Research Article
Two New Conjugate Gradient Methods for Unconstrained Optimization
Table 1
Numerical results for the first group methods.
| | No. | Problems | LMYCD1 | hDY | NHS | JMJ | MDL1 | MDL2 | | Name/n | Itr/NF/NG/Tcpu | Itr/NF/NG/Tcpu | Itr/NF/NG/Tcpu | Itr/NF/NG/Tcpu | Itr/NF/NG/Tcpu | Itr/NF/NG/Tcpu |
| | 1 | cosine 200 | 16/168/98/0.035 | 103/1624/853/0.140 | 23/266/141/0.037 | 23/267/146/0.023 | 21/298/154/0.022 | 21/263/140/0.020 | | 2 | cosine 1000 | 19/198/103/0.037 | NaN/NaN/NaN/NaN | 25/362/180/0.086 | 24/327/168/0.075 | 21/264/151/0.051 | 24/329/166/0.062 | | 3 | cosine 10000 | 20/198/109/0.333 | NaN/NaN/NaN/NaN | 25/396/196/0.586 | 27/393/219/0.634 | 20/201/106/0.284 | 22/165/84/0.251 | | 4 | cosine 100000 | 20/198/109/2.425 | NaN/NaN/NaN/NaN | 55/1352/651/16.743 | 27/365/193/4.301 | 21/164/94/2.135 | 25/202/110/2.653 | | 5 | dixmaanj 3000 | 712/8825/4742/24.677 | 753/10019/5290/27.130 | 819/10004/5376/27.108 | 588/8189/4271/21.668 | NaN/NaN/NaN/NaN | 750/9567/5102/25.574 | | 6 | dixmaank 3000 | 558/8089/4042/21.231 | 706/9188/4868/24.756 | 869/11340/6028/30.712 | 533/6368/3414/17.636 | 1146/17072/8887/45.486 | 792/9774/5205/26.502 | | 7 | dixmaanl 3000 | 533/6689/3598/18.359 | 513/7035/3766/19.059 | 836/10716/5808/29.109 | 466/5473/2934/14.674 | 679/11438/5517/29.298 | 478/5633/3035/15.046 | | 8 | dixon3dq 20 | 200/2293/1258/0.157 | 171/2270/1204/0.117 | 236/2750/1492/0.134 | 213/2865/1493/0.134 | 189/2710/1434/0.138 | 172/2280/1184/0.122 | | 9 | dixon3dq 100 | 1013/12988/6889/0.665 | 1088/13497/7144/0.655 | NaN/NaN/NaN/NaN | 975/12564/6605/0.603 | 1168/16229/8469/0.788 | 1265/16115/8600/0.792 | | 10 | dqrtic 10 | 12/29/32/0.002 | 12/29/32/0.002 | 12/29/32/0.002 | 12/29/32/0.003 | 11/28/30/0.002 | 12/29/32/0.002 | | 11 | dqdrtic 10000 | 60/695/345/0.421 | 100/1409/717/0.692 | 61/723/395/0.385 | 90/1231/666/0.605 | 74/841/438/0.494 | 77/1206/632/0.568 | | 12 | dqdrtic 100000 | 68/713/392/2.755 | 73/954/491/3.383 | 86/970/527/3.722 | 80/996/517/3.817 | NaN/NaN/NaN/NaN | 69/1068/527/3.768 | | 13 | dqdrtic 1000000 | 58/618/358/25.828 | 87/1409/727/54.533 | 110/2431/1025/89.848 | 107/1590/815/63.473 | 83/1038/534/41.895 | 68/916/463/38.205 | | 14 | edensch 50 | 29/275/160/0.019 | 48/697/341/0.043 | 27/213/128/0.021 | 31/303/169/0.020 | 43/660/331/0.041 | 31/310/160/0.024 | | 15 | edensch 1000 | 37/531/270/0.268 | 37/467/254/0.260 | 33/435/239/0.277 | 28/274/146/0.138 | 37/410/225/0.240 | 37/564/281/0.295 | | 16 | edensch 10000 | 33/370/214/1.936 | 51/815/412/3.951 | 35/441/250/2.182 | 36/483/265/2.438 | 42/607/313/2.902 | 30/312/151/1.647 | | 17 | fletchcr 110 | 36/491/251/0.024 | 39/364/205/0.018 | 49/581/320/0.028 | 49/680/313/0.033 | 44/580/304/0.031 | NaN/NaN/NaN/NaN | | 18 | freuroth 5 | 121/1831/964/0.074 | NaN/NaN/NaN/NaN | 335/7995/3349/0.436 | 145/2101/1145/0.101 | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | | 19 | genrose 1400 | 196/2510/1309/0.266 | 204/2531/1337/0.342 | 210/2563/1370/0.348 | NaN/NaN/NaN/NaN | 147/2332/1195/0.246 | 175/2517/1319/0.337 | | 20 | genrose 500000 | 97/1198/625/37.852 | 98/1320/693/41.850 | 215/2405/1318/78.872 | 158/1644/909/53.339 | 112/1162/621/40.218 | 108/1185/648/38.429 | | 21 | genrose 1000000 | 122/1639/881/105.302 | 156/2310/1194/144.932 | 154/2264/1202/140.419 | NaN/NaN/NaN/NaN | 133/1882/989/117.067 | NaN/NaN/NaN/NaN | | 22 | genrose 1500000 | 127/1299/702/127.953 | 151/1740/932/170.937 | 186/2209/1182/209.864 | NaN/NaN/NaN/NaN | 101/1438/751/132.774 | 106/1259/685/118.715 | | 23 | himmelbg 50 | 8/21/28/0.003 | 8/23/30/0.002 | 8/23/30/0.002 | 8/23/30/0.002 | 8/22/29/0.002 | 8/23/30/0.002 | | 24 | himmelbg 5000 | 9/25/33/0.025 | 9/26/34/0.026 | 9/26/34/0.026 | 9/26/34/0.033 | 9/25/33/0.032 | 9/26/34/0.036 | | 25 | himmelbg 20000 | 9/25/33/0.112 | 9/26/34/0.112 | 9/26/34/0.108 | 9/26/34/0.115 | 9/25/33/0.097 | 9/26/34/0.122 | | 26 | himmelbg 500000 | 10/28/37/2.707 | 10/29/38/2.792 | 10/29/38/2.970 | 10/29/38/3.071 | 10/28/37/2.961 | 10/29/38/2.881 | | 27 | mccormak 2 | 18/167/85/0.010 | NaN/NaN/NaN/NaN | 18/189/96/0.012 | NaN/NaN/NaN/NaN | 28/232/127/0.017 | NaN/NaN/NaN/NaN | | 28 | nondquar 5 | 70/725/406/0.070 | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | | 29 | nonscomp 2 | 34/445/227/0.026 | 43/682/343/0.030 | 61/1032/475/0.051 | 53/940/497/0.043 | 47/833/413/0.036 | NaN/NaN/NaN/NaN | | 30 | penalty1 10 | 9/10/13/0.006 | 6/10/10/0.001 | 6/10/10/0.001 | 6/10/10/0.001 | 6/9/9/0.001 | 6/10/10/0.001 | | 31 | penalty1 500 | 7/22/17/0.032 | 6/22/16/0.037 | 5/22/15/0.031 | 5/22/15/0.032 | 5/21/14/0.027 | 5/22/15/0.027 | | 32 | penalty1 1500 | 8/24/19/0.232 | 11/24/22/0.274 | 10/24/21/0.262 | 10/24/21/0.241 | 10/23/20/0.226 | 10/24/21/0.256 | | 33 | penalty1 2500 | 10/26/22/0.708 | 10/26/22/0.739 | 10/26/22/0.768 | 10/26/22/0.744 | 10/25/21/0.701 | 10/26/22/0.768 | | 34 | penalty1 10000 | 10/30/24/12.546 | 11/30/25/12.838 | 11/30/25/12.769 | 11/30/25/12.932 | 11/29/24/12.570 | 11/30/25/12.808 | | 35 | power1 4 | 34/307/171/0.014 | 45/776/398/0.040 | 34/343/191/0.014 | 49/697/380/0.027 | 82/1971/655/0.073 | 34/405/201/0.021 | | 36 | quartc 50 | 14/40/40/0.005 | 15/40/41/0.004 | 15/40/41/0.004 | 15/40/41/0.008 | 16/40/42/0.007 | 15/40/41/0.004 | | 37 | quartc 300 | 16/51/48/0.011 | 18/51/50/0.013 | 18/51/50/0.013 | 18/51/50/0.012 | 21/119/80/0.029 | 18/51/50/0.012 | | 38 | quartc 450 | 18/52/50/0.019 | 19/53/52/0.022 | 19/53/52/0.018 | 19/53/52/0.019 | 33/164/110/0.050 | 19/53/52/0.020 | | 39 | sine 1200 | 22/181/105/0.053 | NaN/NaN/NaN/NaN | 21/296/164/0.062 | 25/363/195/0.065 | 31/491/255/0.087 | 17/171/95/0.033 | | 40 | sine 50000 | 19/179/88/1.320 | NaN/NaN/NaN/NaN | 29/552/277/4.156 | 19/171/95/1.344 | 26/358/174/2.532 | 42/937/438/6.981 | | 41 | sine 1000000 | 19/239/114/31.308 | 51/1197/560/178.174 | 754/23822/9561/3300.832 | NaN/NaN/NaN/NaN | 27/299/171/41.517 | NaN/NaN/NaN/NaN | | 42 | tridia 5 | 58/669/354/0.033 | 69/887/447/0.041 | 66/687/354/0.037 | 51/546/284/0.026 | 53/669/341/0.030 | 50/597/314/0.031 | | 43 | tridia 100 | 397/4862/2584/0.280 | 429/5999/3112/0.389 | 405/5843/3057/0.350 | 328/4146/2239/0.207 | 453/7170/3542/0.465 | 353/4357/2305/0.309 | | 44 | tridia 150 | 480/6130/3248/0.331 | 528/7801/4061/0.537 | 540/6307/3390/0.337 | 463/5441/2963/0.348 | 481/7390/3906/0.507 | 552/7319/3783/0.427 | | 45 | tridia 1150 | 1428/19793/9820/2.478 | 1595/21705/11472/2.759 | NaN/NaN/NaN/NaN | 1247/15946/8435/1.875 | NaN/NaN/NaN/NaN | 1894/26206/13791/2.954 | | 46 | tridia 5000 | 1944/29689/13110/12.150 | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | | 47 | bdexp 10000 | 10/23/32/0.176 | 12/24/35/0.210 | 12/24/35/0.203 | 12/24/35/0.207 | 12/23/34/0.200 | 12/24/35/0.197 | | 48 | bdexp 100000 | 10/23/32/1.414 | 12/24/35/1.668 | 12/24/35/1.611 | 12/24/35/1.674 | 12/23/34/1.633 | 12/24/35/1.587 | | 49 | bdexp 1000000 | 11/27/37/16.882 | 13/26/38/17.484 | 13/26/38/17.751 | 13/26/38/17.919 | 13/25/37/17.563 | 13/26/38/17.759 | | 50 | beale 2 | 41/349/195/0.032 | 52/658/349/0.046 | 49/578/289/0.050 | 44/773/372/0.053 | NaN/NaN/NaN/NaN | 28/365/189/0.020 | | 51 | biggsb1 10 | 45/486/261/0.028 | 78/1240/626/0.058 | 50/575/318/0.025 | 62/931/489/0.042 | 60/721/390/0.034 | 52/796/408/0.038 | | 52 | biggsb1 50 | 205/2667/1419/0.119 | 213/2723/1468/0.131 | 379/4896/2556/0.316 | 196/2531/1355/0.191 | 244/3237/1670/0.191 | 199/2645/1352/0.115 | | 53 | bv 4 | 36/353/203/0.037 | 57/781/407/0.044 | 45/682/361/0.049 | 42/509/264/0.039 | 58/1084/487/0.072 | 53/802/409/0.054 | | 54 | bv 500 | 100/1371/714/2.532 | 267/3700/1928/6.837 | 148/2007/1067/3.837 | 253/3310/1727/6.161 | 115/2026/970/3.662 | 337/4193/2179/7.834 | | 55 | bv 1000 | 22/272/151/1.317 | 41/617/315/2.804 | 24/269/138/1.273 | 40/561/291/2.672 | 17/232/118/1.067 | 40/616/312/2.830 | | 56 | diagonal1 10 | 28/245/123/0.025 | 28/239/151/0.023 | 29/275/159/0.013 | 27/366/184/0.017 | 42/783/389/0.039 | 29/400/206/0.023 | | 57 | diagonal2 50 | 49/426/238/0.023 | 58/769/401/0.039 | 52/518/280/0.033 | 49/343/216/0.018 | 62/775/434/0.065 | 52/648/345/0.048 | | 58 | diagonal2 200 | 114/1235/697/0.135 | 107/1653/889/0.128 | 113/1482/799/0.116 | 98/1141/610/0.090 | 111/1257/717/0.087 | 99/1311/741/0.083 | | 59 | diagonal3 10 | 31/215/118/0.014 | 38/440/214/0.020 | 32/374/188/0.021 | 36/560/292/0.025 | 28/207/117/0.012 | 32/306/158/0.017 | | 60 | diagonal3 50 | 66/588/329/0.050 | NaN/NaN/NaN/NaN | 69/1158/574/0.087 | 75/1027/529/0.085 | NaN/NaN/NaN/NaN | 73/739/438/0.046 | | 61 | exdenschnb 100000 | 15/71/44/0.360 | 3/4/4/0.021 | 15/75/45/0.356 | 16/138/79/0.616 | 21/205/107/0.918 | 15/108/66/0.491 | | 62 | exdenschnf 5000 | 21/115/70/0.083 | 36/496/249/0.274 | 30/407/214/0.211 | 24/179/96/0.091 | 38/633/310/0.388 | 28/308/174/0.186 | | 63 | exdenschnf 50000 | 23/148/88/0.860 | 36/496/249/2.362 | 31/408/216/2.017 | 25/180/98/0.932 | 40/698/329/3.275 | 29/309/176/1.572 | | 64 | exdenschnf 100000 | 23/148/87/1.574 | 37/496/251/4.593 | 31/408/216/3.835 | 26/213/116/1.992 | 41/700/321/6.019 | 29/309/172/2.878 | | 65 | genquartic 50000 | 21/123/79/0.411 | 48/664/334/1.991 | 23/305/165/0.954 | 29/442/228/1.419 | 18/140/74/0.413 | 28/372/202/1.187 | | 66 | genquartic 100000 | 29/327/172/2.023 | 67/858/482/5.113 | 24/280/134/1.684 | 24/215/117/1.324 | 22/268/135/1.546 | 27/277/151/1.605 | | 67 | ie 3 | 12/99/56/0.011 | 12/65/36/0.005 | 12/132/68/0.009 | 12/132/70/0.008 | 13/101/59/0.006 | 11/99/56/0.012 | | 68 | ie 10 | 12/99/59/0.010 | 12/65/41/0.008 | 12/132/66/0.019 | 12/132/63/0.016 | 13/103/54/0.012 | 11/99/47/0.010 | | 69 | ie 50 | 13/99/57/0.140 | 12/65/41/0.101 | 12/132/75/0.187 | 12/132/76/0.185 | 15/166/85/0.197 | 11/99/56/0.116 | | 70 | ie 150 | 13/99/60/1.252 | 13/65/43/0.870 | 12/132/74/1.550 | 12/132/76/1.509 | 15/166/81/1.874 | 12/99/55/1.145 | | 71 | ie 200 | 13/99/60/2.073 | 13/65/43/1.516 | 13/132/73/2.618 | 13/132/75/2.685 | 15/166/91/3.305 | 12/99/58/2.059 | | 72 | ie 800 | 13/99/56/31.879 | 13/65/45/23.171 | 13/132/74/40.877 | 13/132/77/41.227 | 16/166/88/51.090 | 12/99/58/32.305 | | 73 | ie 1000 | 14/99/57/51.186 | 13/65/45/35.255 | 13/132/74/65.817 | 13/132/80/65.400 | 16/166/89/81.337 | 13/99/56/50.446 | | 74 | ie 2000 | 14/99/57/203.034 | 13/65/44/144.064 | 13/132/71/251.499 | 14/165/93/320.752 | 16/166/86/319.077 | 13/99/58/202.787 | | 75 | lin 50 | 2/2/2/0.006 | 2/2/2/0.002 | 2/2/2/0.002 | 2/2/2/0.003 | 2/2/2/0.002 | 2/2/2/0.002 | | 76 | lin 100 | 2/2/2/0.004 | 2/2/2/0.004 | 2/2/2/0.004 | 2/2/2/0.006 | 2/2/2/0.006 | 2/2/2/0.007 | | 77 | lin 300 | 2/2/2/0.016 | 2/2/2/0.014 | 2/2/2/0.016 | 2/2/2/0.016 | 2/2/2/0.012 | 2/2/2/0.023 | | 78 | lin 500 | 2/2/2/0.022 | 2/2/2/0.022 | 2/2/2/0.022 | 2/2/2/0.026 | 2/2/2/0.025 | 2/2/2/0.036 | | 79 | lin 1000 | 9/5/9/8.799 | 3/5/3/3.406 | 11/5/11/10.082 | 11/5/11/10.456 | 11/6/11/10.495 | 11/5/11/10.403 | | 80 | lin 2000 | 14/129/45/1399.218 | 3/32/16/340.750 | 12/33/25/497.186 | 12/33/25/501.923 | 12/34/25/510.829 | 12/33/25/508.992 | | 81 | lin 3000 | 7/9/7/645.228 | 10/9/10/828.686 | 10/9/10/831.544 | 10/9/10/825.478 | 10/10/10/850.265 | 10/9/10/819.908 | | 82 | pen1 50 | 195/2534/1365/0.222 | 109/1676/850/0.123 | 202/3459/1695/0.253 | 374/3067/1653/0.245 | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | | 83 | raydan1 100 | 90/1064/555/0.055 | 91/1122/661/0.058 | 99/1283/709/0.069 | 86/1168/616/0.065 | 98/1466/700/0.088 | 90/733/439/0.039 | | 84 | raydan1 550 | 211/2880/1399/0.220 | 230/3337/1658/0.263 | 221/3130/1638/0.228 | 213/2560/1313/0.208 | NaN/NaN/NaN/NaN | NaN/NaN/NaN/NaN | | 85 | raydan1 850 | 353/5182/2719/0.479 | 427/6444/3162/0.617 | 426/6577/3383/0.540 | NaN/NaN/NaN/NaN | 331/5592/2799/0.492 | 292/4371/2299/0.443 | | 86 | raydan1 1550 | 406/5155/2639/0.608 | 539/8573/4322/1.068 | 515/7443/3835/0.928 | 411/5995/3045/0.730 | NaN/NaN/NaN/NaN | 561/8552/4439/1.044 | | 87 | raydan2 8000 | 16/258/126/0.216 | 17/269/151/0.191 | 15/226/127/0.186 | 16/289/157/0.250 | 15/225/134/0.192 | 16/258/144/0.215 | | 88 | raydan2 10000 | 17/324/158/0.343 | NaN/NaN/NaN/NaN | 14/194/113/0.225 | 14/194/113/0.216 | 15/225/129/0.237 | 14/194/113/0.218 | | 89 | trid 40 | 40/438/242/0.054 | 89/1164/627/0.155 | 41/566/308/0.062 | 39/335/184/0.031 | 51/851/431/0.070 | 42/470/275/0.043 | | 90 | trid 150 | 85/1024/521/0.250 | 97/1372/703/0.376 | 91/1003/530/0.288 | 79/757/403/0.184 | 83/814/442/0.206 | 86/1100/599/0.285 | | 91 | trid 200 | 84/1033/553/0.362 | 92/1216/640/0.359 | 97/1480/767/0.466 | 88/1044/533/0.305 | 94/1791/855/0.509 | 93/1173/642/0.390 | | 92 | trid 400 | 86/971/534/1.345 | 112/1479/771/1.978 | 100/1264/677/1.717 | 89/885/499/1.248 | 92/1382/717/1.844 | 101/1207/639/1.629 | | 93 | trid 900 | 75/947/459/3.721 | 95/1377/712/5.347 | 86/1096/564/4.310 | 97/1181/620/4.736 | 164/3703/1426/13.257 | 76/994/546/3.986 | | 94 | trid 1500 | 86/807/438/7.548 | 88/1168/614/10.417 | 93/1347/727/12.063 | 82/982/512/8.789 | 101/1462/737/12.781 | 77/771/429/7.194 | | 95 | trid 2000 | 75/828/446/12.334 | 94/1344/668/19.059 | 91/842/461/12.558 | 84/950/530/14.230 | 100/1365/711/19.965 | 84/811/419/11.853 | | 96 | trid 2500 | 89/973/529/21.609 | 106/1285/681/28.453 | 97/1151/599/25.250 | 84/848/466/19.354 | 130/2409/1039/49.331 | 111/1719/891/39.398 | | 97 | trid 5000 | 83/1099/555/90.581 | 107/1555/791/124.931 | 95/1227/658/100.214 | 84/1023/569/84.970 | 106/1540/776/123.150 | 85/1038/572/85.971 | | 98 | vardim 2 | 10/11/13/0.006 | 12/11/15/0.002 | 12/11/15/0.001 | 12/11/15/0.001 | 12/10/14/0.001 | 12/11/15/0.001 | | 99 | vardim 8 | 9/30/19/0.002 | 11/30/21/0.002 | 11/30/21/0.002 | 11/30/21/0.004 | 11/29/20/0.004 | 11/30/21/0.003 | | 100 | watson 2 | 14/75/44/0.012 | 15/136/79/0.014 | 18/175/107/0.025 | 14/109/64/0.016 | 17/144/83/0.012 | 19/146/82/0.013 |
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