Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 4519274, 9 pages
https://doi.org/10.1155/2020/4519274

Research Article

Convergence Analysis of an Improved BFGS Method and Its Application in the Muskingum Model

Tianshan Yang,1,2 Pengyuan Li,³ and Xiaoliang Wang⁴

¹School of Business, Guangxi University, Nanning, Guangxi, China
²School of Finance and Economics, Nanning College for Vocational Technology, Nanning, Guangxi, China
³College of Mathematics and Information Science, Guangxi Center for Mathematical Research, Guangxi University, Nanning, Guangxi, China
⁴School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, China

Correspondence should be addressed to Xiaoliang Wang; xliangwang@126.com

Received 28 June 2020; Accepted 28 July 2020; Published 18 August 2020

Guest Editor: Weijun Zhou

Copyright © 2020 Tianshan Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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:

where , . To solve (1), the following iterative formula is widely used. Given the starting point , the iterative scheme is

where is the current iteration point, is the next iteration point, is the step length, and is the search direction that is obtained by solving the quasi-Newton equation:

where is the gradient of at the point , is the quasi-Newton updating matrix or its approximation, and the sequence satisfies the standard secant equation . The updating matrix of can be defined by

where , , and is symmetric and positive definite.

Formula (4) is the famous standard BFGS update formula, which is one of the most effective quasi-Newton methods. For a convex function, using exact line search or some special inexact line search, the global convergence (see [1, 2]) and superlinear convergence (see [3, 4]) of the BFGS method were obtained. For general functions, the BFGS method may fail under inexact line search techniques. This fact has been proven by Dai [5], and Mascarenhas [6] has also proven that the BFGS method is not convergent, even under the exact line search techniques. Although the convergence of the BFGS method under general nonconvex functions has some shortcomings, its high efficiency and great numerical stability have motivated many scholars [7–12] to study and improve the BFGS method. The improvements achieved by scholars are as follows.

Formula 1. (see [7]). The BFGS update formula is modified by

where and function satisfies (i) for all ; (ii) if and only if ; (iii) if is in a bounded set, is bounded. Li and Fukushima discussed its global convergence without the convexity assumption on .

Formula 2. (see [8]) The BFGS update formula is modified by

where and . Moreover, scholars [8, 13] have proven that this method is better than the original BFGS method.

Formula 3. (see [9]) The BFGS update formula is modified by

where and . According to , it is clear that the method contains both gradient and function value information. In addition, the modulated quasi-Newton method with superlinear convergence constructed by formula 3 is studied in [9].

Formula 4. (see [14]) The BFGS update formula is modified by

where and . The global convergence of the improved BFGS method (MBFGS) is discussed by Li et al. [14]. Meanwhile, they also compared the three methods in numerical experiments. The results show that the algorithm based on this method is superior to the other three methods.
In many optimization algorithms, scholars often use the weak Wolfe–Powell (WWP) line search technique to find the step length . The WWP line search technique is determined by

where , , and .
In order to get more interesting properties of WWP line search, many scholars have improved the line search technique. Yuan et al. [15] improved the WWP line search technique and studied the new line search technique that has global convergence in the BFGS and PRP methods. Their improved line search technique (MWWP) is formulated as follows:

where , , , and . The detailed line search is elaborated in [15]. Some research results based on this improved line search can be found in [16, 17]. The above discussion motivate us to seek an improved BFGS method which may obtain better numerical performance.
In this article, we will discuss our work in the following sections. In Section 2, using (8) and the MWWP line search technique, algorithms are constructed to solve optimization problems. In Section 3, we study convergence of the modified BFGS method. In Section 4, the numerical results of the algorithm are reported. In the last section, the conclusion is presented.

2. Algorithm

The corresponding modified BFGS algorithm is called Algorithm 1 and can be presented as follows.

Algorithm 1.

(i)Step 1: choose an initial point , , , , and . Given an initial symmetric and positive definite matrix , set .(ii)Step 2: when , stop. Otherwise, take the next step.(iii)Step 3: solve to obtain .(iv)Step 4: the step length is determined by (10) and (11).(v)Step 5: set a new iteration point of . Update by (8).(vi)Step 6: let and return to Step 2.

Remark 1. The step length , generated by the proposed new line search technique, has a great numerical performance. And the rationality proof of the MWWP line search has been given in [15].

3. Convergence Analysis

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

Assumption 1.

(i)The level set of is bounded.(ii)The objective function is convex on .(iii) is twice continuously differentiable and bounded below, with a Lipschitz continuous gradient function . It means that there exists a positive constant , such that

Next, we will give the global convergence. The positive definite of is presented in the following lemma.

Lemma 1. Let the sequence be generated by (8); then, the matrix is positive definite for all .

Proof. Induction is used to prove the positive definiteness of . For , it is obvious that the matrix is positive definite. For , by and (11), we have

where the last inequality holds by and . Therefore, the matrix is positive definite. The proof is completed.

Lemma 2. Let Assumption 1 hold, and the sequence is generated in the Algorithm 1. Then,

Proof. By MWWP line search (8) and formula (12), we obtain

Therefore, the following bound holds:

By (10) and Assumption 1 (iii), we have

Adding these inequalities from to , we obtain

Combining the above inequality with (16), we obtain (14). Therefore, Lemma 2 has been proven.

Remark 2. It is obvious that there are two values of the . Therefore, the MWWP line search has two situations. In this paper, we discuss the situation of .

Lemma 3. Let and Assumption 1 hold. Then, there exists a positive constant such that

hold for at least values of with any positive integer .

Proof. By the , if , then . Lemma 3 holds (see [15]).
If , then . It is similar to the result of Yuan and Wei [18]. According to the convexity of objective function , we obtain

The above two inequalities and the definition of indicate that

Then, we obtain

Therefore, by the above analysis, it follows

By the definition of , it follows that

Then, we have that

The proof of Theorem 2.1 of [2] implies that Lemma 3 holds.
Based on the above conclusions, the global convergence is analysed in the following theorem.

Theorem 1. If the conditions of Lemma 3 hold, then we obtain

Proof. By Lemma 2, we obtain

Since , we have

Combining (19) with (20), we obtain

Thus, , , and , and we obtain

Therefore, (27) holds. The proof is complete.

4. Numerical Results

In this section, we will study the numerical performance of the MBFGS-MWWP algorithm established in Section 2. To verify the algorithm’s effectiveness, we divide the experiments into two parts: we first compare our algorithm with the standard BFGS method with the weak Wolfe–Powell line search technique (BFGS-WWP) in 74 academic problems listed in Table 1 with the dimension varying from 300 to 2700 and then apply our algorithm to the Muskingum engineering model.

Table 1: 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

4.1. Unconstrained Optimisation Problems

In this section, we compare Algorithm 1 with the BFGS-WWP algorithms for the 74 academic problems listed in Table 1. The codes are written with MATLAB R2014a and run on a PC with an Inter (R) Core (TM) i5-4210U CPU @ 1.70 GHz, 8.00 GB of RAM and the Windows 10 operating system, and the parameters are chosen as , , , and . The numerical results and comparison are shown in Tables 2–6. The columns in Tables 2–6 have the following meaning:

(i): the index of the tested problems(ii): the dimension of the tested problem(iii): the iteration number consumed(iv): the total number both of the gradient and the function value(v): the time consumed in corresponding algorithms in seconds

Table 2: 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

Table 3: 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

Table 4: 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

Table 5: 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

Table 6: 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 [19] to show the performance of different algorithms. From the strategy in [19], the higher the line in the figure is, the better the numerical results are. Figures 1–3 show the NI, NFG, and CPU time performances, respectively, of the new algorithm and the standard BFGS method. From the results shown in the figures, the NI, NFG, and CPU time of the algorithm constructed in this paper are generally better than those of the standard BFGS algorithm. Therefore, our algorithm is interesting and reliable from this perspective.

Figure 1: Performance profiles of these methods (NI).

Figure 2: Performance profiles of these methods (NFG).

Figure 3: Performance profiles of these methods (CPU time).

4.2. Muskingum Model

In this section, the main work is to use Algorithm 1 to numerically estimate the Muskingum model [20], whose definition is as follows:

where is the total time, is the observed inflow discharge, is the observed outflow discharge, denotes the storage time constant, denotes the weight coefficient, denotes an extra parameter, and is the time step at time (). The observation data in the experiment are from the process of flood runoff from Chenggouwan and Linqing of Nanyunhe in the Haihe Basin, Tianjin, China, and the initial point . In addition, the time step are selected, and the detailed values of and for the years 1960, 1961, and 1964 are given in [21].

Figures 4–6 and Table 7 imply the following conclusions: (1) Similar to the BFGS method and the HIWO method, the MBFGS method, combined with the Muskingum model, has interesting experimental results. (2) The final points (, , and ) of the MBFGS method are more competitive than other similar methods. (3) Due to the difference in the final points of these three methods, the Muskingum model may have more approximate optimal points.

Figure 4: Performance of Algorithm 1 in 1960.

Figure 5: Performance of Algorithm 1 in 1961.

Figure 6: Performance of Algorithm 1 in 1964.

Table 7: Results of the three algorithms.

Algorithm

BFGS [22]
10.8156
0.9826
1.0219

HIWO [20]
13.2813
0.8001
0.9933

MBFGS
11.1832
0.9999
0.9996

5. Conclusion

In this paper, we study the improved BFGS method with the line search technique [14, 15] and mainly discuss its convergence on a convex function. The numerical results show that the proposed algorithm has a better problem-solving capability than that of the standard BFGS algorithm based on WWP line search. As for the further work, we have several points to consider: (i) That whether the improved BFGS method, based on other line search, also has convergence property. (ii) The combination of line search techniques (10) and (11) with other quasi-Newton methods is worth studying. (iii) Similar applications of nonlinear conjugate gradient algorithm, especially the PRP method, are also worthy of attention.

Data Availability

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.

Conflicts of Interest

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

Acknowledgments

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

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