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September
2021, 11(3): 377-389.
doi: 10.3934/naco.2020032
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.
Keywords:
Unconstrained optimization,
hybrid conjugate gradient,
convex combination,
sufficient descent,
global convergence.
Mathematics Subject Classification:
49M07; 49M10; 90C06; 65K05; 90C26; 65H10.
Citation:
Sarra Delladji, Mohammed Belloufi, Badreddine Sellami. Behavior of the combination of PRP and HZ methods for unconstrained optimization. Numerical Algebra, Control and Optimization,
2021, 11
(3)
: 377-389.
doi: 10.3934/naco.2020032
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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.
|
| [6] |
N. Chenna,
Comments on "New Hybrid Conjugate Gradient Method as a Convex Combination of FR and PRP Methods", FILOMAT, 33 (2019), 3083-3100.
|
| [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.
|
| [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.
|
| [6] |
N. Chenna,
Comments on "New Hybrid Conjugate Gradient Method as a Convex Combination of FR and PRP Methods", FILOMAT, 33 (2019), 3083-3100.
|
| [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.
|
| [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 |
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