817101 | 2.056868 | 28.300200 |
105 | 7.925998 | 15.507716 | 57.672689 | 14.110938 | 98.215964 | |
Ex.PSC1 | 104 | 0.114309 | 0.116711 | 0.137112 | 0.121277 | 0.120501 |
105 | 1.172552 | 0.942523 | 1.131255 | 1.341338 | 1.095562 | |
106 | 13.738276 | 10.067906 | 9.391618 | 12.643046 | 10.754309 | |
Ex.B.Diag.BD1 | 104 | 0.481994 | 0.093661 | 7.119658 | 0.153198 | 0.184555 |
105 | 4.780717 | 0.828624 | 63.779966 | 1.219591 | 1.886973 | |
106 | 45.244221 | 7.678318 | F | 13.094154 | 17.652279 | |
Ex.Maratos | 104 | 7.298980 | 0.215135 | 7.417174 | 0.400084 | 1.720770 |
105 | 72.871313 | 2.399795 | 69.118730 | 2.917748 | 15.965102 | |
106 | F | 24.968870 | F | 37.905524 | 188.771631 | |
Qua.Diag.Pert. | 104 | 20.021639 | 20.021639 | 23.190386 | 6.862350 | 20.0167895 |
Ex.Wood | 104 | 3.626209 | 0.449196 | 2.218795 | 0.481428 | 1.548869 |
105 | 30.297717 | 5.710406 | 25.452955 | 5.599862 | 24.178781 | |
106 | F | 83.358482 | F | 50.926588 | 80.176211 | |
Ex.Hiebert | 104 | 12.325250 | 0.423625 | 4.209053 | 0.505931 | 4.111580 |
105 | 61.420777 | 7.301559 | 44.310328 | 4.483430 | 43.492047 | |
106 | F | 98.625677 | F | 40.245693 | F | |
Ex.Qua.Pena.QP1 | 104 | 0.209738 | 0.131130 | 0.120703 | 0.213959 | 0.120383 |
105 | 0.931057 | 2.044439 | 0.928968 | 1.702844 | 1.680728 | |
106 | 19.710059 | 13.914293 | 5.142260 | 6.478819 | 13.674789 | |
Ex.Qua.Pena.QP2 | 104 | F | 0.368244 | 7.436318 | 0.329816 | 1.646625 |
105 | F | 3.211811 | 214.833356 | 3.229403 | 14.055916 | |
106 | F | 37.820369 | F | 31.182550 | 183.466287 | |
Qua.QF2 | 104 | 6.548727 | 6.315666 | 10.817725 | 6.994623 | 6.088805 |
FLETCBV3 | 1000 | 3.391656 | 3.208908 | 3.322490 | 9.722141 | 10.174606 |
FLETCHCR | 5000 | 17.258755 | 14.429972 | 16.787908 | 14.802347 | 10.449042 |
BDQRTIC | 104 | 83.808804 | F | 11.207452 | 9.377903 | 75.611094 |
TRIDIA | 104 | 20.207208 | 7.295958 | 13.794344 | 31.899388 | 85.858224 |
ARGLINB | 104 | 0.110820 | 0.107058 | 0.077671 | 0.068173 | 0.104807 |
105 | 0.722081 | 5.972367 | 0.946074 | 0.622368 | 0.790848 | |
106 | 7.334012 | 8.595424 | 6.361517 | 13.706507 | 5.689481 | |
ARWHEAD | 104 | 0.370001 | 0.077944 | 0.132404 | 0.090989 | 0.101133 |
105 | 1.579815 | 0.563831 | 1.544622 | 1.490355 | 1.243687 | |
106 | 5.837702 | 5.826076 | 4.586703 | 6.565910 | 5.671701 | |
NONBIA | 104 | 0.537552 | 0.102890 | 1.058223 | 0.103249 | 0.8355213 |
105 | 2.747465 | 1.337234 | 2.297892 | 2.992103 | 3.579316 | |
106 | 5.671371 | 22.742947 | 3.420499 | 6.936500 | 29.025324 | |
NONDQUAR | 104 | 0.442060 | 0.190346 | 0.158307 | 0.400302 | 0.649100 |
105 | 0.796334 | 2.503826 | 0.704471 | 1.828935 | 7.990690 | |
106 | 5.778999 | 20.906151 | 4.532799 | 17.272681 | 67.882148 | |
EG2 | 104 | 0.994080 | 0.313402 | 1.554111 | 1.337630 | 2.278667 |
105 | 7.160837 | 35.861218 | 3.615775 | 7.705992 | 10.859091 | |
106 | 43.235295 | 14.166499 | 11.586350 | 10.875278 | 50.416843 | |
DIXMAANA | 3*104 | 1.271498 | 0.762316 | 1.282368 | 0.736608 | 0.756586 |
3*105 | 11.612663 | 10.441716 | 11.562110 | 8.106027 | 7.118575 | |
3*106 | 114.760219 | 102.835981 | 125.107882 | 79.990697 | 75.923830 | |
DIXMAANB | 3*104 | 0.517920 | 0.385757 | 0.542675 | 0.537647 | 0.354278 |
3*105 | 4.638275 | 7.027447 | 9.318548 | 7.237380 | 4.477140 | |
3*106 | 82.225445 | 70.181132 | 91.019613 | 71.300978 | 43.957930 | |
DIXMAANC | 3*104 | 1.841306 | 0.543875 | 1.021507 | 0.939805 | 0.525989 |
3*105 | 16.686708 | 5.288750 | 9.791455 | 8.450863 | 5.193175 | |
3*106 | 218.619095 | 51.116314 | 98.113856 | 83.487215 | 50.139642 | |
DIXMAAND | 3*104 | 1.368970 | 0.791623 | 1.046740 | 0.935925 | 0.779501 |
3*105 | 12.372986 | 7.759932 | 10.334092 | 9.122740 | 7.526088 | |
3*106 | 120.560906 | 114.310394 | 101.432738 | 89.735515 | 83.011910 | |
DIXMAANE | 3*104 | 32.531045 | F | 34.405608 | 47.489975 | 296.898199 |
DIXMAANF | 3*104 | 35.843660 | F | 29.699410 | 62.706209 | 283.669710 |
DIXMAANG | 3*104 | 26.189755 | F | 36.058009 | 36.637936 | 293.464782 |
DIXMAANH | 3*104 | 47.59872 | F | 25.163339 | 33.846983 | 227.372189 |
DIXMAANI | 3*104 | 108.335661 | F | 139.441850 | 199.234093 | 271.533107 |
DIXMAANJ | 3*104 | 33.383252 | F | 51.195703 | 46.553180 | 207.800703 |
DIXMAANK | 3*104 | 29.999585 | F | 48.466471 | 48.972313 | 193.059611 |
DIXMAANL | 3*104 | 29.847873 | F | 46.188729 | 24.687342 | 240.765512 |
LIARWHD | 104 | 0.293234 | 0.167451 | 0.602854 | 0.194428 | 1.350052 |
105 | 23.934621 | 5.071408 | 4.936705 | 1.784358 | 20.476402 | |
106 | F | 17.774764 | 84.730927 | 20.451442 | 286.401251 | |
ENGVAL1 | 104 | 2.945172 | 1.918970 | 2.481500 | 1.763492 | 2.291727 |
105 | 25.852729 | 21.748152 | 16.928397 | 14.810821 | 14.884762 | |
106 | 165.309436 | 138.481046 | 194.509697 | 198.810671 | 190.808154 | |
EDENSCH | 104 | 0.847047 | 0.669810 | 0.844103 | 0.561064 | 0.506037 |
105 | 7.170456 | 6.061741 | 7.142879 | 5.470726 | 5.411347 | |
106 | 60.312014 | 53.737460 | 70.688592 | 46.010424 | 51.866055 | |
VARDIM | 104 | 0.208694 | 0.744199 | 0.206950 | 0.186933 | 0.177388 |
105 | 1.870152 | 1.437232 | 1.864668 | 2.162191 | 2.020566 | |
106 | 28.081020 | 23.333807 | 27.601484 | 22.723724 | 22.667290 | |
QUARTC | 104 | 105.093070 | 50.764779 | 4.301317 | 54.501230 | 0.196144 |
105 | F | F | 63.741145 | F | 3.083412 | |
106 | F | F | F | F | 50.399742 | |
SINQUAD | 5000 | 31.929046 | 49.651847 | 131.459433 | 29.357854 | 119.515009 |
DENSCHNB | 104 | 0.053052 | 0.039400 | 0.044617 | 0.042022 | 0.039368 |
105 | 0.331962 | 0.373278 | 0.470038 | 0.411125 | 0.381222 | |
106 | 3.100415 | 3.535380 | 5.128739 | 4.312925 | 3.585630 | |
DENSCHNF | 104 | 0.099564 | 0.084990 | 0.105827 | 0.120146 | 0.111866 |
105 | 0.904616 | 0.758068 | 0.936884 | 1.111403 | 1.071840 | |
106 | 9.215575 | 8.414838 | 9.944450 | 11.481290 | 10.755364 | |
COSINE | 104 | 0.172759 | 0.482814 | 0.222049 | 0.133286 | 0.170851 |
105 | 1.422683 | 4.501162 | 1.686364 | 1.103350 | 1.410227 | |
106 | 13.332279 | 39.732100 | 17.426555 | 10.334436 | 13.239945 |
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