Research Article
Global Convergence of a Spectral Conjugate Gradient Method for Unconstrained Optimization
Table 1
The numerical results of SCD method, CD method, FR method, and PRP method.
| | Problem | Dim | SCD method | CD method | FR method | PRP method |
| | Extended Freudenstein and Roth | 1000 | 815/0.58 | 947/1.20 | 499/0.56 | 13/0.02 | | 3000 | 1450/3.26 | 1677/6.35 | 406/1.50 | 1542/6.10 | | 5000 | 1447/6.22 | 1700/10.92 | 537/3.18 | 1566/10.26 | | Extended trigonometric | 5000 | 32/0.03 | 817/1.13 | 342/0.25 | 73/0.07 | | 3000 | 31/0.08 | 119/0.22 | 284/0.56 | 87/0.19 | | 5000 | 36/0.16 | 310/0.97 | 346/1.10 | 38/0.15 | | Extended Beale | 1000 | 24/0.00 | 44/0.00 | 24/0.00 | 13/0.02 | | 3000 | 33/0.02 | 17/0.02 | 24/0.00 | 13/0.00 | | 5000 | 30/0.03 | 43/0.05 | 20/0.04 | 19/0.02 | | Extended penalty | 1000 | 65/0.05 | 29/0.01 | 16/0.00 | 96/0.11 | | 3000 | 10/0.00 | 16/0.00 | 15/0.00 | 14/0.00 | | 5000 | 11/0.02 | 22/0.05 | 35/0.13 | 12/0.01 | | Perturbed quadratic | 1000 | 334/0.05 | 1847/0.26 | 1088/0.17 | 407/0.08 | | 3000 | 679/0.32 | 1736/0.74 | 3191/1.27 | 705/0.31 | | 5000 | 922/0.68 | 1494/1.02 | 2830/1.84 | 1081/0.80 | | Raydan 1 | 1000 | 331/0.12 | 680/0.25 | 799/0.25 | 450/0.18 | | 3000 | 601/0.77 | 1542/1.57 | 1044/0.94 | 760/0.94 | | 5000 | 1229/2.35 | 1857/3.04 | 4192/6.34 | 1058/2.16 | | Raydan 2 | 1000 | 4/0.00 | 4/0.00 | 4/0.00 | 4/0.00 | | 3000 | 4/0.01 | 4/0.02 | 4/0.02 | 4/0.00 | | 5000 | 4/0.02 | 4/0.01 | 4/0.01 | 4/0.02 | | Hager | 1000 | 434/4.23 | 699/2.53 | 1450/16.01 | 183/1.48 | | 3000 | 1404/47.23 | 1591/33.20 | 2193/74.72 | 1019/34.11 | | 5000 | 3163/171.28 | 946/39.49 | 4332/234.59 | 2583/142.81 | | Generalized tridiagonal 1 | 1000 | 26/0.00 | 72/0.07 | 87/0.05 | 43/0.04 | | 3000 | 55/0.14 | 59/0.22 | 167/0.62 | 62/0.20 | | 5000 | 89/0.56 | 61/0.31 | 38/0.09 | 34/0.11 | | Extended tridiagonal 1 | 1000 | 34/0.00 | 15/0.00 | 86/0.02 | 14/0.00 | | 3000 | 10/0.02 | 67/0.03 | 22/0.01 | 12/0.00 | | 5000 | 9/0.02 | 79/0.06 | 16/0.01 | 16/0.00 | | Extended three expo terms | 1000 | 14/0.01 | 97/1.71 | 23/0.08 | 8/0.00 | | 3000 | 15/0.05 | 86/4.27 | 49/1.72 | 8/0.03 | | 5000 | 64/4.88 | 123/11.48 | 247/23.04 | 8/0.06 | | Generalized tridiagonal 2 | 1000 | 67/0.01 | 370/0.07 | 234/0.05 | 58/0.00 | | 3000 | 47/0.04 | 770/0.40 | 281/0.18 | 54/0.04 | | 5000 | 62/0.04 | 261/0.22 | 139/0.11 | 60/0.06 | | Diagonal 4 | 1000 | 7/0.02 | 4/0.00 | 6/0.00 | 4/0.00 | | 3000 | 4/0.00 | 4/0.00 | 6/0.02 | 4/0.00 | | 5000 | 9/0.00 | 6/0.02 | 7/0.00 | 4/0.00 | | Diagonal 5 | 1000 | 4/0.01 | 4/0.00 | 4/0.01 | 4/0.02 | | 3000 | 4/0.02 | 4/0.04 | 4/0.01 | 4/0.01 | | 5000 | 4/0.03 | 4/0.05 | 4/0.04 | 4/0.03 | | Extended Himmelblau | 1000 | 10/0.00 | 17/0.00 | 17/0.00 | 23/0.02 | | 3000 | 11/0.00 | 131/0.06 | 19/0.01 | 24/0.02 | | 5000 | 12/0.02 | 103/0.07 | 20/0.01 | 24/0.02 | | Extended PSC1 | 1000 | 12/0.02 | 40/0.28 | 17/0.04 | 11/0.00 | | 3000 | 11/0.03 | 51/1.18 | 91/2.36 | 18/0.22 | | 5000 | 9/0.03 | 67/2.75 | 39/1.28 | 11/0.03 | | Extended block-diagonal BD1 | 1000 | 67/0.02 | 71/0.02 | 32/0.02 | 26/0.01 | | 3000 | 69/0.06 | 24/0.03 | 29/0.03 | 31/0.07 | | 5000 | 71/0.13 | 23/0.05 | 35/0.06 | 28/0.05 | | Extended Maratos | 1000 | 981/0.14 | 227/0.03 | 617/0.08 | 59/0.00 | | 3000 | 781/0.35 | 313/0.12 | 672/0.27 | 55/0.02 | | 5000 | 715/0.55 | 262/0.17 | 690/0.45 | 49/0.05 | | Extended Cliff | 1000 | 41/0.01 | 48/0.03 | 154/0.58 | 27/0.01 | | 3000 | 51/0.09 | 303/2.22 | 206/4.23 | 39/0.06 | | 5000 | 17/0.04 | 61/1.08 | 113/2.89 | 20/0.06 | | Quadratic diagonal perturbed | 1000 | 234/0.03 | 456/0.08 | 499/0.08 | 400/0.06 | | 3000 | 978/0.45 | 2999/1.57 | 702/0.36 | 817/0.46 | | 5000 | 807/0.64 | 1157/1.03 | 1092/0.91 | 1023/0.89 | | Extended Wood | 1000 | 661/0.13 | 68/0.02 | 96/0.02 | 107/0.03 | | 3000 | 491/0.25 | 113/0.06 | 60/0.03 | 150/0.08 | | 5000 | 514/0.42 | 93/0.08 | 67/0.05 | 206/0.17 | | Quadratic QF1 | 1000 | 344/0.05 | 1120/0.14 | 949/0.11 | 363/0.05 | | 3000 | 607/0.25 | 1677/0.64 | 3576/1.29 | 731/0.28 | | 5000 | 1038/0.72 | 1606/1.04 | 2467/1.47 | 1076/0.72 | | Extended quadratic enalty QP2 | 1000 | 521/0.36 | 98/0.06 | 2476/1.17 | 28/0.01 | | 3000 | 813/1.65 | 191/0.30 | 251/0.37 | 40/0.10 | | 5000 | 561/1.89 | 156/0.39 | 242/0.61 | 35/0.14 | | Quadratic QF2 | 1000 | 1188/0.19 | 1469/0.20 | 1679/0.21 | 433/0.06 | | 3000 | 2949/1.34 | 1867/0.75 | 2952/1.13 | 929/0.40 | | 5000 | 4161/3.25 | 2709/1.78 | 3840/2.39 | 1236/0.08 | | Extended EP1 | 1000 | 2/V0 | 2/0.00 | 2/0.00 | 2/0.02 | | 3000 | 3/0.00 | 3/0.00 | 3/0.00 | 3/0.00 | | 5000 | 3/0.00 | 3/0.00 | 3/0.00 | 3/0.00 | | Extended tridiagonal 2 | 1000 | 41/0.00 | 137/0.10 | 78/0.01 | 36/0.00 | | 3000 | 179/0.74 | 865/1.86 | 236/0.99 | 109/0.44 | | 5000 | 125/0.76 | 381/2.75 | 335/2.47 | 202/1.58 | | ARWHEAD | 1000 | 5/0.00 | 58/0.03 | 44/0.01 | 5/0.02 | | 3000 | 7/0.00 | 81/0.11 | 31/0.03 | 15/0.04 | | 5000 | 13/0.03 | 50/0.18 | 54/0.16 | 31/0.14 | | NONDIA | 1000 | 16/0.00 | 47/0.02 | 50/0.00 | 10/0.01 | | 3000 | 16/0.00 | 10/0.02 | 11/0.00 | 12/0.00 | | 5000 | 11/0.02 | 11/0.01 | 14/0.00 | 13/0.01 | | DQDRTIC | 1000 | 33/0.00 | 30/0.00 | 7/0.00 | 16/0.00 | | 3000 | 41/0.02 | 10/0.00 | 7/0.00 | 11/0.00 | | 5000 | 35/0.02 | 34/0.02 | 7/0.00 | 9/0.02 | | DIXMAANA | 1000 | 7/0.02 | 11/0.03 | 12/0.03 | 9/0.03 | | 3000 | 8/0.02 | 11/0.01 | 13/0.02 | 9/0.01 | | 5000 | 9/0.02 | 11/0.02 | 13/0.01 | 9/0.01 | | DIXMAANB | 1000 | 11/0.00 | 12/0.00 | 12/0.02 | 12/0.02 | | 3000 | 12/0.00 | 12/0.02 | 12/0.01 | 12/0.02 | | 5000 | 12/0.03 | 12/0.01 | 12/0.03 | 13/0.03 | | DIXMAANC | 1000 | 14/0.00 | 14/0.00 | 16/0.00 | 15/0.00 | | 3000 | 15/0.02 | 17/0.02 | 17/0.01 | 16/0.01 | | 5000 | 15/0.01 | 16/0.01 | 17/0.03 | 16/0.04 | | DIXMAANE | 1000 | 273/0.09 | 1021/0.31 | 579/0.16 | 246/0.07 | | 3000 | 521/0.80 | 756/0.70 | 712/0.55 | 484/0.47 | | 5000 | 680/1.44 | 1066/1.50 | 1013/1.28 | 666/1.09 | | Partial perturbed quadratic PPQ1 | 1000 | 371/3.19 | 707/5.31 | 664/3.86 | 450/4.08 | | 3000 | 370/30.39 | 543/45.59 | 802/49.48 | 517/44.19 | | 5000 | 287/65.78 | 523/113.13 | 787/133.55 | 276/63.87 | | Broyden tridiagonal | 1000 | 41/0.00 | 346/0.08 | 2167/0.34 | 45/0.02 | | 3000 | 88/0.05 | 429/0.20 | 497/0.22 | 86/0.05 | | 5000 | 86/0.08 | 734/0.58 | 344/0.26 | 81/0.06 | | Almost perturbed quadratic | 1000 | 410/0.06 | 809/0.11 | 990/0.13 | 384/0.06 | | 3000 | 708/0.31 | 1436/0.57 | 2255/0.85 | 768/0.31 | | 5000 | 848/0.61 | 1770/1.19 | 2400/1.49 | 967/0.66 | | Tridiagonal perturbed quadratic | 1000 | 351/0.06 | 1028/0.15 | 1265/0.19 | 358/0.06 | | 3000 | 663/0.33 | 1330/0.59 | 2074/0.87 | 549/0.25 | | 5000 | 987/0.95 | 1667/1.22 | 3034/2.11 | 1260/0.97 | | EDENSCH | 1000 | 162/0.28 | 261/0.25 | 128/0.21 | 93/0.16 | | 3000 | 45/0.11 | 178/0.89 | 145/0.72 | 47/0.14 | | 5000 | 65/0.42 | 318/1.32 | 164/1.38 | 54/0.25 | | VARDIM | 1000 | 16/0.02 | 16/0.01 | 16/0.02 | 16/0.00 | | 3000 | 19/0.01 | 19/0.02 | 19/0.01 | 19/0.01 | | 5000 | 14/0.01 | 14/0.02 | 14/0.01 | 14/0.01 | | Diagonal 6 | 1000 | 4/0.00 | 4/0.00 | 4/0.02 | 4/0.01 | | 3000 | 4/0.02 | 4/0.02 | 4/0.01 | 4/0.02 | | 5000 | 4/0.02 | 4/0.00 | 4/0.02 | 4/0.01 | | DIXMAANF | 1000 | 279/0.16 | 587/0.23 | 338/0.12 | 239/0.11 | | 3000 | 673/1.44 | 869/0.83 | 599/0.46 | 348/0.36 | | 5000 | 742/2.45 | 692/1.01 | 2588/6.44 | 584/0.95 | | DIXMAANG | 1000 | 293/0.12 | 673/0.29 | 368/0.11 | 257/0.08 | | 3000 | 989/1.48 | 1449/1.35 | 739/0.62 | 448/0.47 | | 5000 | 507/0.97 | 1671/3.17 | 997/1.39 | 742/1.29 | | DIXMAANH | 1000 | 272/0.11 | 632/1.41 | 1010/0.28 | 247/0.10 | | 3000 | 733/1.43 | 653/0.64 | 513/0.41 | 1771/29.08 | | 5000 | 898/3.06 | 604/1.44 | 1296/1.74 | 588/2.11 | | DIXMAANI | 1000 | 231/0.08 | 876/0.25 | 426/0.11 | 268/0.09 | | 3000 | 439/0.50 | 1261/1.06 | 628/0.50 | 542/0.52 | | 5000 | 468/0.81 | 1071/1.59 | 943/1.22 | 654/1.05 | | DIXMAANJ | 1000 | 266/0.13 | 488/0.14 | 362/0.11 | 281/0.09 | | 3000 | 768/1.56 | 720/3.23 | 516/0.41 | 385/0.39 | | 5000 | 471/0.83 | 1250/2.15 | 1589/2.08 | 547/1.64 | | DIXMAANK | 1000 | 260/0.11 | 1036/0.36 | 500/0.14 | 278/0.11 | | 3000 | 420/0.47 | 1103/1.13 | 774/0.62 | 428/0.42 | | 5000 | 620/1.59 | 1386/2.66 | 784/1.06 | 620/1.04 | | ENGVAL1 | 1000 | 32/0.00 | 242/0.14 | 268/0.38 | 53/0.03 | | 3000 | 80/0.28 | 366/1.06 | 950/4.67 | 229/1.05 | | 5000 | 340/2.56 | 267/1.64 | 325/2.37 | 181/1.30 | | ENSCHNB | 1000 | 10/0.00 | 9/0.00 | 9/0.00 | 9/0.00 | | 3000 | 8/0.01 | 9/0.01 | 9/0.02 | 9/0.01 | | 5000 | 9/0.02 | 9/0.02 | 9/0.02 | 8/0.00 | | ENSCHNF | 1000 | 27/0.00 | 45/0.00 | 1206/1.51 | 23/0.00 | | 3000 | 28/0.02 | 85/0.06 | 487/1.33 | 26/0.04 | | 5000 | 27/0.04 | 76/0.07 | 158/0.23 | 24/0.03 |
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