From: A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems
Sl. no. | Test problem | Starting point | PRP | |||||
---|---|---|---|---|---|---|---|---|
Using standard Wolfe | Using strong Wolfe | |||||||
\(x^{*}\) | \(f(x^{*})\) | NI | \(x^{*}\) | \(f(x^{*})\) | NI | |||
1 | Ackley | \((2.000E-01,1.000E-01)^{T}\) | \((8.0700E-15,-1.127E-12)^{T}\) | 3.1872E − 12 | 35 | \((-1.691E-02,-3.888E-02)^{T}\) | 4.2395E − 07 | 14 |
2 | Beale | \((-1.370E+00,-1.433E+00)^{T}\) | \((3.0000E+00,5.000E-01)^{T}\) | 9.0393E − 14 | 59 | \((3.000E+00,5.000E-01)^{T}\) | 1.3600E − 13 | 34 |
3 | Beale | \((1.882E+00,-3.130E-01)^{T}\) | \((3.0000E+00,5.000E-01)^{T}\) | 2.2124E − 13 | 43 | \((3.000E+00,5.000E-01)^{T}\) | 2.4689E − 13 | 47 |
4 | Beale | \((1.476E+00,5.286E-01)^{T}\) | \((3.0000E+00,5.000E-01)^{T}\) | 2.1515E − 14 | 42 | \((3.000E+00,5.000E-01)^{T}\) | 9.4967E − 14 | 54 |
5 | Beale | \((6.077E-01,-1.118E+00)^{T}\) | \((3.0000E+00,5.000E-01)^{T}\) | 1.8879E − 10 | 43 | \((3.000E+00,5.000E-01)^{T}\) | 1.5070E − 16 | 21 |
6 | Booth | \((1.000E+00,4.000E+00)^{T}\) | \((1.0000E+00,3.000E+00)^{T}\) | 3.8496E − 15 | 25 | \((1.000E+00,3.000E+00)^{T}\) | 6.0707E − 21 | 2 |
7 | Camel: Three Hump | \((1.472E-01,-4.025E+00)^{T}\) | \((5.7530E-09,-2.537E-07)^{T}\) | 6.2992E − 14 | 27 | \((2.026E-08,-5.473E-08)^{T}\) | 2.7071E − 15 | 7 |
8 | Camel: Three Hump | \((4.058E+00,-2.215E+00)^{T}\) | \((-3.0604E-07,3.778E-07)^{T}\) | 2.1442E − 13 | 38 | \((2.385E-08,-1.432E-07)^{T}\) | 1.8236E − 14 | 13 |
9 | Camel: Three Hump | \((-3.730E+00,4.688E-01)^{T}\) | \((1.2745E-07,-8.439E-08)^{T}\) | 2.8855E − 14 | 33 | \((-1.373E-07,2.043E-07)^{T}\) | 5.1397E − 14 | 9 |
10 | Camel: Three Hump | \((4.134E+00,4.575E+00)^{T}\) | \((-1.6209E-07,3.520E-07)^{T}\) | 1.1940E − 13 | 33 | \((4.917E-09,-4.958E-08)^{T}\) | 2.2623E − 15 | 10 |
11 | Camel: Three Hump | \((1.324E+00,4.649E+00)^{T}\) | \((-7.3313E-08,1.078E-07)^{T}\) | 1.4461E − 14 | 26 | \((2.647E-09,-6.108E-09)^{T}\) | 3.5154E − 17 | 7 |
12 | Camel: Six Hump | \((0.000E+00,9.000E-01)^{T}\) | \((-8.9842E-02,7.127E-01)^{T}\) | −1.0316E + 00 | 19 | \((-8.984E-02,7.127E-01)^{T}\) | −1.0316E + 00 | 8 |
13 | Camel: Six Hump | \((1.000E+00,-1.000E+00)^{T}\) | \((8.9842E-02,-7.127E-01)^{T}\) | −1.0316E + 00 | 29 | \((8.984E-02,-7.127E-01)^{T}\) | −1.0316E + 00 | 11 |
14 | Camel: Six Hump | \((-4.864E-01,-8.967E-01)^{T}\) | \((-8.9842E-02,7.127E-01)^{T}\) | −1.0316E + 00 | 26 | \((-8.984E-02,7.127E-01)^{T}\) | −1.0316E + 00 | 9 |
15 | Camel: Six Hump | \((4.561E-01,1.377E+00)^{T}\) | \((8.9842E-02,-7.127E-01)^{T}\) | −1.0316E + 00 | 25 | \((-8.984E-02,7.127E-01)^{T}\) | −1.0316E + 00 | 11 |
16 | Camel: Six Hump | \((1.036E+00,-7.086E-01)^{T}\) | \((8.9842E-02,-7.127E-01)^{T}\) | −1.0316E + 00 | 31 | \((-8.984E-02,7.127E-01)^{T}\) | −1.0316E + 00 | 13 |
17 | Chung Reynolds | \((-1.225E+00,-2.047E-01)^{T}\) | \((-5.0624E-03,-4.786E-03)^{T}\) | 1.1816E − 09 | 12 | \((4.931E-03,-5.851E-03)^{T}\) | 1.7631E − 09 | 12 |
18 | Cube | \((-1.000E+00,2.000E+00)^{T}\) | \((1.0180E+00,1.054E+00)^{T}\) | 3.6294E − 04 | 14 | \((1.000E+00,1.000E+00)^{T}\) | 1.9171E − 13 | 38 |
19 | Cube | \((-7.000E-01,1.700E+00)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 1.2254E − 13 | 76 | \((-2.669E-07,-1.256E-07)^{T}\) | 1.6958E − 13 | 40 |
20 | Dixon–Price | \((1.892E+00,4.133E+00)^{T}\) | \((9.9172E-01,7.070E-01)^{T}\) | 2.0024E − 04 | 17 | \((1.000E+00,-7.071E-01)^{T}\) | 7.3023E − 15 | 12 |
21 | Dixon–Price | \((2.482E+00,-1.476E+00)^{T}\) | \((1.0000E+00,7.071E-01)^{T}\) | 2.8514E − 15 | 24 | \((1.000E+00,7.071E-01)^{T}\) | 4.3728E − 16 | 12 |
22 | Egg Crate | \((1.000E-01,5.000E-01)^{T}\) | \((-6.4083E-10,1.830E-08)^{T}\) | 8.7177E − 15 | 18 | \((1.759E-11,2.400E-12)^{T}\) | 8.1916E − 21 | 5 |
23 | Himmelblau | \((1.236E+00,1.941E+00)^{T}\) | \((3.0000E+00,2.000E+00)^{T}\) | 1.8810E − 16 | 21 | \((3.000E+00,2.000E+00)^{T}\) | 7.6171E − 15 | 12 |
24 | Himmelblau | \((2.809E+00,1.169E+00)^{T}\) | \((3.0000E+00,2.000E+00)^{T}\) | 5.6991E − 15 | 23 | \((3.000E+00,2.000E+00)^{T}\) | 2.9048E − 17 | 12 |
25 | Himmelblau | \((1.339E+00,3.097E+00)^{T}\) | \((3.0000E+00,2.000E+00)^{T}\) | 4.5473E − 15 | 20 | \((3.000E+00,2.000E+00)^{T}\) | 1.4686E − 16 | 16 |
26 | Himmelblau | \((2.723E+00,2.701E+00)^{T}\) | \((3.0000E+00,2.000E+00)^{T}\) | 3.8989E − 15 | 32 | \((3.000E+00,2.000E+00)^{T}\) | 2.6335E − 18 | 11 |
27 | Leon | \((5.007E-01,1.187E+00)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 4.7101E − 13 | 43 | \((1.000E+00,1.000E+00)^{T}\) | 2.4088E − 14 | 20 |
28 | Leon | \((1.083E+00,1.080E+00)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 3.4102E − 14 | 51 | \((1.000E+00,1.000E+00)^{T}\) | 6.3897E − 13 | 30 |
29 | Leon | \((9.337E-01,4.431E-01)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 5.8582E − 15 | 53 | \((1.000E+00,1.000E+00)^{T}\) | 9.0974E − 13 | 41 |
30 | Rotated Ellipse 2 | \((2.242E+00,-2.944E+00)^{T}\) | \((-6.8708E-07,-6.668E-07)^{T}\) | 4.5830E − 13 | 20 | \((-5.928E-07,-5.525E-07)^{T}\) | 3.2912E − 13 | 4 |
31 | Rotated Ellipse 2 | \((-8.821E-01,3.536E+00)^{T}\) | \((-2.1865E-07,5.103E-08)^{T}\) | 6.1570E − 14 | 20 | \((-3.800E-08,-3.864E-08)^{T}\) | 1.4688E − 15 | 10 |
32 | Rotated Ellipse 2 | \((-2.067E+00,3.649E+00)^{T}\) | \((1.9903E-08,-3.484E-07)^{T}\) | 1.2873E − 13 | 25 | \((8.176E-08,-2.497E-08)^{T}\) | 9.3490E − 15 | 7 |
33 | Rotated Ellipse 2 | \((-7.687E-01,6.017E-01)^{T}\) | \((1.6098E-07,-2.524E-07)^{T}\) | 1.3026E − 13 | 17 | \((-8.662E-08,-9.228E-08)^{T}\) | 8.0248E − 15 | 4 |
34 | Hosaki | \((3.235E+00,1.287E+00)^{T}\) | \((4.0000E+00,2.000E+00)^{T}\) | −2.3458E + 00 | 24 | \((4.000E+00,2.000E+00)^{T}\) | −2.3458E + 00 | 9 |
35 | Leon | \((1.238E+00,7.653E-01)^{T}\) | \((9.7925E-01,9.571E-01)^{T}\) | 7.6735E − 04 | 19 | \((1.000E+00,1.000E+00)^{T}\) | 2.2956E − 13 | 23 |
36 | Leon | \((7.768E-01,-5.870E-01)^{T}\) | \((9.0943E-01,7.994E-01)^{T}\) | 8.4879E − 02 | 33 | \((1.000E+00,1.000E+00)^{T}\) | 1.9307E − 15 | 33 |
37 | Full Hessian 2 | \((1.000E+00,1.000E+00)^{T}\) | \((5.0000E+00,-4.000E+00)^{T}\) | 1.4611E − 14 | 18 | \((5.000E+00,-4.000E+00)^{T}\) | 2.5310E − 20 | 2 |
38 | Full Hessian 2 | \((2.231E+00,1.675E+00)^{T}\) | \((5.0000E+00,-4.000E+00)^{T}\) | 1.0410E − 14 | 19 | \((5.000E+00,-4.000E+00)^{T}\) | 2.8648E − 20 | 2 |
39 | Full Hessian 1 | \((-2.244E+00,-1.776E+00)^{T}\) | \((3.0000E+00,-2.997E+00)^{T}\) | 2.4935E − 10 | 61 | \((3.000E+00,-2.996E+00)^{T}\) | 4.3440E − 10 | 52 |
40 | Full Hessian 1 | \((2.864E+00,2.857E+00)^{T}\) | \((2.9994E+00,-3.022E+00)^{T}\) | 2.8657E − 06 | 27 | \((3.000E+00,-3.004E+00)^{T}\) | 3.2194E − 10 | 40 |
41 | Full Hessian 1 | \((1.683E+00,-1.268E+00)^{T}\) | \((3.0000E+00,-2.996E+00)^{T}\) | 5.4192E − 10 | 49 | \((3.000E+00,-2.991E+00)^{T}\) | 1.3520E − 08 | 41 |
42 | Extended Maratos | \((3.683E+00,4.268E+00)^{T}\) | \((-5.0000E-03,-2.646E-09)^{T}\) | −1.0000E + 02 | 17 | \((-5.000E-03,6.277E-09)^{T}\) | −1.0000E + 02 | 2 |
43 | Generalized White & Holst | \((1.010E+00,2.000E-02)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 1.3010E − 14 | 100 | \((1.000E+00,1.000E+00)^{T}\) | 9.0708E − 13 | 37 |
44 | Fletcher | \((3.150E+00,4.693E+00)^{T}\) | \((-2.9958E+00,4.979E+00)^{T}\) | 5.2363E − 17 | 14 | \((-2.615E+00,3.223E+00)^{T}\) | 8.9434E − 19 | 4 |
45 | Zakharov | \((1.786E+00,8.985E-02)^{T}\) | \((-9.5498E-04,1.184E-03)^{T}\) | 2.8119E − 06 | 7 | \((-5.000E-03,-1.585E-02)^{T}\) | 6.1296E − 04 | 6 |
46 | Zakharov | \((-1.812E+00,7.856E-01)^{T}\) | \((-8.0501E-08,7.469E-08)^{T}\) | 1.3245E − 14 | 13 | \((-2.701E-03,-1.436E-02)^{T}\) | 4.6014E − 04 | 3 |
47 | HIMMELH | \((1.200E+00,1.000E-01)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | −1.0000E + 00 | 17 | \((1.000E+00,1.000E+00)^{T}\) | −1.0000E + 00 | 9 |
48 | HIMMELH | \((1.200E+00,-2.100E+00)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | −1.0000E + 00 | 21 | \((1.000E+00,1.000E+00)^{T}\) | −1.0000E + 00 | 8 |
49 | DIXON3DQ | \((2.245E+00,2.306E+00)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 2.8005E − 14 | 17 | \((1.000E+00,1.000E+00)^{T}\) | 5.6225E − 19 | 2 |
50 | DIXON3DQ | \((7.012E-01,1.999E+00)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 6.5167E − 14 | 47 | \((1.000E+00,1.000E+00)^{T}\) | 1.1851E − 20 | 2 |
51 | DIXON3DQ | \((1.236E+00,5.929E-02)^{T}\) | \((1.0000E+00,1.000E+00)^{T}\) | 7.6903E − 14 | 42 | \((1.000E+00,1.000E+00)^{T}\) | 2.4317E − 21 | 2 |