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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