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Behavior of the combination of PRP and HZ methods for unconstrained optimization

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  • 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.

    Mathematics Subject Classification: 49M07; 49M10; 90C06; 65K05; 90C26; 65H10.

    Citation:

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  • Figure 1.   

    Figure 2.   

    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
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