
- Previous Article
- NACO Home
- This Issue
-
Next Article
Application of survival theory in Mining industry
Behavior of the combination of PRP and HZ methods for unconstrained optimization
Laboratory Informatics and Mathematics (LiM), Mohamed Cherif Messaadia University, Souk Ahras, 41000, Algeria |
To achieve a conjugate gradient method which is strong in theory and efficient in practice for solving unconstrained optimization problem, we propose a hybridization of the Hager and Zhang (HZ) and Polak-Ribière and Polyak (PRP) conjugate gradient methods which possesses an important property of the well known PRP method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, averting a sequence of tiny steps from happening, the new scalar $ \beta_k $ is obtained by convex combination of PRP and HZ under the wolfe line search we prove the sufficient descent and the global convergence. Numerical results are reported to show the effectiveness of our procedure.
References:
[1] |
M. Al-Baali,
Descent property and global convergence of the Fletcher–Reeves method with inexact line search, IMA Journal of Numerical Analysis, 5 (1985), 121-124.
doi: 10.1093/imanum/5.1.121. |
[2] |
N. Andrei,
Another hybrid conjugate gradient algorithm for unconstrained optimization, Numerical Algorithms, 47 (2008), 143-156.
doi: 10.1007/s11075-007-9152-9. |
[3] |
N. Andrei, New hybrid conjugate gradient algorithms for unconstrained optimization, in Encyclopedia of Optimization(eds. A. F. Christodoulos and M. P. Panos), 2009.
doi: 10.1007/978-0-387-74759-0_441. |
[4] |
B. Balaram, M. Narayanan and P. Rajendrakumar,
Optimal design of multi-parametric nonlinear systems using a parametric continuation based genetic algorithm approach, Nonlinear Dynamics, 67 (2012), 2759-2777.
doi: 10.1007/s11071-011-0187-z. |
[5] |
I. Bongartz, A. R. Conn, N. Gould and P. L. Toint, CUTE: constrained and unconstrained testing environments, ACM Transactions on Mathematical Software (TOMS), 21 (1995), 123-160. Google Scholar |
[6] |
N. Chenna, Comments on "New Hybrid Conjugate Gradient Method as a Convex Combination of FR and PRP Methods", FILOMAT, 33 (2019), 3083-3100. Google Scholar |
[7] |
Y.-H. Dai and Y. Yuan,
A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10 (1999), 177-182.
doi: 10.1137/S1052623497318992. |
[8] |
S. S. Djordjević,
New hybrid conjugate gradient method as a convex combination of FR and PRP methods, Filomat, 30 (2016), 3083-3100.
doi: 10.2298/FIL1611083D. |
[9] |
E. D. Dolan and J. J. Moré,
Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.
doi: 10.1007/s101070100263. |
[10] |
W. W. Hager and H. 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. |
[11] |
X. Z. Jiang, G.-D. Ma and J.-B. Jian,
A new global convergent conjugate gradient method with Wolfe line search, Chinese Journal of Engineering Mathematics, 28 (2011), 779-786.
|
[12] |
J. Liu and S. Li,
New hybrid conjugate gradient method for unconstrained optimization, Applied Mathematics and Computation, 245 (2014), 36-43.
doi: 10.1016/j.amc.2014.07.096. |
[13] |
E. Polak and G. Ribière, Note sur la convergence de méthodes de directions conjuguées, Revue Française D'informatique et De Recherche Opérationnelle, Série Rouge, 3 (1969), 35–43. |
[14] |
B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112. Google Scholar |
[15] |
M. Powell, Nonconvex minimization calculations and the conjugate gradient method, Numerical Analysis(ed. D. F.Griffiths), Volume 1066 of Lecture Notes in Math., Dundee, (1984), 122–141.
doi: 10.1007/BFb0099521. |
[16] |
Z. Wei, G. Li and L. Qi,
New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems, Applied Mathematics and Computation, 179 (2006), 407-430.
doi: 10.1016/j.amc.2005.11.150. |
[17] |
G. Yuan, Z. Wei and Q. Zhao,
A modified Polak–Ribière–Polyak conjugate gradient algorithm for large-scale optimization problems, IIE Transactions, 46 (2014), 397-413.
doi: 10.1080/01630563.2013.777350. |
[18] |
G. Zoutendijk, Nonlinear programming, computational methods, Integer and Nonlinear Programming, (1970), 37–86. |
show all references
References:
[1] |
M. Al-Baali,
Descent property and global convergence of the Fletcher–Reeves method with inexact line search, IMA Journal of Numerical Analysis, 5 (1985), 121-124.
doi: 10.1093/imanum/5.1.121. |
[2] |
N. Andrei,
Another hybrid conjugate gradient algorithm for unconstrained optimization, Numerical Algorithms, 47 (2008), 143-156.
doi: 10.1007/s11075-007-9152-9. |
[3] |
N. Andrei, New hybrid conjugate gradient algorithms for unconstrained optimization, in Encyclopedia of Optimization(eds. A. F. Christodoulos and M. P. Panos), 2009.
doi: 10.1007/978-0-387-74759-0_441. |
[4] |
B. Balaram, M. Narayanan and P. Rajendrakumar,
Optimal design of multi-parametric nonlinear systems using a parametric continuation based genetic algorithm approach, Nonlinear Dynamics, 67 (2012), 2759-2777.
doi: 10.1007/s11071-011-0187-z. |
[5] |
I. Bongartz, A. R. Conn, N. Gould and P. L. Toint, CUTE: constrained and unconstrained testing environments, ACM Transactions on Mathematical Software (TOMS), 21 (1995), 123-160. Google Scholar |
[6] |
N. Chenna, Comments on "New Hybrid Conjugate Gradient Method as a Convex Combination of FR and PRP Methods", FILOMAT, 33 (2019), 3083-3100. Google Scholar |
[7] |
Y.-H. Dai and Y. Yuan,
A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10 (1999), 177-182.
doi: 10.1137/S1052623497318992. |
[8] |
S. S. Djordjević,
New hybrid conjugate gradient method as a convex combination of FR and PRP methods, Filomat, 30 (2016), 3083-3100.
doi: 10.2298/FIL1611083D. |
[9] |
E. D. Dolan and J. J. Moré,
Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.
doi: 10.1007/s101070100263. |
[10] |
W. W. Hager and H. 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. |
[11] |
X. Z. Jiang, G.-D. Ma and J.-B. Jian,
A new global convergent conjugate gradient method with Wolfe line search, Chinese Journal of Engineering Mathematics, 28 (2011), 779-786.
|
[12] |
J. Liu and S. Li,
New hybrid conjugate gradient method for unconstrained optimization, Applied Mathematics and Computation, 245 (2014), 36-43.
doi: 10.1016/j.amc.2014.07.096. |
[13] |
E. Polak and G. Ribière, Note sur la convergence de méthodes de directions conjuguées, Revue Française D'informatique et De Recherche Opérationnelle, Série Rouge, 3 (1969), 35–43. |
[14] |
B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112. Google Scholar |
[15] |
M. Powell, Nonconvex minimization calculations and the conjugate gradient method, Numerical Analysis(ed. D. F.Griffiths), Volume 1066 of Lecture Notes in Math., Dundee, (1984), 122–141.
doi: 10.1007/BFb0099521. |
[16] |
Z. Wei, G. Li and L. Qi,
New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems, Applied Mathematics and Computation, 179 (2006), 407-430.
doi: 10.1016/j.amc.2005.11.150. |
[17] |
G. Yuan, Z. Wei and Q. Zhao,
A modified Polak–Ribière–Polyak conjugate gradient algorithm for large-scale optimization problems, IIE Transactions, 46 (2014), 397-413.
doi: 10.1080/01630563.2013.777350. |
[18] |
G. Zoutendijk, Nonlinear programming, computational methods, Integer and Nonlinear Programming, (1970), 37–86. |
Problems | n | hPRPHZ | PRP | HZ | |||
time | iter | time | iter | time | iter | ||
FLETCHCR | 5000 | 95.6800 | 34677 | 123.9500 | 456454 | 84.2000 | 40000 |
CURLY30 | 1000 | 8.8600 | 15122 | 8.8700 | 15401 | NaN | NaN |
CURLY20 | 1000 | 10.9100 | 15084 | 6.9600 | 15797 | NaN | NaN |
DIXMAANI | 6000 | 9.4300 | 2661 | 9.0600 | 2261 | 13.9800 | 4720 |
EIGENBLS | 420 | 3.5500 | 4978 | 10.1100 | 5440 | 14.9300 | 9714 |
TRIDIA | 10 000 | 7.3200 | 1116 | 3.1900 | 1116 | 3.8900 | 2231 |
NONDQUAR | 5000 | 4.2400 | 5099 | 7.5000 | 5058 | 9.4700 | 10058 |
CURLY10 | 1000 | 4.2700 | 14406 | 4.0600 | 13659 | NaN | NaN |
EIGENCLS | 462 | 4.2500 | 1802 | 4.1000 | 1883 | 5.9900 | 3312 |
SPARSINE | 1000 | 2.5700 | 4516 | 4.3200 | 4483 | 6.5900 | 8793 |
EIGENALS | 420 | 3.9700 | 1344 | 2.4900 | 1306 | 4.7400 | 2998 |
FLETCHCR | 1000 | 6.0300 | 7479 | 4.9300 | 9139 | 3.5700 | 8986 |
GENHUMPS | 1000 | 2.2400 | 3555 | 5.8400 | 3435 | 7.5500 | 5807 |
FMINSURF | 5625 | 1.0000 | 492 | 3.4700 | 669 | 3.3900 | 949 |
TRIDIA | 5000 | 1.0900 | 783 | 1.0700 | 783 | 1.3100 | 1565 |
DIXMAANE | 6000 | 1.2200 | 303 | 1.2600 | 306 | 2.1300 | 620 |
DIXMAANJ | 6000 | 23.8000 | 296 | 1.1800 | 275 | 2.1700 | 557 |
BDQRTIC | 5000 | 1.3500 | 8726 | 7.6400 | 2428 | NaN | NaN |
DIXMAANK | 6000 | 1.8100 | 264 | 1.1100 | 248 | 1.8000 | 587 |
NONCVXU2 | 1000 | 1.5600 | 2055 | 1.9200 | 2015 | 3.6400 | 3919 |
DIXMAANL | 6000 | 0.9700 | 245 | 1.3200 | 215 | 3.0100 | 702 |
SENSORS | 100 | 1.0700 | 44 | 0.9700 | 45 | 1.3600 | 66 |
DIXMAANF | 6000 | 1.0400 | 230 | 1.1200 | 230 | 1.6200 | 437 |
DIXMAANG | 6000 | 1.3400 | 227 | 1.0800 | 227 | 1.4500 | 420 |
DIXMAANH | 6000 | 0.9900 | 224 | 1.1600 | 224 | 2.6400 | 825 |
FLETCBV2 | 1000 | 1.4000 | 1055 | 1.0000 | 1044 | 1.2900 | 1886 |
SCHMVETT | 10 000 | 2.3800 | 60 | 1.5000 | 64 | 2.5900 | 105 |
GENHUMPS | 500 | 1.0100 | 2258 | 2.1500 | 2531 | 2.7000 | 4147 |
CRAGGLVY | 5000 | 0.7400 | 143 | 0.9900 | 138 | NaN | NaN |
MOREBV | 10 000 | 1.1900 | 97 | 0.8900 | 97 | 1.2800 | 201 |
WOODS | 10 000 | 0.8400 | 257 | 1.1700 | 230 | 2.1400 | 487 |
NONDQUAR | 1000 | 0.3800 | 3147 | 1.4500 | 4900 | 1.6300 | 8128 |
SPARSQUR | 10 000 | 0.3500 | 23 | 0.3800 | 23 | 1.1300 | 131 |
POWER | 5000 | 0.6500 | 259 | 0.6100 | 408 | 0.4000 | 514 |
MANCINO | 100 | 0.3500 | 12 | 0.6000 | 11 | 1.1500 | 27 |
CRAGGLVY | 2000 | 0.3300 | 132 | 0.3700 | 142 | NaN | NaN |
CURLY30 | 200 | 0.4800 | 2819 | 0.3600 | 3066 | NaN | NaN |
LIARWHD | 10 000 | 0.5700 | 41 | 0.4600 | 39 | 0.4800 | 46 |
BDQRTIC | 1000 | 0.4600 | 1025 | 0.4900 | 798 | NaN | NaN |
GENROSE | 500 | 0.2900 | 1309 | 0.4900 | 1624 | 0.4600 | 2278 |
VARDIM | 10 000 | 0.2700 | 62 | 0.2900 | 57 | NaN | NaN |
CURLY20 | 200 | 0.7100 | 2951 | 0.3000 | 2835 | NaN | NaN |
FREUROTH | 5000 | 0.4000 | 96 | 0.5900 | 76 | NaN | NaN |
ENGVAL1 | 10 000 | 0.2800 | 35 | 0.4100 | 34 | NaN | NaN |
POWELLSG | 10 000 | 0.2500 | 77 | 0.2300 | 49 | 0.7200 | 362 |
DIXON3DQ | 1000 | 0.3100 | 1002 | 0.2700 | 1002 | 0.3300 | 2005 |
BRYBND | 5000 | 0.4500 | 39 | 0.3200 | 40 | 0.3800 | 66 |
HILBERTA | 200 | 0.7100 | 50 | 0.3700 | 25 | 0.3800 | 38 |
TQUARTIC | 10 000 | 0.1900 | 61 | 0.6500 | 52 | 0.5800 | 38 |
CURLY10 | 200 | 0.2100 | 3100 | 0.2000 | 3182 | NaN | NaN |
FLETCBV2 | 500 | 0.2600 | 480 | 0.2200 | 482 | 0.3600 | 962 |
FMINSURF | 1024 | 0.1200 | 238 | 0.2400 | 300 | 0.2800 | 455 |
VARDIM | 5000 | 0.2000 | 44 | 0.1300 | 47 | NaN | NaN |
FMINSRF2 | 1024 | 0.1400 | 282 | 0.2600 | 355 | 0.2900 | 517 |
SPMSRTLS | 1000 | 0.2400 | 151 | 0.1500 | 151 | 0.2000 | 281 |
LIARWHD | 5000 | 0.2600 | 32 | 0.3000 | 48 | 0.2500 | 46 |
NONDIA | 10 000 | 0.2600 | 16 | 0.2300 | 10 | 0.3100 | 26 |
POWELLSG | 5000 | 0.5500 | 187 | 0.1100 | 53 | 0.3200 | 346 |
ARWHEAD | 10 000 | 0.1600 | 15 | 0.5300 | 12 | NaN | NaN |
SROSENBR | 10 000 | 0.1900 | 17 | 0.1700 | 19 | 0.1700 | 26 |
TQUARTIC | 5000 | 0.1700 | 38 | 0.2100 | 54 | 0.1700 | 32 |
PENALTY1 | 5000 | 0.2500 | 62 | 0.2200 | 80 | 0.3400 | 152 |
DQDRTIC | 10 000 | 0.1300 | 8 | 0.2600 | 8 | 0.2700 | 15 |
NONDIA | 5000 | 0.2200 | 22 | 0.1400 | 26 | 0.1300 | 26 |
ARGLINB | 300 | 0.1300 | 23 | 0.2000 | 17 | NaN | NaN |
DIXMAAND | 6000 | 0.2500 | 13 | 0.1300 | 12 | 0.1600 | 25 |
ARGLINC | 300 | 0.0800 | 19 | 0.2700 | 25 | NaN | NaN |
DQRTIC | 5000 | 0.0900 | 34 | 0.1000 | 34 | 0.1000 | 66 |
QUARTC | 5000 | 0.0900 | 34 | 0.0900 | 34 | 0.1000 | 66 |
EIGENALS | 110 | 0.0400 | 389 | 0.0800 | 359 | 0.1600 | 806 |
SINQUAD | 500 | 0.0800 | 111 | 0.0400 | 93 | NaN | NaN |
SPARSINE | 200 | 0.0600 | 445 | 0.0800 | 445 | 0.1300 | 917 |
DIXON3DQ | 500 | 0.2400 | 500 | 0.0600 | 500 | 0.0800 | 1003 |
DIXMAANC | 6000 | 0.2200 | 11 | 0.2400 | 11 | 0.2600 | 23 |
HILBERTB | 200 | 0.2100 | 6 | 0.2200 | 6 | 0.2500 | 13 |
BROWNAL | 400 | 0.0700 | 13 | 0.2000 | 7 | 0.2700 | 37 |
EIGENCLS | 90 | 0.2500 | 360 | 0.0700 | 350 | 0.1100 | 743 |
ARGLINA | 300 | 0.2300 | 2 | 0.2500 | 2 | 0.2600 | 5 |
EXTROSNB | 50 | 0.1300 | 5819 | 0.1900 | 5294 | 0.2400 | 7808 |
PENALTY2 | 200 | 0.1800 | 365 | 0.1400 | 417 | NaN | NaN |
FREUROTH | 1000 | 0.0700 | 187 | 0.1600 | 137 | NaN | NaN |
BRYBND | 1000 | 0.0600 | 52 | 0.0600 | 35 | 0.0800 | 73 |
DIXMAANB | 3000 | 0.0400 | 10 | 0.0600 | 10 | 0.0700 | 23 |
NONCVXU2 | 100 | 0.0600 | 396 | 0.0300 | 414 | 0.0500 | 801 |
DIXMAANA | 3000 | 0.2100 | 10 | 0.0500 | 9 | 0.0700 | 20 |
TOINTGSS | 10 000 | 0.0300 | 5 | 0.2100 | 5 | 0.3800 | 20 |
POWER | 1000 | 0.0600 | 117 | 0.0600 | 222 | 0.0400 | 236 |
DECONVU | 61 | 0.0200 | 462 | 0.0600 | 460 | 0.0700 | 581 |
GENROSE | 100 | 0.0200 | 347 | 0.0200 | 392 | 0.0300 | 626 |
COSINE | 1000 | 0.0300 | 24 | 0.0200 | 24 | 0.0300 | 29 |
DIXMAANB | 1500 | 0.0100 | 10 | 0.0300 | 10 | 0.0400 | 24 |
CHNROSNB | 50 | 0.0300 | 273 | 0.0200 | 285 | 0.0100 | 500 |
DIXMAANA | 1500 | 0.0100 | 10 | 0.0300 | 9 | 0.0300 | 22 |
FMINSRF2 | 121 | 0.0300 | 115 | 0.0100 | 124 | 0.0100 | 250 |
ARWHEAD | 1000 | 0.0100 | 16 | 0.0300 | 19 | NaN | NaN |
COSINE | 500 | 0.0200 | 23 | 0 | 22 | 0.0100 | 26 |
DQDRTIC | 1000 | 0.0600 | 8 | 0.0200 | 8 | 0.0300 | 15 |
ERRINROS | 50 | 0.0200 | 1444 | 0.0900 | 2416 | NaN | NaN |
EG2 | 1000 | 0.0100 | 6 | 0.0100 | 6 | NaN | NaN |
TESTQUAD | 100 | 0.0100 | 321 | 0.0100 | 303 | 0.0100 | 925 |
TOINTGOR | 50 | 0.8800 | 151 | 0.0100 | 155 | 0.0100 | 250 |
SPARSINE | 5000 | 0.1300 | 370 | 1.5700 | 544 | 1.1200 | 719 |
FMINSRF2 | 10 000 | 0.2800 | 26 | 0.1200 | 23 | 0.1300 | 27 |
FMINSRF2 | 15 625 | 1.1300 | 28 | 0.2600 | 23 | 0.2800 | 28 |
FMINSRF2 | 5625 | 3.0500 | 227 | 1.3100 | 214 | 1.8900 | 430 |
NONDQUAR | 10 000 | 1.3200 | 234 | 2.4200 | 225 | 3.5200 | 440 |
POWER | 10 000 | 43.7500 | 142 | 0.7100 | 62 | NaN | NaN |
ARWHEAD | 5000 | 0.2100 | 7298 | 36.8400 | 6398 | NaN | NaN |
COSINE | 5000 | 59.1900 | 37 | 0.2000 | 35 | NaN | NaN |
COSINE | 10 000 | 3.6600 | 8476 | 31.5200 | 4721 | 53.2500 | 8965 |
FMINSURF | 10 000 | 0.6400 | 8771 | 2.1400 | 5022 | 2.4100 | 6779 |
FMINSURF | 15 625 | 0.3600 | 108 | 0.4700 | 62 | NaN | NaN |
BROYDN7D | 1000 | 5.3900 | 498 | 0.2700 | 371 | NaN | NaN |
SPMSRTLS | 4999 | 0.0010 | 2232 | 5.4700 | 2183 | 6.4500 | 4093 |
SPMSRTLS | 10 000 | 0.0010 | NaN | NaN | NaN | 0.2800 | NaN |
FREUROTH | 10 000 | 0.0010 | NaN | NaN | NaN | 1.8900 | NaN |
FLETCBV2 | 500 | 0.0010 | NaN | NaN | NaN | 3.5200 | NaN |
BDQRTIC | 10 000 | 0.0010 | 1 | NaN | NaN | 0.2800 | NaN |
VAREIGVL | 10 000 | 0.0010 | 1 | NaN | NaN | 1.8900 | NaN |
ENGVAL1 | 5000 | NaN | 1 | NaN | NaN | 3.5200 | NaN |
BRYBND | 10 000 | 0.1000 | 34677 | 0.5000 | 456454 | 0.9000 | 40000 |
EIGENBLS | 930 | 0.1000 | 15122 | 0.0500 | 15401 | 0.9000 | NaN |
NONCVXUN | 500 | 0.1000 | 15084 | 0.0500 | 15797 | 0.9000 | NaN |
GENROSE | 1000 | 0.1000 | 2661 | 0.5000 | 2261 | 0.9000 | 4720 |
GENROSE | 5000 | 0.1000 | 4978 | 0.0500 | 5440 | 0.9000 | 9714 |
EIGENALS | 930 | 0.1000 | 1116 | 0.0500 | 1116 | 0.9000 | 2231 |
SINQUAD | 5000 | 0.1000 | 5099 | 0.5000 | 5058 | 0.9000 | 10058 |
SINQUAD | 10 000 | 0.1000 | 14406 | 0.0500 | 13659 | 0.9000 | NaN |
GENHUMPS | 5000 | 0.1000 | 1802 | 0.0500 | 1883 | 0.9000 | 3312 |
CHAINWOO | 1000 | 0.1000 | 4516 | 0.5000 | 4483 | 0.9000 | 8793 |
TESTQUAD | 1000 | 0.1000 | 1344 | 0.0500 | 1306 | 0.9000 | 2998 |
TESTQUAD | 10 000 | 0.1000 | 7479 | 0.0500 | 9139 | 0.9000 | 8986 |
TESTQUAD | 5000 | 0.1000 | 3555 | 0.5000 | 3435 | 0.9000 | 5807 |
FLETCHCR | 5000 | 0.1000 | 492 | 0.0500 | 669 | 0.9000 | 949 |
CURLY30 | 1000 | 0.1000 | 783 | 0.0500 | 783 | 0.9000 | 1565 |
CURLY20 | 1000 | 0.1000 | 303 | NaN | 306 | 0.9000 | 620 |
DIXMAANI | 6000 | 0.1000 | 296 | NaN | 275 | 0.9000 | 557 |
EIGENBLS | 420 | 0.1000 | 8726 | NaN | 2428 | 0.9000 | NaN |
Problems | n | hPRPHZ | PRP | HZ | |||
time | iter | time | iter | time | iter | ||
FLETCHCR | 5000 | 95.6800 | 34677 | 123.9500 | 456454 | 84.2000 | 40000 |
CURLY30 | 1000 | 8.8600 | 15122 | 8.8700 | 15401 | NaN | NaN |
CURLY20 | 1000 | 10.9100 | 15084 | 6.9600 | 15797 | NaN | NaN |
DIXMAANI | 6000 | 9.4300 | 2661 | 9.0600 | 2261 | 13.9800 | 4720 |
EIGENBLS | 420 | 3.5500 | 4978 | 10.1100 | 5440 | 14.9300 | 9714 |
TRIDIA | 10 000 | 7.3200 | 1116 | 3.1900 | 1116 | 3.8900 | 2231 |
NONDQUAR | 5000 | 4.2400 | 5099 | 7.5000 | 5058 | 9.4700 | 10058 |
CURLY10 | 1000 | 4.2700 | 14406 | 4.0600 | 13659 | NaN | NaN |
EIGENCLS | 462 | 4.2500 | 1802 | 4.1000 | 1883 | 5.9900 | 3312 |
SPARSINE | 1000 | 2.5700 | 4516 | 4.3200 | 4483 | 6.5900 | 8793 |
EIGENALS | 420 | 3.9700 | 1344 | 2.4900 | 1306 | 4.7400 | 2998 |
FLETCHCR | 1000 | 6.0300 | 7479 | 4.9300 | 9139 | 3.5700 | 8986 |
GENHUMPS | 1000 | 2.2400 | 3555 | 5.8400 | 3435 | 7.5500 | 5807 |
FMINSURF | 5625 | 1.0000 | 492 | 3.4700 | 669 | 3.3900 | 949 |
TRIDIA | 5000 | 1.0900 | 783 | 1.0700 | 783 | 1.3100 | 1565 |
DIXMAANE | 6000 | 1.2200 | 303 | 1.2600 | 306 | 2.1300 | 620 |
DIXMAANJ | 6000 | 23.8000 | 296 | 1.1800 | 275 | 2.1700 | 557 |
BDQRTIC | 5000 | 1.3500 | 8726 | 7.6400 | 2428 | NaN | NaN |
DIXMAANK | 6000 | 1.8100 | 264 | 1.1100 | 248 | 1.8000 | 587 |
NONCVXU2 | 1000 | 1.5600 | 2055 | 1.9200 | 2015 | 3.6400 | 3919 |
DIXMAANL | 6000 | 0.9700 | 245 | 1.3200 | 215 | 3.0100 | 702 |
SENSORS | 100 | 1.0700 | 44 | 0.9700 | 45 | 1.3600 | 66 |
DIXMAANF | 6000 | 1.0400 | 230 | 1.1200 | 230 | 1.6200 | 437 |
DIXMAANG | 6000 | 1.3400 | 227 | 1.0800 | 227 | 1.4500 | 420 |
DIXMAANH | 6000 | 0.9900 | 224 | 1.1600 | 224 | 2.6400 | 825 |
FLETCBV2 | 1000 | 1.4000 | 1055 | 1.0000 | 1044 | 1.2900 | 1886 |
SCHMVETT | 10 000 | 2.3800 | 60 | 1.5000 | 64 | 2.5900 | 105 |
GENHUMPS | 500 | 1.0100 | 2258 | 2.1500 | 2531 | 2.7000 | 4147 |
CRAGGLVY | 5000 | 0.7400 | 143 | 0.9900 | 138 | NaN | NaN |
MOREBV | 10 000 | 1.1900 | 97 | 0.8900 | 97 | 1.2800 | 201 |
WOODS | 10 000 | 0.8400 | 257 | 1.1700 | 230 | 2.1400 | 487 |
NONDQUAR | 1000 | 0.3800 | 3147 | 1.4500 | 4900 | 1.6300 | 8128 |
SPARSQUR | 10 000 | 0.3500 | 23 | 0.3800 | 23 | 1.1300 | 131 |
POWER | 5000 | 0.6500 | 259 | 0.6100 | 408 | 0.4000 | 514 |
MANCINO | 100 | 0.3500 | 12 | 0.6000 | 11 | 1.1500 | 27 |
CRAGGLVY | 2000 | 0.3300 | 132 | 0.3700 | 142 | NaN | NaN |
CURLY30 | 200 | 0.4800 | 2819 | 0.3600 | 3066 | NaN | NaN |
LIARWHD | 10 000 | 0.5700 | 41 | 0.4600 | 39 | 0.4800 | 46 |
BDQRTIC | 1000 | 0.4600 | 1025 | 0.4900 | 798 | NaN | NaN |
GENROSE | 500 | 0.2900 | 1309 | 0.4900 | 1624 | 0.4600 | 2278 |
VARDIM | 10 000 | 0.2700 | 62 | 0.2900 | 57 | NaN | NaN |
CURLY20 | 200 | 0.7100 | 2951 | 0.3000 | 2835 | NaN | NaN |
FREUROTH | 5000 | 0.4000 | 96 | 0.5900 | 76 | NaN | NaN |
ENGVAL1 | 10 000 | 0.2800 | 35 | 0.4100 | 34 | NaN | NaN |
POWELLSG | 10 000 | 0.2500 | 77 | 0.2300 | 49 | 0.7200 | 362 |
DIXON3DQ | 1000 | 0.3100 | 1002 | 0.2700 | 1002 | 0.3300 | 2005 |
BRYBND | 5000 | 0.4500 | 39 | 0.3200 | 40 | 0.3800 | 66 |
HILBERTA | 200 | 0.7100 | 50 | 0.3700 | 25 | 0.3800 | 38 |
TQUARTIC | 10 000 | 0.1900 | 61 | 0.6500 | 52 | 0.5800 | 38 |
CURLY10 | 200 | 0.2100 | 3100 | 0.2000 | 3182 | NaN | NaN |
FLETCBV2 | 500 | 0.2600 | 480 | 0.2200 | 482 | 0.3600 | 962 |
FMINSURF | 1024 | 0.1200 | 238 | 0.2400 | 300 | 0.2800 | 455 |
VARDIM | 5000 | 0.2000 | 44 | 0.1300 | 47 | NaN | NaN |
FMINSRF2 | 1024 | 0.1400 | 282 | 0.2600 | 355 | 0.2900 | 517 |
SPMSRTLS | 1000 | 0.2400 | 151 | 0.1500 | 151 | 0.2000 | 281 |
LIARWHD | 5000 | 0.2600 | 32 | 0.3000 | 48 | 0.2500 | 46 |
NONDIA | 10 000 | 0.2600 | 16 | 0.2300 | 10 | 0.3100 | 26 |
POWELLSG | 5000 | 0.5500 | 187 | 0.1100 | 53 | 0.3200 | 346 |
ARWHEAD | 10 000 | 0.1600 | 15 | 0.5300 | 12 | NaN | NaN |
SROSENBR | 10 000 | 0.1900 | 17 | 0.1700 | 19 | 0.1700 | 26 |
TQUARTIC | 5000 | 0.1700 | 38 | 0.2100 | 54 | 0.1700 | 32 |
PENALTY1 | 5000 | 0.2500 | 62 | 0.2200 | 80 | 0.3400 | 152 |
DQDRTIC | 10 000 | 0.1300 | 8 | 0.2600 | 8 | 0.2700 | 15 |
NONDIA | 5000 | 0.2200 | 22 | 0.1400 | 26 | 0.1300 | 26 |
ARGLINB | 300 | 0.1300 | 23 | 0.2000 | 17 | NaN | NaN |
DIXMAAND | 6000 | 0.2500 | 13 | 0.1300 | 12 | 0.1600 | 25 |
ARGLINC | 300 | 0.0800 | 19 | 0.2700 | 25 | NaN | NaN |
DQRTIC | 5000 | 0.0900 | 34 | 0.1000 | 34 | 0.1000 | 66 |
QUARTC | 5000 | 0.0900 | 34 | 0.0900 | 34 | 0.1000 | 66 |
EIGENALS | 110 | 0.0400 | 389 | 0.0800 | 359 | 0.1600 | 806 |
SINQUAD | 500 | 0.0800 | 111 | 0.0400 | 93 | NaN | NaN |
SPARSINE | 200 | 0.0600 | 445 | 0.0800 | 445 | 0.1300 | 917 |
DIXON3DQ | 500 | 0.2400 | 500 | 0.0600 | 500 | 0.0800 | 1003 |
DIXMAANC | 6000 | 0.2200 | 11 | 0.2400 | 11 | 0.2600 | 23 |
HILBERTB | 200 | 0.2100 | 6 | 0.2200 | 6 | 0.2500 | 13 |
BROWNAL | 400 | 0.0700 | 13 | 0.2000 | 7 | 0.2700 | 37 |
EIGENCLS | 90 | 0.2500 | 360 | 0.0700 | 350 | 0.1100 | 743 |
ARGLINA | 300 | 0.2300 | 2 | 0.2500 | 2 | 0.2600 | 5 |
EXTROSNB | 50 | 0.1300 | 5819 | 0.1900 | 5294 | 0.2400 | 7808 |
PENALTY2 | 200 | 0.1800 | 365 | 0.1400 | 417 | NaN | NaN |
FREUROTH | 1000 | 0.0700 | 187 | 0.1600 | 137 | NaN | NaN |
BRYBND | 1000 | 0.0600 | 52 | 0.0600 | 35 | 0.0800 | 73 |
DIXMAANB | 3000 | 0.0400 | 10 | 0.0600 | 10 | 0.0700 | 23 |
NONCVXU2 | 100 | 0.0600 | 396 | 0.0300 | 414 | 0.0500 | 801 |
DIXMAANA | 3000 | 0.2100 | 10 | 0.0500 | 9 | 0.0700 | 20 |
TOINTGSS | 10 000 | 0.0300 | 5 | 0.2100 | 5 | 0.3800 | 20 |
POWER | 1000 | 0.0600 | 117 | 0.0600 | 222 | 0.0400 | 236 |
DECONVU | 61 | 0.0200 | 462 | 0.0600 | 460 | 0.0700 | 581 |
GENROSE | 100 | 0.0200 | 347 | 0.0200 | 392 | 0.0300 | 626 |
COSINE | 1000 | 0.0300 | 24 | 0.0200 | 24 | 0.0300 | 29 |
DIXMAANB | 1500 | 0.0100 | 10 | 0.0300 | 10 | 0.0400 | 24 |
CHNROSNB | 50 | 0.0300 | 273 | 0.0200 | 285 | 0.0100 | 500 |
DIXMAANA | 1500 | 0.0100 | 10 | 0.0300 | 9 | 0.0300 | 22 |
FMINSRF2 | 121 | 0.0300 | 115 | 0.0100 | 124 | 0.0100 | 250 |
ARWHEAD | 1000 | 0.0100 | 16 | 0.0300 | 19 | NaN | NaN |
COSINE | 500 | 0.0200 | 23 | 0 | 22 | 0.0100 | 26 |
DQDRTIC | 1000 | 0.0600 | 8 | 0.0200 | 8 | 0.0300 | 15 |
ERRINROS | 50 | 0.0200 | 1444 | 0.0900 | 2416 | NaN | NaN |
EG2 | 1000 | 0.0100 | 6 | 0.0100 | 6 | NaN | NaN |
TESTQUAD | 100 | 0.0100 | 321 | 0.0100 | 303 | 0.0100 | 925 |
TOINTGOR | 50 | 0.8800 | 151 | 0.0100 | 155 | 0.0100 | 250 |
SPARSINE | 5000 | 0.1300 | 370 | 1.5700 | 544 | 1.1200 | 719 |
FMINSRF2 | 10 000 | 0.2800 | 26 | 0.1200 | 23 | 0.1300 | 27 |
FMINSRF2 | 15 625 | 1.1300 | 28 | 0.2600 | 23 | 0.2800 | 28 |
FMINSRF2 | 5625 | 3.0500 | 227 | 1.3100 | 214 | 1.8900 | 430 |
NONDQUAR | 10 000 | 1.3200 | 234 | 2.4200 | 225 | 3.5200 | 440 |
POWER | 10 000 | 43.7500 | 142 | 0.7100 | 62 | NaN | NaN |
ARWHEAD | 5000 | 0.2100 | 7298 | 36.8400 | 6398 | NaN | NaN |
COSINE | 5000 | 59.1900 | 37 | 0.2000 | 35 | NaN | NaN |
COSINE | 10 000 | 3.6600 | 8476 | 31.5200 | 4721 | 53.2500 | 8965 |
FMINSURF | 10 000 | 0.6400 | 8771 | 2.1400 | 5022 | 2.4100 | 6779 |
FMINSURF | 15 625 | 0.3600 | 108 | 0.4700 | 62 | NaN | NaN |
BROYDN7D | 1000 | 5.3900 | 498 | 0.2700 | 371 | NaN | NaN |
SPMSRTLS | 4999 | 0.0010 | 2232 | 5.4700 | 2183 | 6.4500 | 4093 |
SPMSRTLS | 10 000 | 0.0010 | NaN | NaN | NaN | 0.2800 | NaN |
FREUROTH | 10 000 | 0.0010 | NaN | NaN | NaN | 1.8900 | NaN |
FLETCBV2 | 500 | 0.0010 | NaN | NaN | NaN | 3.5200 | NaN |
BDQRTIC | 10 000 | 0.0010 | 1 | NaN | NaN | 0.2800 | NaN |
VAREIGVL | 10 000 | 0.0010 | 1 | NaN | NaN | 1.8900 | NaN |
ENGVAL1 | 5000 | NaN | 1 | NaN | NaN | 3.5200 | NaN |
BRYBND | 10 000 | 0.1000 | 34677 | 0.5000 | 456454 | 0.9000 | 40000 |
EIGENBLS | 930 | 0.1000 | 15122 | 0.0500 | 15401 | 0.9000 | NaN |
NONCVXUN | 500 | 0.1000 | 15084 | 0.0500 | 15797 | 0.9000 | NaN |
GENROSE | 1000 | 0.1000 | 2661 | 0.5000 | 2261 | 0.9000 | 4720 |
GENROSE | 5000 | 0.1000 | 4978 | 0.0500 | 5440 | 0.9000 | 9714 |
EIGENALS | 930 | 0.1000 | 1116 | 0.0500 | 1116 | 0.9000 | 2231 |
SINQUAD | 5000 | 0.1000 | 5099 | 0.5000 | 5058 | 0.9000 | 10058 |
SINQUAD | 10 000 | 0.1000 | 14406 | 0.0500 | 13659 | 0.9000 | NaN |
GENHUMPS | 5000 | 0.1000 | 1802 | 0.0500 | 1883 | 0.9000 | 3312 |
CHAINWOO | 1000 | 0.1000 | 4516 | 0.5000 | 4483 | 0.9000 | 8793 |
TESTQUAD | 1000 | 0.1000 | 1344 | 0.0500 | 1306 | 0.9000 | 2998 |
TESTQUAD | 10 000 | 0.1000 | 7479 | 0.0500 | 9139 | 0.9000 | 8986 |
TESTQUAD | 5000 | 0.1000 | 3555 | 0.5000 | 3435 | 0.9000 | 5807 |
FLETCHCR | 5000 | 0.1000 | 492 | 0.0500 | 669 | 0.9000 | 949 |
CURLY30 | 1000 | 0.1000 | 783 | 0.0500 | 783 | 0.9000 | 1565 |
CURLY20 | 1000 | 0.1000 | 303 | NaN | 306 | 0.9000 | 620 |
DIXMAANI | 6000 | 0.1000 | 296 | NaN | 275 | 0.9000 | 557 |
EIGENBLS | 420 | 0.1000 | 8726 | NaN | 2428 | 0.9000 | NaN |
[1] |
Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020176 |
[2] |
Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115 |
[3] |
Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328 |
[4] |
M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 |
[5] |
Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122 |
[6] |
Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100 |
[7] |
Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 |
[8] |
Hedy Attouch, Aïcha Balhag, Zaki Chbani, Hassan Riahi. Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021010 |
[9] |
Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139 |
[10] |
Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077 |
[11] |
Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140 |
[12] |
George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 |
[13] |
Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 |
[14] |
Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 |
[15] |
Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 |
[16] |
Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020296 |
[17] |
Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020465 |
[18] |
Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020367 |
[19] |
Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 |
[20] |
Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]