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Journal of Inequalities and Applications

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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