Skip to main content

Table 5 Numerical results using classical PRP

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
\