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The BFGS method is one of the most effective quasi-Newton algorithms for minimization-optimization problems. In this paper, an improved BFGS method with a modified weak Wolfe–Powell line search technique is used to solve convex minimization problems and its convergence analysis is established. Seventy-four academic test problems and the Muskingum model are implemented in the numerical experiment. The numerical results show that our algorithm is comparable to the usual BFGS algorithm in terms of the number of iterations and the time consumed, which indicates our algorithm is effective and reliable.

With the development of the economy and society, a large number of optimization problems have been emerged in the fields of economic management, aerospace, transportation, national defense and so on. It is very necessary and meaningful for us to discuss, analyse the problems, and find some effective methods to solve them. Let us consider the optimization model:

Formula (

(see [

(see [

(see [

(see [

In many optimization algorithms, scholars often use the weak Wolfe–Powell (WWP) line search technique to find the step length

In order to get more interesting properties of WWP line search, many scholars have improved the line search technique. Yuan et al. [

In this article, we will discuss our work in the following sections. In Section

The corresponding modified BFGS algorithm is called Algorithm

Step 1: choose an initial point

Step 2: when

Step 3: solve

Step 4: the step length

Step 5: set a new iteration point of

Step 6: let

The step length

The global convergence analysis of the improved BFGS method will be introduced in this section, and the following assumptions are needed.

The level set of

The objective function

Next, we will give the global convergence. The positive definite of

Let the sequence

Induction is used to prove the positive definiteness of

Let Assumption

By MWWP line search (

Therefore, the following bound holds:

By (

Adding these inequalities from

Combining the above inequality with (

It is obvious that there are two values of the

Let

By the

If

The above two inequalities and the definition of

Then, we obtain

Therefore, by the above analysis, it follows

By the definition of

Then, we have that

The proof of Theorem 2.1 of [

Based on the above conclusions, the global convergence is analysed in the following theorem.

If the conditions of Lemma

By Lemma

Since

Combining (

Thus,

Therefore, (

In this section, we will study the numerical performance of the MBFGS-MWWP algorithm established in Section

The test problems.

Test problem | |
---|---|

1 | Extended Freudenstein and Roth function |

2 | Extended trigonometric function |

3 | Extended Rosenbrock function |

4 | Extended White and Holst function |

5 | Extended Beale function |

6 | Extended penalty function |

7 | Perturbed quadratic function |

8 | Raydan 1 function |

9 | Raydan 2 function |

10 | Diagonal 1 function |

11 | Diagonal 2 function |

12 | Diagonal 3 function |

13 | Hager function |

14 | Generalized Tridiagonal-1 function |

15 | Extended Tridiagonal-1 function |

16 | Extended three exponential terms function |

17 | Generalized Tridiagonal-2 function |

18 | Diagonal 4 function |

19 | Diagonal 5 function |

20 | Extended Himmelblau function |

21 | Generalized PSC1 function |

22 | Extended PSC1 function |

23 | Extended Powell function |

24 | Extended block diagonal BD1 function |

25 | Extended Maratos function |

26 | Extended Cliff function |

27 | Quadratic diagonal perturbed function |

28 | Extended Wood function |

29 | Extended Hiebert function |

30 | Quadratic function QF1 function |

31 | Extended quadratic penalty QP1 function |

32 | Extended quadratic penalty QP2 function |

33 | A quadratic function QF2 function |

34 | Extended EP1 function |

35 | Extended Tridiagonal-2 function |

36 | BDQRTIC function (CUTE) |

37 | TRIDIA function (CUTE) |

38 | ARWHEAD function (CUTE) |

39 | NONDIA function (CUTE) |

40 | NONDQUAR function (CUTE) |

41 | DQDRTIC function (CUTE) |

42 | EG2 function (CUTE) |

43 | DIXMAANA function (CUTE) |

44 | DIXMAANB function (CUTE) |

45 | DIXMAANC function (CUTE) |

46 | DIXMAANE function (CUTE) |

47 | Partial perturbed quadratic function |

48 | Broyden Tridiagonal function |

49 | Almost perturbed quadratic function |

50 | Tridiagonal perturbed quadratic function |

51 | EDENSCH function (CUTE) |

52 | VARDIM function (CUTE) |

53 | STAIRCASE S1 function |

54 | LIARWHD function (CUTE) |

55 | DIAGONAL 6 function |

56 | DIXON3DQ function (CUTE) |

57 | DIXMAANF function (CUTE) |

58 | DIXMAANG function (CUTE) |

59 | DIXMAANH function (CUTE) |

60 | DIXMAANI function (CUTE) |

61 | DIXMAANJ function (CUTE) |

62 | DIXMAANK function (CUTE) |

63 | DIXMAANL function (CUTE) |

64 | DIXMAAND function (CUTE) |

65 | ENGVAL1 function (CUTE) |

66 | FLETCHCR function (CUTE) |

67 | COSINE function (CUTE) |

68 | Extended DENSCHNB function (CUTE) |

69 | Extended DENSCHNF function (CUTE) |

70 | SINQUAD function (CUTE) |

71 | BIGGSB1 function (CUTE) |

72 | Partial perturbed quadratic PPQ2 function |

73 | Scaled quadratic SQ1 function |

74 | Scaled quadratic SQ2 function |

In this section, we compare Algorithm

The numerical results for problems 1–17.

N0 | Dim | MBFGS-MWWP | BFGS-WWP | ||||
---|---|---|---|---|---|---|---|

NI | NFG | CPU time | NI | NFG | CPU time | ||

1 | 300 | 9 | 26 | 0.1251 | 9 | 26 | 1.0625 |

1 | 900 | 7 | 19 | 2.4063 | 7 | 19 | 2.4375 |

1 | 2700 | 7 | 21 | 34.6561 | 7 | 21 | 34.0131 |

2 | 300 | 279 | 598 | 4.6563 | 254 | 547 | 4.0052 |

2 | 900 | 660 | 1381 | 186.2312 | 700 | 1489 | 203.3942 |

2 | 2700 | 7 | 16 | 30.4691 | 7 | 16 | 30.5030 |

3 | 300 | 628 | 1471 | 8.7344 | 676 | 1472 | 9.2969 |

3 | 900 | 1000 | 2429 | 286.1406 | 1000 | 2598 | 275.7656 |

3 | 2700 | 1000 | 3086 | 5405.7969 | 1000 | 2999 | 5390.2344 |

4 | 300 | 733 | 1636 | 9.9219 | 733 | 1610 | 10.9375 |

4 | 900 | 1000 | 2371 | 278.4063 | 1000 | 2421 | 285.8281 |

4 | 2700 | 1000 | 2110 | 5408.3906 | 1000 | 2116 | 5375.0313 |

5 | 300 | 13 | 39 | 0.1875 | 15 | 45 | 0.2031 |

5 | 900 | 15 | 46 | 4.0313 | 15 | 46 | 3.8125 |

5 | 2700 | 15 | 46 | 74.1875 | 15 | 46 | 73.2188 |

6 | 300 | 8 | 33 | 0.0938 | 8 | 33 | 0.1406 |

6 | 900 | 13 | 49 | 3.4844 | 13 | 49 | 3.0625 |

6 | 2700 | 4 | 17 | 10.5001 | 4 | 17 | 10.3906 |

7 | 300 | 198 | 401 | 2.5625 | 198 | 401 | 2.5781 |

7 | 900 | 409 | 823 | 112.9531 | 409 | 823 | 111.6563 |

7 | 2700 | 883 | 1768 | 4735.7969 | 883 | 1768 | 4768.6094 |

8 | 300 | 22 | 48 | 0.4063 | 22 | 48 | 0.3281 |

8 | 900 | 31 | 66 | 8.3594 | 31 | 66 | 8.5156 |

8 | 2700 | 57 | 118 | 294.2188 | 57 | 118 | 292.7812 |

9 | 300 | 6 | 16 | 0.0625 | 13 | 28 | 0.1563 |

9 | 900 | 6 | 16 | 1.4063 | 13 | 28 | 3.2656 |

9 | 2700 | 6 | 16 | 24.7813 | 14 | 30 | 63.6563 |

10 | 300 | 2 | 9 | 0.0313 | 2 | 9 | 0 |

10 | 900 | 2 | 9 | 0.0469 | 2 | 9 | 0.0469 |

10 | 2700 | 2 | 9 | 0.1563 | 2 | 9 | 0.1563 |

11 | 300 | 46 | 94 | 0.6094 | 46 | 94 | 0.5781 |

11 | 900 | 67 | 136 | 18.0312 | 66 | 134 | 19.0021 |

11 | 2700 | 82 | 200 | 422.8906 | 97 | 196 | 533.4688 |

12 | 300 | 38 | 78 | 0.5313 | 38 | 78 | 0.5469 |

12 | 900 | 46 | 94 | 12.7031 | 46 | 94 | 12.6253 |

12 | 2700 | 66 | 134 | 340.4375 | 66 | 134 | 343.1406 |

13 | 300 | 14 | 34 | 0.1562 | 14 | 34 | 0.2031 |

13 | 900 | 15 | 42 | 3.9688 | 15 | 42 | 4.0156 |

13 | 2700 | 49 | 153 | 257.4219 | 49 | 151 | 250.4063 |

14 | 300 | 11 | 25 | 0.2813 | 11 | 25 | 0.3438 |

14 | 900 | 10 | 23 | 2.8594 | 10 | 23 | 2.5625 |

14 | 2700 | 6 | 15 | 25.1094 | 6 | 15 | 23.5313 |

15 | 300 | 13 | 32 | 0.3281 | 13 | 32 | 0.3906 |

15 | 900 | 14 | 34 | 3.8125 | 14 | 34 | 3.9219 |

15 | 2700 | 15 | 37 | 72.0156 | 15 | 37 | 75.4844 |

16 | 300 | 7 | 18 | 0.0625 | 7 | 18 | 0.0938 |

16 | 900 | 6 | 16 | 1.3281 | 6 | 16 | 1.2521 |

16 | 2700 | 6 | 18 | 24.4531 | 6 | 16 | 23.8906 |

17 | 300 | 98 | 199 | 1.2656 | 98 | 199 | 1.3281 |

17 | 900 | 88 | 182 | 23.9848 | 87 | 180 | 23.3281 |

17 | 2700 | 82 | 170 | 415.3125 | 82 | 171 | 405.2813 |

The numerical results for problems 18–34.

N0 | Dim | MBFGS-MWWP | BFGS-WWP | ||||
---|---|---|---|---|---|---|---|

NI | NFG | CPU time | NI | NFG | CPU time | ||

18 | 300 | 3 | 10 | 0.0312 | 3 | 10 | 0.0312 |

18 | 900 | 3 | 10 | 0.5468 | 3 | 10 | 0.4844 |

18 | 2700 | 3 | 10 | 9.3438 | 3 | 10 | 9.0032 |

19 | 300 | 3 | 9 | 0.0312 | 3 | 10 | 0.0312 |

19 | 900 | 3 | 9 | 0.5468 | 3 | 10 | 0.5312 |

19 | 2700 | 3 | 9 | 10.1718 | 3 | 10 | 9.6254 |

20 | 300 | 11 | 32 | 0.1875 | 8 | 27 | 0.0625 |

20 | 900 | 13 | 55 | 2.4687 | 10 | 29 | 1.8281 |

20 | 2700 | 16 | 48 | 65.7812 | 13 | 40 | 45.4531 |

21 | 300 | 47 | 122 | 0.5781 | 50 | 114 | 0.7187 |

21 | 900 | 58 | 141 | 16.2101 | 50 | 125 | 13.2567 |

21 | 2700 | 58 | 149 | 295.1718 | 42 | 108 | 207.6562 |

22 | 300 | 6 | 21 | 0.0937 | 7 | 30 | 0.1093 |

22 | 900 | 6 | 21 | 1.3906 | 7 | 30 | 1.5937 |

22 | 2700 | 6 | 21 | 25.1562 | 7 | 30 | 29.0781 |

23 | 300 | 114 | 393 | 1.8593 | 113 | 378 | 1.5937 |

23 | 900 | 152 | 523 | 42.8906 | 190 | 628 | 52.0468 |

23 | 2700 | 245 | 854 | 1304.2521 | 209 | 693 | 1105.5625 |

24 | 300 | 94 | 338 | 1.1562 | 31 | 161 | 0.2813 |

24 | 900 | 93 | 361 | 22.9531 | 17 | 127 | 1.9375 |

24 | 2700 | 24 | 129 | 74.7656 | 25 | 136 | 79.4375 |

25 | 300 | 686 | 1823 | 8.7812 | 734 | 1908 | 9.7031 |

25 | 900 | 1000 | 3200 | 280.9687 | 1000 | 2465 | 272.8751 |

25 | 2700 | 969 | 2867 | 5167.4062 | 1000 | 2791 | 5395.4062 |

26 | 300 | 4 | 15 | 0.0312 | 4 | 15 | 0.0312 |

26 | 900 | 4 | 15 | 0.6093 | 4 | 15 | 0.6406 |

26 | 2700 | 4 | 15 | 10.4681 | 4 | 15 | 10.1093 |

27 | 300 | 7 | 18 | 0.0937 | 7 | 18 | 0.1563 |

27 | 900 | 13 | 30 | 3.3125 | 13 | 30 | 3.7031 |

27 | 2700 | 28 | 60 | 139.2031 | 28 | 60 | 149.8593 |

28 | 300 | 88 | 255 | 1.3593 | 84 | 245 | 1.4843 |

28 | 900 | 103 | 288 | 28.2187 | 102 | 295 | 27.3125 |

28 | 2700 | 131 | 328 | 694.8125 | 128 | 330 | 669.3593 |

29 | 300 | 4 | 15 | 0.0312 | 4 | 15 | 0.0312 |

29 | 900 | 4 | 15 | 0.6562 | 4 | 15 | 0.4218 |

29 | 2700 | 4 | 15 | 10.3906 | 4 | 15 | 6.2968 |

30 | 300 | 207 | 419 | 2.7521 | 207 | 419 | 2.8281 |

30 | 900 | 444 | 893 | 122.4375 | 444 | 893 | 120.4687 |

30 | 2700 | 863 | 1728 | 4608.9843 | 863 | 1728 | 4734.1093 |

31 | 300 | 11 | 31 | 0.1406 | 11 | 31 | 0.1562 |

31 | 900 | 13 | 49 | 3.3593 | 13 | 49 | 3.8281 |

31 | 2700 | 16 | 58 | 79.8751 | 16 | 58 | 91.4531 |

32 | 300 | 13 | 44 | 0.1562 | 13 | 45 | 0.2031 |

32 | 900 | 12 | 33 | 3.0468 | 12 | 33 | 3.2521 |

32 | 2700 | 72 | 223 | 367.9531 | 83 | 235 | 435.4687 |

33 | 300 | 4 | 11 | 0.0312 | 4 | 11 | 0.0312 |

33 | 900 | 4 | 11 | 0.7187 | 4 | 11 | 0.6562 |

33 | 2700 | 4 | 11 | 12.5156 | 4 | 11 | 12.7812 |

34 | 300 | 3 | 9 | 0 | 3 | 9 | 0.0312 |

34 | 900 | 3 | 9 | 0.5625 | 3 | 9 | 0.5121 |

34 | 2700 | 4 | 10 | 14.7187 | 4 | 10 | 14.2031 |

The numerical results for problems 35–51.

N0 | Dim | MBFGS-MWWP | BFGS-WWP | ||||
---|---|---|---|---|---|---|---|

NI | NFG | CPU time | NI | NFG | CPU time | ||

35 | 300 | 5 | 13 | 0.0625 | 5 | 13 | 0.0625 |

35 | 900 | 5 | 13 | 1.0937 | 5 | 13 | 1.0781 |

35 | 2700 | 5 | 13 | 19.5937 | 5 | 13 | 18.1093 |

36 | 300 | 45 | 118 | 0.8751 | 45 | 116 | 0.9375 |

36 | 900 | 47 | 122 | 13.5937 | 55 | 145 | 15.4375 |

36 | 2700 | 64 | 171 | 324.6562 | 64 | 167 | 316.7968 |

37 | 300 | 251 | 507 | 3.5625 | 251 | 507 | 3.7031 |

37 | 900 | 577 | 1159 | 160.8751 | 577 | 1159 | 176.6251 |

37 | 2700 | 869 | 1741 | 4684.3281 | 869 | 1741 | 4743.0937 |

38 | 300 | 5 | 15 | 0.0625 | 5 | 15 | 0.0625 |

38 | 900 | 5 | 15 | 1.0468 | 5 | 15 | 1.1718 |

38 | 2700 | 5 | 15 | 19.7343 | 5 | 15 | 18.9218 |

39 | 300 | 19 | 72 | 0.2522 | 19 | 67 | 0.3281 |

39 | 900 | 23 | 87 | 6.2656 | 21 | 81 | 5.2521 |

39 | 2700 | 34 | 118 | 173.1123 | 35 | 120 | 181.7812 |

40 | 300 | 485 | 1162 | 7.2187 | 520 | 1055 | 7.4375 |

40 | 900 | 506 | 1162 | 143.3437 | 731 | 1468 | 198.0625 |

40 | 2700 | 528 | 1270 | 2713.4218 | 829 | 1664 | 4167.6718 |

41 | 300 | 12 | 34 | 0.1718 | 12 | 34 | 0.1718 |

41 | 900 | 12 | 34 | 2.9062 | 12 | 34 | 3.0468 |

41 | 2700 | 12 | 34 | 56.5231 | 12 | 34 | 53.7968 |

42 | 300 | 18 | 60 | 0.2187 | 16 | 55 | 0.1875 |

42 | 900 | 4 | 21 | 0.0781 | 4 | 21 | 0.0468 |

42 | 2700 | 4 | 21 | 0.2968 | 4 | 21 | 0.3125 |

43 | 300 | 15 | 38 | 0.2656 | 15 | 38 | 0.2343 |

43 | 900 | 18 | 44 | 5.4062 | 18 | 44 | 4.6875 |

43 | 2700 | 19 | 46 | 94.0312 | 20 | 48 | 92.6562 |

44 | 300 | 29 | 71 | 0.4375 | 27 | 70 | 0.3752 |

44 | 900 | 47 | 120 | 13.0937 | 83 | 204 | 22.3593 |

44 | 2700 | 87 | 203 | 443.7031 | 99 | 224 | 494.0937 |

45 | 300 | 56 | 131 | 0.7812 | 63 | 146 | 0.9531 |

45 | 900 | 75 | 178 | 20.8281 | 90 | 209 | 25.4843 |

45 | 2700 | 81 | 190 | 414.4218 | 124 | 265 | 615.4062 |

46 | 300 | 70 | 157 | 0.9375 | 90 | 196 | 1.67187 |

46 | 900 | 121 | 276 | 34.6718 | 117 | 258 | 32.5468 |

46 | 2700 | 174 | 401 | 916.8752 | 199 | 416 | 1049.3437 |

47 | 300 | 110 | 225 | 2.2968 | 110 | 225 | 2.3752 |

47 | 900 | 219 | 446 | 73.8437 | 219 | 446 | 68.9843 |

47 | 2700 | 346 | 702 | 1940.3437 | 346 | 702 | 1948.3437 |

48 | 300 | 90 | 183 | 1.6252 | 90 | 183 | 1.7031 |

48 | 900 | 75 | 157 | 20.5625 | 75 | 157 | 20.0468 |

48 | 2700 | 144 | 291 | 741.3751 | 144 | 291 | 716.9375 |

49 | 300 | 197 | 399 | 2.6718 | 197 | 399 | 3.0781 |

49 | 900 | 422 | 849 | 117.1562 | 422 | 849 | 116.6562 |

49 | 2700 | 882 | 1766 | 4726.1875 | 882 | 1766 | 4725.7031 |

50 | 300 | 185 | 375 | 4.1718 | 185 | 375 | 4.0312 |

50 | 900 | 395 | 795 | 116.1562 | 395 | 795 | 112.3906 |

50 | 2700 | 859 | 1720 | 4628.0468 | 859 | 1720 | 4585.6251 |

51 | 300 | 24 | 58 | 0.3125 | 24 | 58 | 0.3125 |

51 | 900 | 21 | 56 | 5.7031 | 30 | 78 | 8.2523 |

51 | 2700 | 33 | 80 | 163.7812 | 27 | 72 | 127.3593 |

The numerical results for problems 52–68.

N0 | Dim | MBFGS-MWWP | BFGS-WWP | ||||
---|---|---|---|---|---|---|---|

NI | NFG | CPU time | NI | NFG | CPU time | ||

52 | 300 | 3 | 13 | 0 | 3 | 13 | 0.0468 |

52 | 900 | 3 | 13 | 0.3437 | 3 | 13 | 0.2968 |

52 | 2700 | 3 | 13 | 5.6406 | 3 | 13 | 5.4062 |

53 | 300 | 314 | 634 | 4.1093 | 404 | 812 | 5.4063 |

53 | 900 | 927 | 1860 | 261.9531 | 1000 | 2004 | 270.2188 |

53 | 2700 | 1000 | 2004 | 5316.6406 | 1000 | 2002 | 5250.6406 |

54 | 300 | 25 | 78 | 0.3281 | 24 | 74 | 0.2969 |

54 | 900 | 28 | 88 | 7.3281 | 28 | 85 | 7.1406 |

54 | 2700 | 33 | 107 | 166.2181 | 37 | 116 | 183.0625 |

55 | 300 | 10 | 24 | 0.1406 | 20 | 42 | 0.2812 |

55 | 900 | 11 | 26 | 2.9062 | 21 | 44 | 5.2187 |

55 | 2700 | 11 | 26 | 50.9687 | 23 | 48 | 105.5123 |

56 | 300 | 311 | 626 | 3.7812 | 354 | 710 | 4.6252 |

56 | 900 | 921 | 1846 | 252.8593 | 1000 | 2002 | 271.5312 |

56 | 2700 | 1000 | 2002 | 5193.1251 | 1000 | 2000 | 4871.1241 |

57 | 300 | 77 | 199 | 1.2187 | 96 | 232 | 1.4531 |

57 | 900 | 7 | 29 | 1.6406 | 40 | 140 | 11.0781 |

57 | 2700 | 7 | 30 | 30.4531 | 15 | 75 | 65.8594 |

58 | 300 | 67 | 144 | 0.8751 | 73 | 154 | 1.2656 |

58 | 900 | 103 | 214 | 29.8431 | 103 | 214 | 28.1875 |

58 | 2700 | 28 | 88 | 124.1406 | 196 | 407 | 1044.9688 |

59 | 300 | 3 | 15 | 0.0312 | 3 | 15 | 0.0313 |

59 | 900 | 11 | 58 | 2.8125 | 11 | 58 | 3.2656 |

59 | 2700 | 24 | 111 | 107.3125 | 24 | 108 | 109.1093 |

60 | 300 | 71 | 166 | 0.9687 | 79 | 173 | 1.1875 |

60 | 900 | 133 | 295 | 42.2031 | 182 | 384 | 50.2031 |

60 | 2700 | 190 | 433 | 1024.2812 | 182 | 388 | 947.9062 |

61 | 300 | 68 | 172 | 0.9375 | 26 | 87 | 0.4375 |

61 | 900 | 7 | 29 | 1.7521 | 9 | 45 | 1.7656 |

61 | 2700 | 7 | 30 | 31.2812 | 19 | 90 | 82.3906 |

62 | 300 | 359 | 877 | 5.7181 | 391 | 908 | 6.3125 |

62 | 900 | 625 | 1452 | 182.9687 | 618 | 1415 | 168.5625 |

62 | 2700 | 622 | 1473 | 3372.6406 | 156 | 520 | 809.6251 |

63 | 300 | 157 | 371 | 2.5156 | 166 | 391 | 2.6562 |

63 | 900 | 236 | 571 | 67.6406 | 267 | 592 | 74.8906 |

63 | 2700 | 541 | 1216 | 2961.6875 | 493 | 1114 | 2569.2656 |

64 | 300 | 198 | 482 | 3.2131 | 202 | 485 | 3.2187 |

64 | 900 | 276 | 629 | 82.1562 | 21 | 112 | 5.4218 |

64 | 2700 | 3 | 12 | 9.9375 | 3 | 12 | 9.1406 |

65 | 300 | 51 | 114 | 1.0312 | 48 | 108 | 0.9062 |

65 | 900 | 51 | 122 | 15.4218 | 42 | 109 | 11.6562 |

65 | 2700 | 55 | 126 | 290.4844 | 77 | 179 | 379.0781 |

66 | 300 | 527 | 1187 | 10.2968 | 534 | 1211 | 10.1875 |

66 | 900 | 1000 | 2415 | 294.6251 | 1000 | 2442 | 285.8281 |

66 | 2700 | 1000 | 2494 | 5153.8281 | 1000 | 2544 | 4975.5468 |

67 | 300 | 6 | 21 | 0.0312 | 6 | 21 | 0 |

67 | 900 | 100 | 408 | 21.3281 | 12 | 33 | 0.1562 |

67 | 2700 | 10 | 29 | 0.9062 | 13 | 59 | 0.9843 |

68 | 300 | 11 | 26 | 0.1875 | 11 | 26 | 0.2031 |

68 | 900 | 11 | 26 | 3.2187 | 12 | 28 | 3.5156 |

68 | 2700 | 12 | 28 | 60.4375 | 12 | 28 | 52.7656 |

The numerical results for problems 69–74.

N0 | Dim | MBFGS-MWWP | BFGS-WWP | ||||
---|---|---|---|---|---|---|---|

NI | NFG | CPU time | NI | NFG | CPU time | ||

69 | 300 | 43 | 142 | 0.6875 | 26 | 81 | 0.3752 |

69 | 900 | 65 | 200 | 17.8906 | 28 | 89 | 7.5938 |

69 | 2700 | 73 | 222 | 399.7343 | 34 | 107 | 170.4375 |

70 | 300 | 358 | 847 | 9.2031 | 50 | 151 | 0.9218 |

70 | 900 | 44 | 159 | 13.8125 | 71 | 250 | 19.2812 |

70 | 2700 | 263 | 764 | 1469.7031 | 45 | 152 | 207.3438 |

71 | 300 | 163 | 332 | 2.0312 | 163 | 334 | 2.0938 |

71 | 900 | 483 | 975 | 147.3593 | 488 | 986 | 130.2656 |

71 | 2700 | 1000 | 2000 | 5118.8751 | 1000 | 2000 | 4951.7343 |

72 | 300 | 35 | 80 | 0.8281 | 43 | 94 | 0.7968 |

72 | 900 | 58 | 128 | 18.2656 | 57 | 126 | 18.1875 |

72 | 2700 | 96 | 208 | 535.1406 | 98 | 210 | 531.1562 |

73 | 300 | 133 | 271 | 2.2812 | 133 | 271 | 1.7031 |

73 | 900 | 225 | 455 | 63.8593 | 225 | 455 | 60.2521 |

73 | 2700 | 471 | 947 | 2572.9681 | 443 | 891 | 2325.0156 |

74 | 300 | 43 | 112 | 0.7187 | 43 | 112 | 0.5625 |

74 | 900 | 184 | 373 | 57.3593 | 184 | 373 | 50.5781 |

74 | 2700 | 262 | 529 | 1460.0468 | 262 | 529 | 1372.9375 |

For intuitive effect, we adopt the performance technique in [

Performance profiles of these methods (NI).

Performance profiles of these methods (NFG).

Performance profiles of these methods (CPU time).

In this section, the main work is to use Algorithm

Figures

Performance of Algorithm

Performance of Algorithm

Performance of Algorithm

Results of the three algorithms.

Algorithm | |||
---|---|---|---|

BFGS [ | 10.8156 | 0.9826 | 1.0219 |

HIWO [ | 13.2813 | 0.8001 | 0.9933 |

MBFGS | 11.1832 | 0.9999 | 0.9996 |

In this paper, we study the improved BFGS method with the line search technique [

The data used to support the findings of this study are available in tables in this paper and also can be obtained from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Basic Ability Promotion Project of Guangxi Young and Middle-Aged Teachers (No. 2020KY30018).