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

* Corresponding author

Received  January 2020 Revised  April 2020 Published  May 2020

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.

Citation: Sarra Delladji, Mohammed Belloufi, Badreddine Sellami. Behavior of the combination of PRP and HZ methods for unconstrained optimization. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020032
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.  Google Scholar

[2]

N. Andrei, Another hybrid conjugate gradient algorithm for unconstrained optimization, Numerical Algorithms, 47 (2008), 143-156.  doi: 10.1007/s11075-007-9152-9.  Google Scholar

[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.  Google Scholar

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B. BalaramM. 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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

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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.  Google Scholar

[11]

X. Z. JiangG.-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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Z. WeiG. 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.  Google Scholar

[17]

G. YuanZ. 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.  Google Scholar

[18]

G. Zoutendijk, Nonlinear programming, computational methods, Integer and Nonlinear Programming, (1970), 37–86.  Google Scholar

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.  Google Scholar

[2]

N. Andrei, Another hybrid conjugate gradient algorithm for unconstrained optimization, Numerical Algorithms, 47 (2008), 143-156.  doi: 10.1007/s11075-007-9152-9.  Google Scholar

[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.  Google Scholar

[4]

B. BalaramM. 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.  Google Scholar

[5]

I. BongartzA. R. ConnN. 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.  Google Scholar

[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.  Google Scholar

[9]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[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.  Google Scholar

[11]

X. Z. JiangG.-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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Z. WeiG. 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.  Google Scholar

[17]

G. YuanZ. 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.  Google Scholar

[18]

G. Zoutendijk, Nonlinear programming, computational methods, Integer and Nonlinear Programming, (1970), 37–86.  Google Scholar

Table 1.   
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
BRYBND50000.4500390.3200400.380066
HILBERTA2000.7100500.3700250.380038
TQUARTIC10 0000.1900610.6500520.580038
CURLY102000.210031000.20003182NaNNaN
FLETCBV25000.26004800.22004820.3600962
FMINSURF10240.12002380.24003000.2800455
VARDIM50000.2000440.130047NaNNaN
FMINSRF210240.14002820.26003550.2900517
SPMSRTLS10000.24001510.15001510.2000281
LIARWHD50000.2600320.3000480.250046
NONDIA10 0000.2600160.2300100.310026
POWELLSG50000.55001870.1100530.3200346
ARWHEAD10 0000.1600150.530012NaNNaN
SROSENBR10 0000.1900170.1700190.170026
TQUARTIC50000.1700380.2100540.170032
PENALTY150000.2500620.2200800.3400152
DQDRTIC10 0000.130080.260080.270015
NONDIA50000.2200220.1400260.130026
ARGLINB3000.1300230.200017NaNNaN
DIXMAAND60000.2500130.1300120.160025
ARGLINC3000.0800190.270025NaNNaN
DQRTIC50000.0900340.1000340.100066
QUARTC50000.0900340.0900340.100066
EIGENALS1100.04003890.08003590.1600806
SINQUAD5000.08001110.040093NaNNaN
SPARSINE2000.06004450.08004450.1300917
DIXON3DQ5000.24005000.06005000.08001003
DIXMAANC60000.2200110.2400110.260023
HILBERTB2000.210060.220060.250013
BROWNAL4000.0700130.200070.270037
EIGENCLS900.25003600.07003500.1100743
ARGLINA3000.230020.250020.26005
EXTROSNB500.130058190.190052940.24007808
PENALTY22000.18003650.1400417NaNNaN
FREUROTH10000.07001870.1600137NaNNaN
BRYBND10000.0600520.0600350.080073
DIXMAANB30000.0400100.0600100.070023
NONCVXU21000.06003960.03004140.0500801
DIXMAANA30000.2100100.050090.070020
TOINTGSS10 0000.030050.210050.380020
POWER10000.06001170.06002220.0400236
DECONVU610.02004620.06004600.0700581
GENROSE1000.02003470.02003920.0300626
COSINE10000.0300240.0200240.030029
DIXMAANB15000.0100100.0300100.040024
CHNROSNB500.03002730.02002850.0100500
DIXMAANA15000.0100100.030090.030022
FMINSRF21210.03001150.01001240.0100250
ARWHEAD10000.0100160.030019NaNNaN
COSINE5000.0200230220.010026
DQDRTIC10000.060080.020080.030015
ERRINROS500.020014440.09002416NaNNaN
EG210000.010060.01006NaNNaN
TESTQUAD1000.01003210.01003030.0100925
TOINTGOR500.88001510.01001550.0100250
SPARSINE50000.13003701.57005441.1200719
FMINSRF210 0000.2800260.1200230.130027
FMINSRF215 6251.1300280.2600230.280028
FMINSRF256253.05002271.31002141.8900430
NONDQUAR10 0001.32002342.42002253.5200440
POWER10 00043.75001420.710062NaNNaN
ARWHEAD50000.2100729836.84006398NaNNaN
COSINE500059.1900370.200035NaNNaN
COSINE10 0003.6600847631.5200472153.25008965
FMINSURF10 0000.640087712.140050222.41006779
FMINSURF15 6250.36001080.470062NaNNaN
BROYDN7D10005.39004980.2700371NaNNaN
SPMSRTLS49990.001022325.470021836.45004093
SPMSRTLS10 0000.0010NaNNaNNaN0.2800NaN
FREUROTH10 0000.0010NaNNaNNaN1.8900NaN
FLETCBV25000.0010NaNNaNNaN3.5200NaN
BDQRTIC10 0000.00101NaNNaN0.2800NaN
VAREIGVL10 0000.00101NaNNaN1.8900NaN
ENGVAL15000NaN1NaNNaN3.5200NaN
BRYBND10 0000.1000346770.50004564540.900040000
EIGENBLS9300.1000151220.0500154010.9000NaN
NONCVXUN5000.1000150840.0500157970.9000NaN
GENROSE10000.100026610.500022610.90004720
GENROSE50000.100049780.050054400.90009714
EIGENALS9300.100011160.050011160.90002231
SINQUAD50000.100050990.500050580.900010058
SINQUAD10 0000.1000144060.0500136590.9000NaN
GENHUMPS50000.100018020.050018830.90003312
CHAINWOO10000.100045160.500044830.90008793
TESTQUAD10000.100013440.050013060.90002998
TESTQUAD10 0000.100074790.050091390.90008986
TESTQUAD50000.100035550.500034350.90005807
FLETCHCR50000.10004920.05006690.9000949
CURLY3010000.10007830.05007830.90001565
CURLY2010000.1000303NaN3060.9000620
DIXMAANI60000.1000296NaN2750.9000557
EIGENBLS4200.10008726NaN24280.9000NaN
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
BRYBND50000.4500390.3200400.380066
HILBERTA2000.7100500.3700250.380038
TQUARTIC10 0000.1900610.6500520.580038
CURLY102000.210031000.20003182NaNNaN
FLETCBV25000.26004800.22004820.3600962
FMINSURF10240.12002380.24003000.2800455
VARDIM50000.2000440.130047NaNNaN
FMINSRF210240.14002820.26003550.2900517
SPMSRTLS10000.24001510.15001510.2000281
LIARWHD50000.2600320.3000480.250046
NONDIA10 0000.2600160.2300100.310026
POWELLSG50000.55001870.1100530.3200346
ARWHEAD10 0000.1600150.530012NaNNaN
SROSENBR10 0000.1900170.1700190.170026
TQUARTIC50000.1700380.2100540.170032
PENALTY150000.2500620.2200800.3400152
DQDRTIC10 0000.130080.260080.270015
NONDIA50000.2200220.1400260.130026
ARGLINB3000.1300230.200017NaNNaN
DIXMAAND60000.2500130.1300120.160025
ARGLINC3000.0800190.270025NaNNaN
DQRTIC50000.0900340.1000340.100066
QUARTC50000.0900340.0900340.100066
EIGENALS1100.04003890.08003590.1600806
SINQUAD5000.08001110.040093NaNNaN
SPARSINE2000.06004450.08004450.1300917
DIXON3DQ5000.24005000.06005000.08001003
DIXMAANC60000.2200110.2400110.260023
HILBERTB2000.210060.220060.250013
BROWNAL4000.0700130.200070.270037
EIGENCLS900.25003600.07003500.1100743
ARGLINA3000.230020.250020.26005
EXTROSNB500.130058190.190052940.24007808
PENALTY22000.18003650.1400417NaNNaN
FREUROTH10000.07001870.1600137NaNNaN
BRYBND10000.0600520.0600350.080073
DIXMAANB30000.0400100.0600100.070023
NONCVXU21000.06003960.03004140.0500801
DIXMAANA30000.2100100.050090.070020
TOINTGSS10 0000.030050.210050.380020
POWER10000.06001170.06002220.0400236
DECONVU610.02004620.06004600.0700581
GENROSE1000.02003470.02003920.0300626
COSINE10000.0300240.0200240.030029
DIXMAANB15000.0100100.0300100.040024
CHNROSNB500.03002730.02002850.0100500
DIXMAANA15000.0100100.030090.030022
FMINSRF21210.03001150.01001240.0100250
ARWHEAD10000.0100160.030019NaNNaN
COSINE5000.0200230220.010026
DQDRTIC10000.060080.020080.030015
ERRINROS500.020014440.09002416NaNNaN
EG210000.010060.01006NaNNaN
TESTQUAD1000.01003210.01003030.0100925
TOINTGOR500.88001510.01001550.0100250
SPARSINE50000.13003701.57005441.1200719
FMINSRF210 0000.2800260.1200230.130027
FMINSRF215 6251.1300280.2600230.280028
FMINSRF256253.05002271.31002141.8900430
NONDQUAR10 0001.32002342.42002253.5200440
POWER10 00043.75001420.710062NaNNaN
ARWHEAD50000.2100729836.84006398NaNNaN
COSINE500059.1900370.200035NaNNaN
COSINE10 0003.6600847631.5200472153.25008965
FMINSURF10 0000.640087712.140050222.41006779
FMINSURF15 6250.36001080.470062NaNNaN
BROYDN7D10005.39004980.2700371NaNNaN
SPMSRTLS49990.001022325.470021836.45004093
SPMSRTLS10 0000.0010NaNNaNNaN0.2800NaN
FREUROTH10 0000.0010NaNNaNNaN1.8900NaN
FLETCBV25000.0010NaNNaNNaN3.5200NaN
BDQRTIC10 0000.00101NaNNaN0.2800NaN
VAREIGVL10 0000.00101NaNNaN1.8900NaN
ENGVAL15000NaN1NaNNaN3.5200NaN
BRYBND10 0000.1000346770.50004564540.900040000
EIGENBLS9300.1000151220.0500154010.9000NaN
NONCVXUN5000.1000150840.0500157970.9000NaN
GENROSE10000.100026610.500022610.90004720
GENROSE50000.100049780.050054400.90009714
EIGENALS9300.100011160.050011160.90002231
SINQUAD50000.100050990.500050580.900010058
SINQUAD10 0000.1000144060.0500136590.9000NaN
GENHUMPS50000.100018020.050018830.90003312
CHAINWOO10000.100045160.500044830.90008793
TESTQUAD10000.100013440.050013060.90002998
TESTQUAD10 0000.100074790.050091390.90008986
TESTQUAD50000.100035550.500034350.90005807
FLETCHCR50000.10004920.05006690.9000949
CURLY3010000.10007830.05007830.90001565
CURLY2010000.1000303NaN3060.9000620
DIXMAANI60000.1000296NaN2750.9000557
EIGENBLS4200.10008726NaN24280.9000NaN
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El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883

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