Research Article
A New Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence
Table 1
List of test problem functions.
| Problem | n | MTTHS | DHS | BZA | NI/CT/GE/FE | NI/CT/GE/FE | NI/CT/GE/FE |
| Extended | 2 | 1323/9.3448/4645/3321 | 1267/10.1753/4756/3488 | 364/3.1079/1365/1000 | Trigonometric -1 | 50 | 4533/28.4686/15849/11315 | 3771/25.5829/14146/10374 | 2376/18.4341/8910/6533 | function | 500 | 11125/67.51/38848/27722 | 8163/55.0526/30616/22452 | 5904/46.5577/22140/16235 |
| HIMMELBH(Cute) | 2 | 9/0.4680/77/67 | 9/0.4538/77/67 | 9/0.4471/77/67 | | 50 | 9/0.4572/77/67 | 9/0.4574/77/67 | 9/0.4620/77/67 | | 5000 | 9/2.0529/77/67 | 9/2.0198/77/67 | 9/1.9930/77/67 |
| Power function | 2 | 3/0.3968/13/9 | 2/0.3881/9/6 | 2/0.3884/9/6 | | 50 | 212/2.0540/849/636 | 3431/24.0407/13725/10293 | 65/0.7751/261/195 | | 1000 | 4612/33.5261/18449/13836 | F/F/F/F | 1565/11.567/6261/4695 |
| DENSCHNF | 2 | 11/0.6773/64/52 | 12/0.5399/67/54 | 8/0.4719/50/41 | | 50 | 14/0.5959/78/63 | 14/0.5260/75/60 | 9/0.4739/54/44 | | 5000 | 10/1.7879/56/45 | 14/2.1633/75/60 | 9/1.6783/54/44 |
| Sum Squares | 2 | 3/0.4203/13/9 | 2/0.3861/9/6 | 2/0.3966/9/6 | function | 50 | 47/0.7252/189/141 | 39/0.6318/157/117 | 39/0.6162/157/117 | | 1000 | 229/2.4542/917/687 | 1333/11.3244/5333/3999 | 191/1.9552/765/573 |
| TRIDIA(Cute) | 2 | 3/0.4056/13/9 | 2/0.3980/9/6 | 2/0.4024/9/6 | | 50 | 100/1.6029/401/300 | 193/1.7752/773/579 | 56/0.7390/225/168 | | 1000 | 604/5.2318/2417/1812 | 16019/122.48/64077/48057 | 349/3.22771397/1047 |
| SINQUAD(Cute) | 2 | 25/0.4804/102/76 | 1734/12.6402/6076/4341 | 25/0.5218/101/75 | | 5000 | F/F/F/F | F/F/F/F | 411/198.6198/2837/2425 | | 10000 | F/F/F/F | F/F/F/F | 417/493.6830/2650/2232 |
| Generalized | 50 | 23/0.5659/101/77 | 25/0.5348/109/83 | 21/0.5187/94/72 | Quartic GQ2 | 100 | 23/0.5588/103/79 | 31/0.5938/133/101 | 22/0.5233/96/73 | function. | 5000 | 26/2.9409/116/89 | 23/2.6517/103/79 | 25/2.7548/110/84 |
| Generalized | 50 | 50/0.7767/248/197 | 43/0.6750/192/148 | 42/0.6934/198/155 | Triagonal-2 | 500 | 43/1.5284/206/162 | 46/1.4306/209/162 | 38/1.4029/182/143 | function | 1000 | 45/2.1144/214/168 | 47/2.1176/214/166 | 39/1.8380/181/141 |
| Extended | 2 | 8/0.4315/32/23 | 9/0.4256/37/27 | 6/0.4158/25/18 | Trigonometric -2 | 50 | 33/0.6514/148/114 | 26/0.5848/124/97 | 22/0.5586/106/83 | function | 7000 | 85/63.5473/626/540 | 71/44.8822/440/368 | 33/25.6757/251/217 |
| LIARWHD | 100 | 20/0.5798/136/115 | 16/0.5065/95/78 | 14/0.4865/84/69 | | 5000 | 50/8.8611/390/339 | 53/8.7426/384/330 | 24/4.4628/189/164 | | 10000 | 33/14.6362/245/211 | 53/19.3591/331/277 | 23/9.7171/165/141 |
| Generalized | 2 | 6/0.4058/27/20 | 6/0.4127/27/20 | 5/0.4008/23/17 | Quartic GQ1 | 6000 | 11/1.8086/47/35 | 10/1.6695/43/32 | 10/1.7246/43/32 | function | 10000 | 11/3.4464/47/35 | 10/3.0683/43/32 | 10/3.0317/43/32 |
| Extended QP2 | 2 | 17/0.8111/98/80 | 21/0.6076/125/103 | 16/0.5702/111/94 | Quadratic Penalty | 5000 | 226/56.4248/2247/2020 | F/F/F/F | 82/25.7327/1018/935 | function | 10000 | 137/95.8189/1457/1319 | F/F/F/F | 82/67.1490/1024/941 |
| Extended Maratos | 2 | 20/0.6479/131/110 | 43/0.8956/225/181 | 30/0.8305/193/162 | function | 1000 | 26/0.8290/176/149 | F/F/F/F | 30/0.8300/193/162 | | 5000 | F/F/F/F | 45/5.8526/233/187 | 32/6.0318/243/210 |
| DENSCHNC | 2 | 6/0.4508/36/29 | 10/0.4624/51/40 | 6/0.4486/34/27 | | 50 | 7/0.4499/38/30 | 10/0.4590/51/40 | 6/0.4696/34/27 | | 600 | 8/0.5625/44/35 | 15/0.6240/130/114 | 7/0.5445/38/30 |
| DIXON3DQ (Cute) | 2 | 3/0.4835/13/9 | 2/0.3904/9/6 | 2/0.3878/9/6 | | 20 | 106/1.2286/424/317 | 262/2.4917/1061/798 | 90/1.0709/360/269 | | 600 | 2266/18.9637/9064/6797 | F/F/F/F | 2185/18.4756/8740/6554 |
| EDENSCH function | 50 | 3094/15.3448/9288/6193 | 3094/16.4610/9288/6193 | 3089/15.3239/9273/6183 | | 1000 | 3094/17.6514/9288/6193 | 3094/18.4918/9288/6193 | 3089/17.6263/9273/6183 | | 5000 | 3094/198.8550/9288/6193 | 3094/199.0048/9288/6193 | 3089/198.2240/9273/6183 |
| Diagonal 2 function | 50 | 749/6.8385/2626/1876 | 495/3.0663/1488/992 | 490/3.3271/1473/982 | | 1000 | 16517/126.4914/59593/43075 | 8569/53.8256/25711/17141 | 8563/53.3166/25693/17129 | | 10000 | F/F/F/F | 16484/283.6569/49456/32971 | 16478/279.44/49438/32959 |
| DQDRTIC(Cute) | 50 | 12/0.4358/49/36 | 5/0.4070/21/15 | 5/0.3940/21/15 | | 5000 | 14/1.6324/57/42 | 5/0.7966/21/15 | 5/0.8084/21/15 | | 10000 | 14/3.7862/57/42 | 5/1.6205/21/15 | 5/1.5979/21/15 |
| Perturbed | 50 | 43/0.6470/173/129 | 38/0.5837/153/114 | 38/0.6313/153/114 | Quadratic function | 1000 | 207/2.1710/829/621 | 1397/10.8367/5589/4196 | 187/1.9258/749/561 | | 5000 | 481/44.4081/1925/1443 | 5125/435.0973/20501/15375 | 425/39.4768/1701/1275 |
| Diagonal 4 function | 50 | 4/0.3919/17/12 | 2/0.4015/9/6 | 2/0.3963/9/6 | | 1000 | 5/0.4493/21/15 | 2/0.4005/9/6 | 2/0.4039/9/6 | | 5000 | 5/0.8522/21/15 | 2/0.5598/9/6 | 2/0.5698/9/6 |
| Extended Beale | 50 | 15/0.4685/74/58 | 68/0.8247/281/212 | 15/0.4890/72/56 | function | 100 | 16/0.4739/74/57 | 68/0.8245/281/212 | 15/0.4855/72/56 | | 500 | 16/0.5450/74/57 | 68/0.8840/281/212 | 15/0.5050/72/56 |
| DENSCHNA | 50 | 9/0.4336/44/34 | 24/0.5609/102/77 | 9/0.4462/44/34 | | 1000 | 10/0.5912/48/37 | 26/0.7584/110/83 | 9/0.5909/44/34 | | 5000 | 10/1.6387/46/35 | 26/3.0388/110/83 | 9/1.4694/44/34 |
| STAIRCASE S1 | 20 | 79/0.9390/316/236 | 202/1.7976/818/615 | 72/0.8546/290/217 | | 50 | 215/2.0539/860/644 | 1117/8.6778/4503/3385 | 183/1.6893/732/548 | | 100 | 426/3.4107/1705/1278 | 3896/29.4293/15737/11840 | 372/3.0840/1490/1117 |
| NONDQUAR | 50 | 28674/216.605/114767/86092 | F/F/F/F | 14756/104.86/59098/44341 | | 100 | 32283/244.026/129157/96873 | F/F/F/F | 17900/127.16/71670/53769 | | 500 | F/F/F/F | F/F/F/F | 32748/241.1/131043/98294 |
| Extended Wood | 500 | 128/1.5299/605/476 | 942/8.5789/4143/3200 | 96/1.4535/551/454 | | 1000 | 146/1.7443/676/529 | 946/8.8946/4100/3153 | 101/1.7151/606/504 | | 10000 | 213/57.8431/979/765 | 956/249.5361/4202/3245 | 89/31.8805/545/455 |
| Extended Penalty | 20 | 33/0.6528/144/110 | 37/0.7031/158/120 | 22/0.5670/98/75 | | 300 | F/F/F/F | F/F/F/F | 64/1.2248/365/300 | | 600 | F/F/F/F | F/F/F/F | 28/0.7428/172/143 |
| Sphere function | 2 | 1/0.4750/5/3 | 1/0.3777/5/3 | 1/0.3902/5/3 | | 1000 | 1/0.3904/5/3 | 1/0.3877/5/3 | 1/0.3860/5/3 | | 5000 | 1/0.4687/5/3 | 1/0.4576/5/3 | 1/0.4616/5/3 |
| ARGLINB(Cute) | 2 | 1/0.3913/5/3 | 1/0.3903/5/3 | 1/0.3840/5/3 | | 50 | 1/0.3868/5/3 | 1/0.3941/5/3 | 1/0.3832/5/3 | | 500 | 2/0.4279/9/6 | 2/0.4266/9/6 | 2/0.4246/9/6 |
| Extended White | 2 | 28/0.7668/192/163 | 69/1.2885/354/284 | 21/0.6586/136/114 | and Holst function | 500 | 29/0.9337/213/183 | 102/1.5134/488/385 | 21/0.6586/136/114 | | 5000 | 23/4.6845/168/144 | 125/14.3950/577/451 | 21/3.8322/136/114 |
| Extended Hiebert | 50 | 2040/33.6457/16304/14263 | F/F/F/F | 838/21.7040/9977/9138 | | 200 | F/F/F/F | F/F/F/F | 901/23.007/10513/9611 | | 1000 | F/F/F/F | F/F/F/F | F/F/F/F |
| Quadratic QF1 | 50 | 44/0.6479/177/132 | 38/0.6068/153/114 | 38/0.6306/153/114 | | 500 | 147/1.4565/589/441 | 574/4.7399/2297/1722 | 131/1.3814/525/393 | | 10000 | 744/179.6695/2977/2232 | F/F/F/F | 606/146.3228/2425/1818 |
| Quartic | 50 | 11579/66.6569/34740/23160 | 11579/66.7991/34740/23160 | 11574/64.593/34725/23150 | | 500 | 24975/148.8391/74928/49952 | 24975/146.0726/74928/49952 | 24968/142.57/74907/49938 | | 1000 | 31473/201.6973/94422/62948 | 31473/218.8475/94422/62948 | 31466/191.54/94401/62934 |
| Shallow function | 1000 | 10/0.5712/45/34 | 48/0.7436/200/151 | 10/0.5557/45/34 | | 5000 | 11/1.4925/49/37 | 48/4.9218/200/151 | 10/1.4117/45/34 | | 10000 | 12/3.6303/53/40 | 50/13.2094/209/158 | 10/3.1569/45/34 |
| VARDIM | 50 | 28/1.1175/322/293 | 143/4.0110/1604/1460 | 16/0.8685/221/204 | | 100 | 20/1.0429/286/265 | 293/5.9711/2604/2310 | 17/0.9508/250/232 | | 500 | 41/2.2545/736/694 | 1557/25.9389/11724/10166 | 34/1.8745/645/610 |
| DIXMAANA | 6000 | 8/1.8623/35/26 | 8/1.8913/35/26 | 8/1.8504/35/26 | | 6015 | 8/1.9276/35/26 | 8/1.8546/35/26 | 8/1.9200/35/26 | | 6030 | 8/1.9301/35/26 | 8/1.9700/35/26 | 8/1.8819/35/26 |
| DIXMAANB | 9 | 7/0.4044/30/22 | 7/0.4141/30/22 | 7/0.4036/30/22 | | 300 | 7/0.4550/30/22 | 7/0.4776/30/22 | 8/0.4415/34/25 | | 6000 | 8/1.8966/34/25 | 8/1.8364/34/25 | 9/2.0362/38/28 |
| DIXMAANC | 9 | 6/0.4033/29/22 | 6/0.4191/29/22 | 6/0.4168/29/22 | | 300 | 7/0.4701/33/25 | 6/0.4721/29/22 | 7/0.4707/33/25 | | 6000 | 7/2.0093/33/25 | 7/2.0148/33/25 | 8/2.1736/37/28 |
| DIXMAAND | 90 | 7/0.4450/34/26 | 8/0.4333/38/29 | 8/0.4263/38/29 | | 300 | 8/0.5272/38/29 | 8/0.5406/38/29 | 8/0.5418/38/29 | | 6000 | 9/2.4166/42/32 | 7/2.0165/34/26 | 7/2.0556/34/26 |
| DIXMAANE | 9 | 19/0.5080/88/68 | 23/0.5720/107/83 | 18/0.4832/84/65 | | 300 | 84/1.5278/421/336 | 1211/7.7407/3707/2495 | 84/2.1712/420/335 | | 6000 | 331/81.2267/1655/1323 | F/F/F/F | 333/83.0952/1685/1351 |
| EG2 | 2 | 5/0.4912/25/19 | 8/0.4128/40/31 | 4/0.4139/20/15 | | 20 | 160/1.4413/642/481 | 1958/14.8104/8106/6147 | 110/1.0755/447/336 | | 50 | F/F/F/F | 48/0.7452/228/179 | 656/5.3312/2773/2116 |
| EG3 | 20 | 21/0.5445/88/66 | 25/0.5611/102/76 | 14/0.4851/60/45 | | 50 | 25/0.5965/110/84 | 43/0.7618/207/163 | 25/0.5236/69/53 | | 100 | F/F/F/F | F/F/F/F | 20/0.5290/88/67 |
| Fletcher function | 50 | 24/0.5879/126/101 | 23/0.5870/123/99 | 22/0.5687/118/95 | | 6000 | 27/5.2874/161/133 | 28/5.4497/168/139 | 26/4.7950/152/125 | | 10000 | 27/10.4312/161/133 | 27/10.6308/163/135 | 26/9.8879/155/128 |
| Extended | 50 | 8/0.4229/37/28 | 11/0.4510/49/37 | 9/0.4370/41/31 | Himmelblau | 1000 | 8/0.4706/39/30 | 12/0. 5342/53/40 | 9/0.4775/41/31 | function | 5000 | 9/1.4066/43/33 | 13/1.6584/57/43 | 9/1.3377/41/31 |
| Extended | 2 | 11/0.5889/75/63 | 33/0.6806/163/129 | 10/0.4929/69/58 | Freudenstein | 50 | F/F/F/F | 36/0.7871/230/193 | 10/0.4700/69/58 | and Roth function | 200 | 16/0.5442/102/85 | 37/0.8501/241/203 | 11/0.4993/73/61 |
| Dixon and Price | 20 | 132/1.6580/615/482 | 333/3.2018/1409/1075 | 131/1.5682/603/471 | function | 50 | 185/2.5157/1034/848 | 478/4.7648/2199/1720 | 165/2.2048/900/734 | | 100 | 523/6.5773/2938/2414 | 1243/11.1703/5672/4428 | 439/5.4568/2422/1982 |
| Raydan 1 function | 2 | 106/1.3363/319/212 | 105/0.8502/316/210 | 99/0.8493/298/198 | | 50 | 59/0.8854/237/177 | 88/1.0277/364/275 | 58/0.7714/233/174 | | 100 | 70/0.9532/282/211 | 155/1.5252/625/469 | 70/0.8892/282/211 |
| Raydan 2 function | 2 | 2/0.4129/9/6 | 2/0.3995/9/6 | 3/0.3971/13/9 | | 50 | 2/0.3991/9/6 | 2/0.3981/9/6 | 3/0.4176/13/9 | | 100 | 2/0.4002/9/6 | 2/0.4123/9/6 | 3/0.3906/13/9 |
| NONDIA(SHANO- | 500 | 9/0.5483/63/53 | 11/0.4937/70/58 | 7/0.4938/55/47 | 78) | 6000 | 14/2.7799/82/67 | 46/6.5297/212/165 | 9/2.3946/70/60 | | 10000 | 16/6.3412/96/79 | 74/21.4446/339/264 | 15/6.2113/92/76 |
| Extended Block | 500 | 30/1.0226/52000/44250 | 32/0.9386/32250/24000 | 39/1.1887/44000/34000 | Diagonal BD1 | 1000 | 39/3.6068/195000/175000 | 33/1.3795/66500/49500 | 39/1.6724/88500/68500 | function | 10000 | 38/22.3738/1070000/875000 | 35/14.5725/705000/525000 | 43/20.1088/965000/745000 |
| SINCOS | 2000 | F/F/F/F | F/F/F/F | 8/0.6960/39/30 | | 5000 | F/F/F/F | F/F/F/F | 8/1.2553/39/30 | | 10000 | F/F/F/F | F/F/F/F | 8/2.7535/39/30 |
| DIXMAANH | 90 | 58/0.8867/290/231 | 388/2.5254/1230/841 | 57/1.0443/286/228 | | 300 | 81/1.5051/408/326 | 1264/7.8924/3872/2607 | 81/1.3140/408/326 | | 600 | 126/4.4107/630/503 | 2407/40.8602/7294/4886 | 119/4.0122/596/476 |
| Quadratic QF2 | 50 | 77/0.9476/330/252 | 107/1.1580/454/346 | 68/0.8775/293/224 | function | 200 | 183/1.7984/787/603 | 337/2.9913/1419/1081 | 139/1.5/610/470 | | 2000 | 1150/34.4704/4862/3711 | 2903/85.3362/12110/9206 | 540/17.4638/2404/1863 |
| Tridiagonal double | 20 | 108/1.1547/438/329 | 1915/14.1311/7667/5751 | 101/1.1136/410/308 | Bordered | 50 | 407/3.3266/1633/1225 | 11157/78.2325/44633/33475 | 355/2.9421/1425/1069 | | 500 | 6278/48.1130/25119/18840 | 13470/103.5476/53887/40416 | 4595/35.3993/18387/13791 |
| Generalized | 2 | 1323/8.3265/4645/3321 | 1267/8.4042/4756/3488 | 364/2.6867/1365/1000 | Triagonal1 function | 50 | 25/0.5255/103/77 | 28/0.504/115/86 | 24/0.5140/99/74 | | 100 | 24/0.5630/99/74 | 27/0.6205/111/83 | 23/0.5242/95/71 |
| Extended QP2 | 50 | 9/0.4269/49/39 | 15/0.4503/71/55 | 9/0.4190/49/39 | Quadratic penalty | 200 | 13/0.4682/70/56 | 14/0.6931/74/59 | 12/0.4620/66/53 | function | 3000 | F/F/F/F | F/F/F/F | 17/1.5340/118/100 |
| Extended | 2 | 5/0.4089/22/16 | 6/0.4254/26/19 | 5/0.4095/22/16 | DENSCHNB | 500 | 5/0.4222/22/16 | 7/0.4313/30/22 | 6/0.4165/26/19 | function | 10000 | 5/1.7240/22/16 | 7/2.1577/30/22 | 6/1.9461/26/19 |
| Extended three- | 2 | 7/0.4187/30/22 | 13/0.4386/54/40 | 7/0.417/30/22 | Exponential terms | 50 | 7/0.4211/750/550 | 14/0.4492/1450/1075 | 7/0.4255/750/550 | | 100 | 7/0.4378/1500/1100 | 14/0.4705/2900/2150 | 7/0.4183/1500/1100 |
| DIXMAANF | 9 | 23/0.6308/109/85 | 23/0.5828/109/85 | 19/0.5129/90/70 | | 90 | 51/0.8509/256/204 | 386/2.4637/1229/842 | 51/1.1421/256/204 | | 300 | 90/1.4444/451/360 | 1260/7.8258/3871/2610 | 90/1.5578/451/360 |
| DIXMAANG | 9 | 19/0.5318/95/75 | 26/0.5670/127/100 | 21/0.5403/103/81 | | 90 | 83/1.2240/424/340 | 379/2.4949/1221/841 | 56/0.8767/280/223 | | 300 | 137/1.8866/643/505 | 1241/14.5812/3834/2592 | 569/5.5495/2477/1907 |
| Extended | 2 | 25/0.7100/159/133 | 24/0.9013/120/95 | 27/0.7803/163/135 | Rosenbrock | 1000 | 14/0.8076/82/67 | 26/0.6537/128/101 | 28/0.7818/167/138 | function | 5000 | 19/2.8693/109/89 | 27/3.2102/132/104 | 30/4.4233/175/144 |
| ARWHEAD | 500 | F/F/F/F | F/F/F/F | 10/0.4429/54/43 | | 3000 | F/F/F/F | F/F/F/F | 5/0.7593/38/32 | | 8000 | F/F/F/F | F/F/F/F | 7/2.5096/49/41 |
| Hager function | 2 | 4/0.4001/17/12 | 6/0.4188/25/18 | 4/0.4008/17/12 | | 50 | 20/0.4789/82/61 | 24/0.5101/98/73 | 20/0.4738/82/61 | | 100 | 24/0.5275/98/73 | F/F/F/F | 24/0.5382/111/86 |
| Extended Powell | 1000 | 4008/108.2/4032500/3030250 | F/F/F/F | 361/10.5/384000/293500 | function | 3000 | 1026/86.1/3124500/2354250 | F/F/F/F | 108/10.8/379500/297750 | | 5000 | 2213/330/11330000/8562500 | F/F/F/F | 306/48/1652500/1268750 |
| BIGGSB1 function | 2 | 1/0.3861/5/3 | 1/0.3866/5/3 | 1/0.3895/5/3 | | 20 | 53/0.7214/213/159 | 20/0.4869/80/59 | 20/0.4764/80/59 | | 50 | 201/1.9444/804/602 | 867/7.1960/3500/2632 | 50/0.7281/200/149 |
| Extended Cliff | 100 | 11657/87/2075150/1492250 | 39907/329/7892000/5896600 | 1674/12/293750/210000 | | 5000 | F/F/F/F | F/F/F/F | 1009/132/8857500/633250 |
| ENGVAL8 | 2 | 7/0.4675/33/25 | 10/0.4709/45/34 | 7/0.4756/36/28 | | 20 | 22/0.5657/99/76 | 32/0.6306/138/105 | 20/0.5382/90/69 |
| Trecanni function | 2 | 6/0.4146/26/19 | 6/0.4212/26/19 | 4/0.3982/18/13 |
| GENROSEN-2 | 2 | 25/0.8096/159/133 | 24/0.5864/120/95 | 27/0.6770/163/135 |
| Generalized Quartic function | 2 | 6/0.4068/27/20 | 6/0.4082/27/20 | 5/0.4081/23/17 |
| Diagonal 1 function | 2 | 6/0.4015/25/18 | 7/0.4085/29/21 | 4/0.4207/18/13 |
| Six Hump function | 2 | 7/0.4112/31/23 | 7/0.4271/31/23 | 6/0.4064/27/20 |
| Three Hump function | 2 | 11/0.4441/52/40 | 11/0.4335/50/38 | 10/0.4397/47/36 |
| Booth function | 2 | 3/0.3930/13/9 | 2/0.4043/9/6 | 2/0.3919/9/6 |
| Zett1 function | 2 | 26/0.5067/105/78 | 30/0.5581/126/95 | 24/0.5046/102/77 |
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