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A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method

  • * Corresponding author: Min Li

    * Corresponding author: Min Li
This work is supported by the NSF (11401242) of China and Scientific Research Fund of Hunan Provincial Education Department(14B139).
Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
  • In this paper, we develop a three-term Polak-Ribière-Polyak conjugate gradient method, in which the search direction is close to the direction in the memoryless BFGS quasi-Newton method. The new scheme reduces to the standard Polak-Ribière-Polyak method when an exact line search is used. For any line search, the method satisfies the sufficient descent condition $g_{k}^{T}d_{k}≤ -{(1-\frac{1}{4}(1+\overline{t})^2})\|g_k\|^2$, where $\overline{t}∈[0,1)$ is a constant. The global convergence results of the new algorithm are established with suitable line search methods. Numerical results show that the proposed method is efficient for the unconstrained problems in the CUTEr library.

    Mathematics Subject Classification: Primary: 90C30; Secondary: 65K05.

    Citation:

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  • Figure 1.  Performance profiles relative to the CPU time (left) and the total number of iterations (right)

    Figure 2.  Performance profiles relative to the total numbers of function evaluations (left) and gradient evaluations (right)

    Table 1.  The problems and their dimensions

    No. Prob Dim No. Prob Dim No. Prob Dim
    No. Prob Dim No. Prob Dim No. Prob Dim
    1 ARGLINA 100 52 DIXMAANK 1500 103 NONDQUAR 1000
    2 ARGLINA 200 53 DIXMAANL 300 104 NONMSQRT 100
    3 ARGLINB 100 54 DIXMAANL 1500 105 OSCIPATH 100
    4 ARGLINB 200 55 DIXON3DQ 100 106 OSCIPATH 500
    5 ARGLINC 50 56 DIXON3DQ 1000 107 PENALTY1 50
    6 ARGLINC 200 57 DQDRTIC 1000 108 PENALTY1 100
    7 ARWHEAD 100 58 DQDRTIC 5000 109 PENALTY2 100
    8 ARWHEAD 1000 59 DQRTIC 500 110 PENALTY2 200
    9 BDQRTIC 100 60 DQRTIC 1000 111 PENALTY3 50
    10 BDQRTIC 500 61 EDENSCH 2000 112 PENALTY3 100
    11 BDQRTIC 1000 62 EG2 1000 113 POWELLSG 100
    12 BOX 100 63 ENGVAL1 1000 114 POWELLSG 10000
    13 BOX 1000 64 ENGVAL1 5000 115 POWER 5000
    14 BROWNAL 200 65 ERRINROS 50 116 POWER 10000
    15 BROYDN7D 100 66 EXTROSNB 100 117 QUARTC 100
    16 BROYDN7D 10000 67 EXTROSNB 1000 118 QUARTC 10000
    17 BRYBND 100 68 FLETCBV2 5000 119 SCHMVETT 100
    18 BRYBND 500 69 FLETCBV2 10000 120 SCHMVETT 10000
    19 CHAINWOO 1000 70 FLETCHBV 100 121 SCOSINE 100
    20 COSINE 1000 71 FLETCHCR 1000 122 SCURLY10 100
    21 COSINE 10000 72 FMINSRF2 5625 123 SCURLY20 100
    22 CRAGGLVY 1000 73 FMINSRF2 10000 124 SCURLY30 100
    23 CRAGGLVY 5000 74 FMINSURF 121 125 SENSORS 100
    24 CURLY10 100 75 FMINSURF 10000 126 SINQUAD 500
    25 CURLY10 1000 76 FREUROTH 100 127 SINQUAD 10000
    26 CURLY20 100 77 FREUROTH 5000 128 SPARSINE 50
    27 CURLY20 1000 78 GENHUMPS 1000 129 SPARSINE 1000
    28 CURLY30 100 79 GENHUMPS 5000 130 SPARSQUR 5000
    29 CURLY30 1000 80 GENROSE 100 131 SPARSQUR 10000
    30 DECONVU 61 81 GENROSE 500 132 SPMSRTLS 4999
    31 DIXMAANA 3000 82 HILBERTA 10 133 SPMSRTLS 10000
    32 DIXMAANA 9000 83 HILBERTB 50 134 SROSENBR 100
    33 DIXMAANB 300 84 HYDC20LS 99 135 SROSENBR 5000
    34 DIXMAANB 9000 85 LIARWHD 100 136 SROSENBR 10000
    35 DIXMAANC 90 86 LIARWHD 10000 137 TESTQUAD 1000
    36 DIXMAANC 9000 87 MANCINO 50 138 TESTQUAD 5000
    37 DIXMAAND 300 88 MANCINO 100 139 TOINTGOR 50
    38 DIXMAAND 1500 89 MODBEALE 200 140 TOINTGSS 100
    39 DIXMAANE 90 90 MODBEALE 2000 141 TOINTGSS 1000
    40 DIXMAANE 1500 91 MOREBV 50 142 TOINTPSP 50
    41 DIXMAANF 1500 92 MOREBV 500 143 TOINTQOR 50
    42 DIXMAANF 9000 93 MSQRTALS 100 144 TQUARTIC 500
    43 DIXMAANG 90 94 MSQRTALS 529 145 TQUARTIC 5000
    44 DIXMAANG 300 95 MSQRTBLS 100 146 TRIDIA 100
    45 DIXMAANH 3000 96 MSQRTBLS 1024 147 TRIDIA 1000
    46 DIXMAANH 9000 97 NONCVXU2 1000 148 VARDIM 50
    47 DIXMAANI 300 98 NONCVXU2 10000 149 VARDIM 100
    48 DIXMAANI 1500 99 NONCVXUN 100 150 VAREIGVL 1000
    49 DIXMAANJ 300 100 NONDIA 5000 151 VAREIGVL 5000
    50 DIXMAANJ 1500 101 NONDIA 10000 152 WOODS 4000
    51 DIXMAANK 300 102 NONDQUAR 500 153 WOODS 10000
     | Show Table
    DownLoad: CSV

    Table 2.  The numerical results

    CG_DESCENT method MPRP method NPRP+ method
    No. Iter/Nf/Ng/Time Iter/Nf/Ng/Time Iter/Nf/Ng/Time
    1 1/3/2/0.001 1/3/2/0.001 1/3/2/0
    2 1/3/2/0.002 1/3/2/0.002 1/3/2/0.001
    3 4/8/7/0.001 5/10/10/0.001 5/10/10/0.000999
    4 9/16/20/0.005999 7/13/15/0.004999 7/13/15/0.004999
    5 3/7/5/0 3/7/5/0 3/7/5/0
    6 8/14/17/0.003999 5/11/11/0.003 5/11/11/0.004
    7 9/21/15/0 12/27/19/0.000999 9/21/15/0
    8 10/24/16/0.002 9/21/14/0.002 8/21/16/0.002
    9 131/255/180/0.003999 126/245/164/0.001999 101/195/173/0.003
    10 375/765/503/0.027996 510/890/740/0.039994 490/700/1025/0.046993
    11 531/1100/718/0.078988 405/861/589/0.06499 238/533/540/0.052992
    12 15/31/23/0.001 11/24/16/0.001 12/25/17/0.001
    13 13/34/28/0.002999 36/60/77/0.007999 18/43/33/0.004999
    14 4/9/6/0.001 15/31/21/0.003999 18/37/26/0.004
    15 82/157/91/0.006999 90/171/101/0.005 79/152/87/0.004
    16 2909/5808/2925/15.238 2819/5621/2840/15.294 2794/5573/2811/15.206
    17 113/227/114/0.002 112/225/114/0.002 106/213/107/0.001999
    18 37/75/38/0.003 27/55/28/0.003999 29/60/31/0.002999
    19 448/834/531/0.061991 436/770/577/0.062991 4517/8797/4945/0.61991
    20 12/28/24/0.003999 10/25/21/0.003999 11/27/22/0.003
    21 12/32/28/0.038994 11/27/25/0.033995 10/25/21/0.029996
    22 103/185/128/0.041994 122/230/159/0.050992 108/194/133/0.043993
    23 110/200/139/0.22697 132/252/184/0.30595 99/179/131/0.24196
    24 991/1777/1387/0.010999 1018/1825/1492/0.009999 936/1718/1286/0.008999
    25 8654/13686/12760/0.84087 9755/14820/15245/0.97685 8911/13925/13444/0.94286
    26 875/1612/1299/0.015998 852/1590/1249/0.016997 894/1658/1277/0.014998
    27 9816/15450/15254/1.3928 10806/16627/17405/1.5678 9776/15443/15149/1.4718
    28 986/1857/1447/0.018997 978/1828/1484/0.017997 989/1833/1419/0.016998
    29 9832/15701/15229/1.7937 9824/15661/15379/1.8267 10778/16930/17516/2.2757
    30 337/676/339/0.005999 385/773/390/0.006999 395/793/400/0.007998
    31 9/19/10/0.003999 7/15/8/0.004999 7/15/8/0.004
    32 9/19/10/0.008999 7/15/8/0.008999 7/15/8/0.006999
    33 9/19/10/0.001 8/17/9/0.001 8/17/9/0.000999
    34 9/19/10/0.008999 8/17/9/0.007999 8/17/9/0.007998
    35 10/21/11/0.001 9/19/10/0.001 9/19/10/0.000999
    36 10/21/11/0.009999 9/19/10/0.008998 9/19/10/0.008999
    37 12/25/13/0.000999 11/23/12/0.001 11/23/12/0.001
    38 12/25/13/0.001999 11/23/12/0.001999 11/23/12/0.001999
    39 48/97/49/0.000999 49/99/50/0.001 48/97/49/0.000999
    40 167/335/168/0.028996 168/337/169/0.024996 169/339/170/0.023996
    41 133/267/134/0.019997 129/259/130/0.018997 127/255/128/0.018997
    42 269/539/270/0.24496 265/531/266/0.24396 263/527/264/0.25996
    43 54/109/55/0.000999 52/105/53/0.001 52/105/53/0.000999
    44 83/167/84/0.002999 81/163/82/0.003 79/159/80/0.002
    45 167/335/168/0.057992 169/339/170/0.058991 164/329/165/0.054992
    46 263/527/264/0.23896 266/533/267/0.24796 256/513/257/0.23396
    47 1046/2093/1047/0.030995 1057/2115/1058/0.032995 938/1877/939/0.027996
    48 2926/5853/2927/0.43093 2914/5829/2915/0.43593 2918/5837/2919/0.42993
    49 635/1271/636/0.018997 607/1215/608/0.018997 599/1199/600/0.018997
    50 1467/2935/1468/0.20897 1478/2957/1479/0.21497 1413/2827/1414/0.21397
    51 606/1213/607/0.019997 602/1205/603/0.022997 481/963/482/0.013998
    52 1434/2869/1435/0.21897 1413/2827/1414/0.20497 1387/2775/1388/0.28396
    53 596/1193/597/0.020997 604/1209/605/0.018997 388/777/389/0.016998
    54 1374/2749/1375/0.20497 1424/2849/1425/0.20697 1356/2713/1357/0.20697
    55 200/401/202/0.001999 200/401/202/0.001999 200/401/202/0.001999
    56 1000/2001/1002/0.048992 1000/2001/1002/0.06199 1000/2001/1002/0.050992
    57 7/15/8/0.002 7/15/8/0.002 6/13/7/0.001999
    58 7/15/8/0.004 7/15/8/0.004999 7/15/8/0.003999
    59 28/57/29/0.001 27/55/28/0.001 27/55/28/0.001
    60 29/59/30/0.000999 29/59/30/0.002 29/59/30/0.002
    61 32/59/40/0.010999 32/60/45/0.009998 31/56/39/0.011998
    62 4/9/6/0.001 4/9/6/0.001 4/9/6/0.001
    63 26/48/33/0.003 26/49/33/0.003 22/42/28/0.002999
    64 27/50/40/0.016997 25/44/34/0.014998 23/42/30/0.013998
    65 1151/2285/1599/0.008998 1150/2270/1564/0.006998 1708/3414/2234/0.013998
    66 5481/11373/6023/0.041993 5510/11246/5845/0.041994 5906/12296/6550/0.045993
    67 6354/13073/6816/0.47393 7741/15623/7929/0.54592 7866/16031/8277/0.61791
    68 0/1/1/0.001999 0/1/1/0.001999 0/1/1/0.001
    69 0/1/1/0.002999 0/1/1/0.003 0/1/1/0.002
    70 F/F/F/F F/F/F/F F/F/F/F
    71 6604/13682/7147/0.6859 4310/8650/4348/0.43493 6879/14253/7386/0.74789
    72 368/739/371/0.25896 305/611/306/0.20897 395/792/397/0.28096
    73 434/869/435/0.54392 375/752/377/0.53992 460/922/462/0.60891
    74 88/178/90/0.001999 92/186/94/0.001999 80/162/82/0.002
    75 603/1209/606/0.85987 454/910/456/0.6459 655/1311/656/0.96185
    76 52/103/80/0.000999 107/205/145/0.003 36/72/65/0.001
    77 53/107/80/0.047993 49/97/74/0.044993 33/68/56/0.032995
    78 2720/5482/2769/1.2288 1895/4067/2224/0.93486 2964/5978/3028/1.3378
    79 6653/13399/6765/14.238 5844/12355/6649/12.715 7015/14081/7077/14.59
    80 297/631/344/0.005999 297/622/337/0.003 319/677/367/0.005999
    81 1257/2553/1321/0.070989 1094/2231/1154/0.054992 1183/2404/1237/0.06899
    82 7/15/10/0.001 7/15/10/0 7/15/10/0.001
    83 5/11/6/0.001 5/11/6/0.001 5/11/6/0.001
    84 F/F/F/F F/F/F/F F/F/F/F
    85 18/37/19/0.000999 19/40/23/0.001 19/39/23/0
    86 23/54/38/0.031995 23/50/32/0.028996 24/50/30/0.026996
    87 9/19/10/0.013998 9/19/10/0.011998 10/21/11/0.012998
    88 11/23/12/0.057991 10/21/11/0.052991 10/21/11/0.052992
    89 334/684/430/0.025996 453/910/511/0.027996 249/501/306/0.015998
    90 1085/2106/1395/0.75888 F/F/F/F 561/1137/584/0.34195
    91 5640/11281/5776/0.024996 7051/14103/7209/0.031995 4262/8525/4365/0.018997
    92 502/1005/503/0.026996 489/979/490/0.020997 484/969/485/0.019997
    93 284/577/295/0.012998 282/573/293/0.008999 282/573/293/0.008998
    94 4753/9513/4762/1.8517 5342/10691/5351/1.9547 5242/10491/5251/1.9357
    95 356/717/362/0.011998 355/716/363/0.017996 356/718/364/0.010999
    96 2309/4624/2316/2.6026 2209/4425/2218/2.4486 2227/4460/2234/2.4826
    97 1801/3346/2059/0.49393 1968/3821/2085/0.48993 1913/3704/2037/0.47193
    98 9063/17431/9760/22.829 8928/17306/9480/22.166 8989/17216/9753/23.235
    99 168/329/179/0.005998 150/298/154/0.004999 160/315/173/0.005999
    100 10/23/16/0.006999 10/37/32/0.010998 9/31/25/0.008999
    101 8/26/22/0.014997 9/22/14/0.011999 7/16/10/0.008999
    102 4004/8014/4257/0.11298 3540/7087/3693/0.10298 1758/3522/1999/0.06199
    103 3027/6061/3162/0.21297 2014/4032/2071/0.11298 1705/3414/1878/0.10898
    104 F/F/F/F F/F/F/F F/F/F/F
    105 13/26/15/0.001 13/28/15/0 13/28/15/0
    106 14/28/16/0.001 15/31/18/0.002 14/29/15/0.001
    107 44/121/84/0.000999 47/140/101/0.001 41/106/73/0.000999
    108 44/134/98/0.001 49/126/87/0.001 44/123/87/0.000999
    109 84/141/143/0.009999 97/160/158/0.009998 81/140/132/0.010999
    110 200/235/367/0.044993 211/245/395/0.042994 191/224/353/0.038994
    111 70/174/124/0.025997 95/214/150/0.024996 70/167/118/0.019997
    112 F/F/F/F F/F/F/F 78/175/124/0.080988
    113 103/214/121/0.001 102/206/112/0.000999 191/383/237/0.001999
    114 224/461/272/0.11898 115/232/124/0.059991 201/403/240/0.10798
    115 258/517/259/0.043993 762/1525/763/0.13298 372/745/373/0.06499
    116 369/739/370/0.12398 1116/2233/1117/0.40794 473/947/474/0.17397
    117 24/49/25/0.001 23/47/24/0.001 23/47/24/0.000999
    118 35/71/36/0.016998 34/69/35/0.016998 34/69/35/0.015997
    119 46/82/58/0.002999 44/79/55/0.003 44/78/56/0.003
    120 39/65/54/0.26096 41/69/56/0.27796 41/69/56/0.27096
    121 F/F/F/F F/F/F/F F/F/F/F
    122 F/F/F/F F/F/F/F F/F/F/F
    123 F/F/F/F F/F/F/F F/F/F/F
    124 F/F/F/F F/F/F/F F/F/F/F
    125 24/57/43/0.19897 32/72/57/0.25396 22/49/32/0.15398
    126 44/108/92/0.010998 54/125/104/0.010998 43/115/95/0.009999
    127 88/166/179/0.35795 68/166/142/0.27996 37/95/83/0.16198
    128 164/331/168/0.003 185/375/192/0.003 165/333/170/0.002999
    129 4499/8999/4500/1.1628 5259/10519/5260/1.3568 4737/9475/4738/1.2478
    130 21/43/22/0.016998 21/43/22/0.016998 21/43/22/0.017998
    131 22/45/23/0.039994 22/45/23/0.038994 22/45/23/0.038994
    132 218/443/227/0.15398 206/419/215/0.14698 201/409/210/0.14298
    133 225/457/234/0.33295 214/435/223/0.33495 207/421/216/0.30895
    134 11/24/14/0.001 10/22/13/0.001 11/24/15/0.001
    135 12/26/16/0.004998 10/22/13/0.002999 11/24/15/0.003999
    136 12/26/17/0.008999 10/22/13/0.006999 11/24/15/0.007999
    137 870/1741/871/0.041994 856/1713/857/0.033995 855/1711/856/0.033995
    138 1641/3283/1642/0.30795 1728/3457/1729/0.33695 1682/3365/1683/0.32395
    139 122/221/153/0.002999 120/221/149/0.001999 121/223/150/0.001999
    140 13/26/15/0.000999 14/28/16/0.001 14/28/16/0.000999
    141 6/13/7/0.003 6/13/7/0.003 6/13/7/0.002999
    142 156/325/213/0.001 130/270/191/0.001 137/282/194/0.002
    143 32/61/41/0.001 32/60/40/0 31/58/37/0.000999
    144 25/62/42/0.002 20/76/63/0.002 18/45/33/0.002
    145 22/54/39/0.015998 18/71/61/0.016997 23/72/56/0.014998
    146 92/185/93/0.000999 92/185/93/0.001 91/183/92/0.000999
    147 338/677/339/0.017997 339/679/340/0.016997 338/677/339/0.016996
    148 21/43/22/0.001 21/43/22/0 21/43/22/0
    149 25/51/26/0.001 25/51/26/0.001 25/51/26/0.001
    150 93/243/150/0.026996 101/271/170/0.024996 81/214/133/0.019997
    151 119/303/184/0.15498 108/291/184/0.14598 106/285/184/0.13598
    152 268/554/301/0.082988 271/562/303/0.084987 157/355/225/0.057991
    153 241/521/308/0.21097 208/462/286/0.18797 161/358/228/0.17597
     | Show Table
    DownLoad: CSV
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