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Table 4 Numerical results using Algorithm 1

From: A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems

Sl. no. Test problem Starting point q-PRP
Using standard Wolfe Using strong Wolfe
\(x^{*}\) \(f(x^{*})\) NI \(x^{*}\) \(f(x^{*})\) NI
1 Ackley \((2.000E-01,1.000E-01)^{T}\) \((-5.553E-08,-1.525E-06)^{T}\) 4.3168E − 06 20 \((-5.782E-08,8.525E-08)^{T}\) 2.9134E − 07 14
2 Beale \((-1.370E+00,-1.433E+00)^{T}\) \((3.173E+00,5.909E-01)^{T}\) 8.6289E − 02 16 \((3.000E+00,5.000E-01)^{T}\) 1.4030E − 09 17
3 Beale \((1.882E+00,-3.130E-01)^{T}\) \((3.003E+00,5.007E-01)^{T}\) 1.0665E − 06 20 \((3.002E+00,5.004E-01)^{T}\) 3.7846E − 07 10
4 Beale \((1.476E+00,5.286E-01)^{T}\) \((3.003E+00,5.007E-01)^{T}\) 1.0818E − 06 16 \((3.000E+00,5.000E-01)^{T}\) 6.5751E − 09 26
5 Beale \((6.077E-01,-1.118E+00)^{T}\) \((3.000E+00,5.001E-01)^{T}\) 1.8662E − 08 37 \((3.000E+00,5.000E-01)^{T}\) 2.8397E − 09 31
6 Booth \((1.000E+00,4.000E+00)^{T}\) \((9.987E-01,3.000E+00)^{T}\) 5.4094E − 06 16 \((1.000E+00,3.000E+00)^{T}\) 1.1595E − 09 4
7 Camel: Three Hump \((1.472E-01,-4.025E+00)^{T}\) \((-1.113E-07,-3.133E-07)^{T}\) 1.5777E − 13 28 \((-5.381E-08,9.050E-08)^{T}\) 9.1115E − 15 7
8 Camel: Three Hump \((4.058E+00,-2.215E+00)^{T}\) \((7.076E-07,-3.795E-07)^{T}\) 8.7677E − 13 26 \((1.160E-07,-6.956E-07)^{T}\) 4.3008E − 13 14
9 Camel: Three Hump \((-3.730E+00,4.688E-01)^{T}\) \((9.617E-07,-1.863E-06)^{T}\) 3.5296E − 12 33 \((-1.593E-07,-1.680E-08)^{T}\) 5.3740E − 14 10
10 Camel: Three Hump \((4.134E+00,4.575E+00)^{T}\) \((8.942E-07,5.196E-06)^{T}\) 3.3240E − 11 36 \((-5.209E-07,-4.531E-07)^{T}\) 9.8400E − 13 9
11 Camel: Three Hump \((1.324E+00,4.649E+00)^{T}\) \((7.574E-08,-1.716E-07)^{T}\) 2.7932E − 14 41 \((-1.350E-07,-8.584E-09)^{T}\) 3.7701E − 14 6
12 Camel: Six Hump \((0.000E+00,9.000E-01)^{T}\) \((-8.985E-02,7.127E-01)^{T}\) −1.0316E + 00 14 \((-8.984E-02,7.127E-01)^{T}\) −1.0316E + 00 6
13 Camel: Six Hump \((1.000E+00,-1.000E+00)^{T}\) \((8.970E-02,-7.125E-01)^{T}\) −1.0316E + 00 22 \((8.987E-02,-7.127E-01)^{T}\) −1.0316E + 00 9
14 Camel: Six Hump \((-4.864E-01,-8.967E-01)^{T}\) \((-8.987E-02,7.127E-01)^{T}\) −1.0316E + 00 16 \((0.000E+00,0.000E+00)^{T}\) 0.0000E + 00 0
15 Camel: Six Hump \((4.561E-01,1.377E+00)^{T}\) \((8.988E-02,-7.127E-01)^{T}\) −1.0316E + 00 15 \((-8.985E-02,7.127E-01)^{T}\) −1.0316E + 00 8
16 Camel: Six Hump \((1.036E+00,-7.086E-01)^{T}\) \((8.985E-02,-7.127E-01)^{T}\) −1.0316E + 00 21 \((-8.983E-02,7.127E-01)^{T}\) −1.0316E + 00 10
17 Chung Reynolds \((-1.225E+00,-2.047E-01)^{T}\) \((-5.060E-03,-4.790E-03)^{T}\) 1.1819E − 09 12 \((4.930E-03,-5.851E-03)^{T}\) 1.7627E − 09 12
18 Cube \((-1.000E+00,2.000E+00)^{T}\) \((1.074E+00,1.240E+00)^{T}\) 5.5657E − 03 9 \((1.067E+00,1.216E+00)^{T}\) 4.5261E − 03 4
19 Cube \((-7.000E-01,1.700E+00)^{T}\) \((1.069E+00,1.223E+00)^{T}\) 4.8337E − 03 19 \((9.790E-01,9.381E-01)^{T}\) 4.4353E − 04 14
20 Dixon–Price \((1.892E+00,4.133E+00)^{T}\) \((1.000E+00,7.072E-01)^{T}\) 2.1449E − 07 25 \((1.000E+00,-7.072E-01)^{T}\) 2.0993E − 08 10
21 Dixon–Price \((2.482E+00,-1.476E+00)^{T}\) \((1.001E+00,7.070E-01)^{T}\) 1.9147E − 06 19 \((1.000E+00,7.072E-01)^{T}\) 1.2528E − 08 11
22 Egg Crate \((1.000E-01,5.000E-01)^{T}\) \((-2.165E-07,-7.486E-07)^{T}\) 1.5788E − 11 18 \((-5.116E-07,-4.764E-07)^{T}\) 1.2706E − 11 6
23 Himmelblau \((1.236E+00,1.941E+00)^{T}\) \((3.000E+00,2.000E+00)^{T}\) 6.8617E − 07 13 \((3.000E+00,2.000E+00)^{T}\) 2.7077E − 07 8
24 Himmelblau \((2.809E+00,1.169E+00)^{T}\) \((3.000E+00,2.000E+00)^{T}\) 1.2192E − 06 12 \((3.000E+00,2.000E+00)^{T}\) 8.0157E − 07 8
25 Himmelblau \((1.339E+00,3.097E+00)^{T}\) \((3.000E+00,2.000E+00)^{T}\) 1.0315E − 06 8 \((3.000E+00,2.000E+00)^{T}\) 5.6783E − 07 10
26 Himmelblau \((2.723E+00,2.701E+00)^{T}\) \((3.000E+00,2.000E+00)^{T}\) 1.8308E − 06 14 \((3.000E+00,2.000E+00)^{T}\) 6.8798E − 07 8
27 Leon \((5.007E-01,1.187E+00)^{T}\) \((9.967E-01,9.902E-01)^{T}\) 1.0976E − 05 17 \((1.027E+00,1.083E+00)^{T}\) 7.3372E − 04 9
28 Leon \((1.083E+00,1.080E+00)^{T}\) \((1.030E+00,1.094E+00)^{T}\) 9.1674E − 04 7 \((1.031E+00,1.095E+00)^{T}\) 9.4325E − 04 2
29 Leon \((9.337E-01,4.431E-01)^{T}\) \((1.007E+00,1.021E+00)^{T}\) 4.9947E − 05 20 \((1.007E+00,1.021E+00)^{T}\) 4.8354E − 05 17
30 Rotated Ellipse 2 \((2.242E+00,-2.944E+00)^{T}\) \((5.030E-07,-1.723E-06)^{T}\) 4.0889E − 12 20 \((-5.710E-08,-7.529E-08)^{T}\) 4.6296E − 15 5
31 Rotated Ellipse 2 \((-8.821E-01,3.536E+00)^{T}\) \((-1.186E-06,-9.251E-07)^{T}\) 1.1653E − 12 18 \((-1.117E-06,-8.961E-07)^{T}\) 1.0495E − 12 9
32 Rotated Ellipse 2 \((-2.067E+00,3.649E+00)^{T}\) \((-4.149E-07,-5.158E-07)^{T}\) 2.2421E − 13 23 \((-6.791E-07,-7.858E-07)^{T}\) 5.4504E − 13 8
33 Rotated Ellipse 2 \((-7.687E-01,6.017E-01)^{T}\) \((-8.170E-08,-7.721E-08)^{T}\) 6.3286E − 15 18 \((4.064E-08,-2.384E-08)^{T}\) 3.1887E − 15 5
34 Hosaki \((3.235E+00,1.287E+00)^{T}\) \((4.000E+00,2.000E+00)^{T}\) −2.3458E + 00 6 \((4.000E+00,2.000E+00)^{T}\) −2.3458E + 00 11
35 Leon \((1.238E+00,7.653E-01)^{T}\) \((1.003E+00,1.006E+00)^{T}\) 9.3647E − 06 19 \((1.002E+00,1.003E+00)^{T}\) 2.6746E − 06 16
36 Leon \((7.768E-01,-5.870E-01)^{T}\) \((9.932E-01,9.881E-01)^{T}\) 3.6659E − 04 32 \((1.002E+00,1.004E+00)^{T}\) 3.3665E − 06 25
37 Full Hessian 2 \((1.000E+00,1.000E+00)^{T}\) \((5.000E+00,-4.000E+00)^{T}\) 3.8588E − 08 16 \((5.000E+00,-4.000E+00)^{T}\) 4.5348E − 09 4
38 Full Hessian 2 \((2.231E+00,1.675E+00)^{T}\) \((5.000E+00,-4.001E+00)^{T}\) 2.3319E − 07 13 \((5.000E+00,-4.000E+00)^{T}\) 3.7499E − 08 3
39 Full Hessian 1 \((-2.244E+00,-1.776E+00)^{T}\) \((3.000E+00,-2.982E+00)^{T}\) 2.5418E − 07 33 \((3.001E+00,-2.977E+00)^{T}\) 6.1723E − 07 20
40 Full Hessian 1 \((2.864E+00,2.857E+00)^{T}\) \((3.000E+00,-2.979E+00)^{T}\) 4.0601E − 07 21 \((3.000E+00,-2.980E+00)^{T}\) 3.4617E − 07 27
41 Full Hessian 1 \((1.683E+00,-1.268E+00)^{T}\) \((3.000E+00,-2.992E+00)^{T}\) 1.2249E − 08 17 \((3.000E+00,-2.982E+00)^{T}\) 2.4862E − 07 17
42 Extended Maratos \((3.683E+00,4.268E+00)^{T}\) \((4.999E-03,1.113E-06)^{T}\) −1.0000E + 02 13 \((-5.000E-03,-1.175E-08)^{T}\) −1.0000E + 02 3
43 Generalized White & Holst \((1.010E+00,2.000E-02)^{T}\) \((1.016E+00,1.050E+00)^{T}\) 2.7167E − 04 20 \((1.063E+00,1.202E+00)^{T}\) 4.0101E − 03 13
44 Fletcher \((3.150E+00,4.693E+00)^{T}\) \((-2.995E+00,4.979E+00)^{T}\) 6.9824E − 04 6 \((-2.611E+00,3.205E+00)^{T}\) 6.0116E − 09 4
45 Zakharov \((1.786E+00,8.985E-02)^{T}\) \((-9.500E-04,1.178E-03)^{T}\) 2.7825E − 06 6 \((-4.934E-03,-1.568E-02)^{T}\) 5.9939E − 04 6
46 Zakharov \((-1.812E+00,7.856E-01)^{T}\) \((5.014E-06,-1.573E-06)^{T}\) 2.8491E − 11 12 \((-2.703E-03,-1.435E-02)^{T}\) 4.6007E − 04 3
47 HIMMELH \((1.200E+00,1.000E-01)^{T}\) \((1.000E+00,9.999E-01)^{T}\) −1.0000E + 00 20 \((1.000E+00,1.000E+00)^{T}\) −1.0000E + 00 9
48 HIMMELH \((1.200E+00,-2.100E+00)^{T}\) \((1.000E+00,1.000E+00)^{T}\) −1.0000E + 00 11 \((1.000E+00,1.000E+00)^{T}\) −1.0000E + 00 7
49 DIXON3DQ \((2.245E+00,2.306E+00)^{T}\) \((1.000E+00,1.000E+00)^{T}\) 1.8385E − 08 13 \((1.000E+00,1.000E+00)^{T}\) 1.3340E − 10 3
50 DIXON3DQ \((7.012E-01,1.999E+00)^{T}\) \((1.000E+00,1.000E+00)^{T}\) 1.3656E − 08 27 \((1.000E+00,1.000E+00)^{T}\) 1.1506E − 10 4
51 DIXON3DQ \((1.236E+00,5.929E-02)^{T}\) \((1.000E+00,1.000E+00)^{T}\) 2.4082E − 09 16 \((1.000E+00,1.000E+00)^{T}\) 2.4809E − 10 4
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