Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 7945467, 12 pages
https://doi.org/10.1155/2020/7945467

Research Article

An Accelerated Conjugate Gradient Algorithm for Solving Nonlinear Monotone Equations and Image Restoration Problems

Haishan Feng and Tingting Li

Business, College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, China

Correspondence should be addressed to Tingting Li; ltt@st.gxu.edu.cn

Received 6 July 2020; Accepted 25 August 2020; Published 5 October 2020

Guest Editor: Wenjie Liu

Copyright © 2020 Haishan Feng and Tingting Li. 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.

Combining the three-term conjugate gradient method of Yuan and Zhang and the acceleration step length of Andrei with the hyperplane projection method of Solodov and Svaiter, we propose an accelerated conjugate gradient algorithm for solving nonlinear monotone equations in this paper. The presented algorithm has the following properties: (i) All search directions generated by the algorithm satisfy the sufficient descent and trust region properties independent of the line search technique. (ii) A derivative-free search technique is proposed along the direction to obtain the step length . (iii) If , then an acceleration scheme is used to modify the step length in a multiplicative manner and create a point. (iv) If the point satisfies the given condition, then it is the next point; otherwise, the hyperplane projection technique is used to obtain the next point. (v) The global convergence of the proposed algorithm is established under some suitable conditions. Numerical comparisons with other conjugate gradient algorithms show that the accelerated computing scheme is more competitive. In addition, the presented algorithm can also be applied to image restoration.

1. Introduction

In this paper, the following nonlinear equation is considered:

where is continuous and monotone, and satisfies

It is not difficult to show that the solution set of monotone equation (1), unless empty, is convex. This problem has many significant applications in applied mathematics, economics, and engineering. For example, the economic equilibrium problem [1] can be transformed into problem (1). Generally, an iteration formula generates the next iteration point by

where is the step length and is a search direction, which are two important factors for solving nonlinear equations. Some derivative-free line search techniques [2–5] were proposed to search for step length . Li and Li [6] presented a derivative-free line search to find such that

where and . represents the Euclidean norm. The line search technique (4) is different from other existing derivative-free line search techniques because it does not use a merit function. If satisfies , the inequality (4) holds for all sufficiently small . As a result, can be obtained by some backtracking processes.

For , it is well known that the Newton methods [7, 8], quasi-Newton methods [9–13], spectral gradient methods [14, 15], and conjugate gradient methods [16, 17] can deal with large-scale nonlinear equations. For solving large-scale optimization problems, the conjugate gradient methods are quite effective since they only calculate and store the gradient value of the objective function. Many scholars have applied conjugate gradient theory to solve nonlinear monotone equations and have achieved good results [17–19]. Classical conjugate gradient methods include HS method [20], FR method [21], PRP method [22, 23], LS method [24], CD method [25], and DY method [26]. In particular, the PRP method, as one of the most effective methods, generates a small step near a minimum point; then, the subsequence generated for the search direction will automatically approach the negative gradient direction, which avoids continuously generating a small step. However, the global convergence of the PRP method is not established under inexact line search techniques for general functions. Many scholars have performed continuous research and have reached satisfactory conclusions. Zhang [27] proposed the MPRP method, where is designed as follows:

where , , , , and . It is easy to obtain from (5) that

The above equation indicates that is a descent direction of at . If the exact line search is used, then we have . Consequently, formula (5) is inferred to be the standard PRP method. Under some mild conditions, the MPRP method is globally convergent under the Armijo-type line search, but global convergence cannot be established under the weak Wolfe–Powell line search. The main reason is that the MPRP method does not satisfy the trust region property. Inspired by the above discussions, Yuan and Zhang [28] proposed a three-term PRP (TTPRP) method in which is defined by

where is a constant. It is worth noting that the denominator in the formula of the MPRP method is adjusted to in the formula of the TTPRP method. The TTPRP method automatically maintains the trust region property. Its global convergence is also established under certain conditions. Numerical results show that the TTPRP method is effective for large-scale nonlinear monotone equations.

In addition, the hyperplane projection method [17, 29–32] is the most effective method for solving large-scale nonlinear monotone equations. The concept of projection was first proposed by Goldstein [33] for convex programming in a Hilbert space. Furthermore, Solodov and Svaiter [34] proposed the hyperplane projection method for solving optimization problems. The specific process of the hyperplane projection method is as follows: let be the current iteration point, and obtain a point along a certain line search direction such that . Due to the monotonicity of , for a certain point that satisfies , it can be deduced that

Obviously, the hyperplane strictly separates the current iteration point from the solution set of the equation (1). The point is projected onto the hyperplane to obtain the next iteration point , i.e.,

The hyperplane projection method has been proved to possess good theoretical properties and numerical performance for nonlinear monotone equations [6, 34].

Furthermore, we know that the search directions tend to be poorly scaled in conjugate gradient methods. Consequently, in the line search, more function evaluations must be carried out to obtain an appropriate step length . Andrei [35] presented an acceleration scheme that modifies the step length in a multiplicative manner to improve the reduction of the function values along the iterations. The step length is defined as follows:

where . , . If , let . A numerical comparison with some conjugate gradient algorithms shows that the computational scheme is effective.

Inspired by the above discussions, we proposed an accelerated conjugate gradient algorithm that combines the TTPRP method, the acceleration step length, and the hyperplane projection method. The main contributions of the algorithm are as follows:

 An accelerated conjugate gradient algorithm is introduced for solving nonlinear monotone equations All search directions of the algorithm satisfy the sufficient descent condition All search directions of the algorithm belong to a trust region The global convergence of the presented algorithm is proved The numerical results show that the proposed algorithm is more effective for nonlinear monotone equations The algorithm can be applied to restore an original image from an image damaged by impulse noise

This paper is organized as follows: in the next section, we discuss the ATTPRP algorithm and global convergence analysis. In Section 3, we report the preliminary numerical experiments to show that the algorithm is efficient for nonlinear monotone equations and applicable to image restoration problems. In Section 4, the conclusion regarding the proposed algorithm is given.

2. Accelerated Algorithm and Convergence Analysis

In this section, we will propose an accelerated algorithm and prove its global convergence. The steps of the given algorithm are as follows.

2.1. Accelerated Three-Term PRP Conjugate Gradient (ATTPRP) Algorithm

 Step 0: choose any as the initial point and constants , , , , and , let . Step 1: stop if . Otherwise, compute by using formula (5). Step 2: choose satisfying the inequality (4). Step 3: if , then . Step 4: let the next iterative value be . Step 5: if , stop and let . Otherwise, determine using formula (9). Step 6: let . Go to Step 1.

The following lemma shows that the search direction designed by using formula (7) has not only the sufficient descent property but also the trust region property independent of the line search.

Lemma 1. is defined by using formula (7); then, we obtain

Proof. If , formulas (7) and (12) are obviously true. If , we obtain from formula (7) that

In addition, by formula (7), we get and

where . Then, the proof is completed.
The following assumption need to be established in order to study some properties of the ATTPRP algorithm:

Assumption 1.

(i)The solution set of the problem (1) is nonempty.(ii)The function is Lipschitz continuous on ; that is, there exists a positive constant satisfying

Remark 1. Assumption 1(ii) implies that is bounded; then, there exists a constant such that

In the following paper, if not specifically stated, we always assume that the conditions in Assumption 1 hold.

Lemma 2. Let and be generated by using the ATTPRP algorithm. The step length generated by the ATTPRP algorithm satisfies

where and .

Proof. By line search (4), assuming , let , by the definition , does not satisfy the line search (4). That is,

Since is Lipschitz continuous and by using formula (11), we have

namely,

This leads to the ideal inequality (17). The proof is completed.
The following lemma is similar to Lemma 1 in the study of Solodov and Svaiter [34], which also holds for the ATTPRP algorithm. Therefore, we only state it as follows but omit its proof.

Lemma 3. Let the sequence be generated by using the ATTPRP algorithm. Suppose that is a solution of problem (1) with . We obtain

In particular, the sequence is bounded, and

Remark 2. The above lemma reveals that the distance from the iterative points to the solution set of the problem (1) decreases along iterations. Otherwise, for any , it is followed from formulas (9) and (4) that

Particularly, we obtain

In the following part, the global convergence and the strong global convergence properties of the ATTPRP algorithm will be proven.

Theorem 1. Let be generated by using the ATTPRP algorithm. Then, we have

Proof. We will prove this theorem by contradiction. Supposing that the equation (25) does not hold, there exists a constant such that holds for all . From formula (12), we have

According to Lemma 3 and equation (24), the sequences and are bounded. By formulas (7) and (16), for all , we obtain

where . Since converges to zero, the last inequality shows that is bounded. By formulas (12) and (16), we obtain

where . Thus, from the formulas (16) and (17), we obtain that

This contradicts with formula (24). Consequently, the proof is completed.
The following theorem indicates the strong global convergence of the ATTPRP algorithm, which is similar to Theorem 1 in [6]. We also give a specific proof for convenience of understanding.

Theorem 2. Let be generated by using the ATTPRP algorithm. Then, the whole sequence converges to a solution of the problem (1).

Proof. Theorem 1 shows that there exists a subsequence of converging to a solution of the problem (1). On the other side, it follows from Lemma 3 that the sequence converges. Therefore, the whole sequence converges to .

3. Numerical Experiments

In this section, the numerical experiments will be divided into two parts for illustration. The first subsection involves normal nonlinear equations, and the second subsection describes image restoration problems. All tests in this section are coded in MATLAB R2017a, run on a PC with Intel (R) Core (TM) i5-4460 3.20 GHz, 8.00 GB of SDRAM memory, and Windows 7 operating system.

3.1. Normal Nonlinear Equations

In this subsection, we perform some numerical experiments to show the effectiveness of the ATTPRP algorithm. Some test problems and their relevant initial points are listed as follows:

Function 1. Exponential Function 1:

Initial guess: .

Function 2. Exponential Function 2:

Initial guess: .

Function 3. Singular function:

Initial guess: .

Function 4. Logarithmic function:

Initial guess: .

Function 5. Broyden tridiagonal function:

Initial guess: .

Function 6. Trigexp function:

Initial guess: .

Function 7. Strictly convex Function 1: is the gradient of .

Initial guess: .

Function 8. Variable dimensioned function:

Initial guess: .

Function 9. Tridiagonal system:

Initial guess: .

Function 10. Five-diagonal system:

Initial guess: .

Function 11. Extended Freudenstein and Roth function ( is even):
For i = 1, 2, …,

Initial guess: .

Function 12. Brent problem:

Initial guess: .
To test the numerical performances of the ATTPRP algorithm, we also perform the experiments with the LS algorithm and the TTPRP algorithm. The columns of Tables 1–3 have the following meanings:

 NO: the serial number of the problem Dim: the variable dimensions NI: the number of iterations NF: the number of iterations of the function value CPU: the calculation time in seconds GN: the final function norm evaluations when the program is stopped Initialization: the parameters are chosen as , , , , and  Stop rule: when the condition or NI is satisfied, we stop the process

From Tables 1–3, it is obvious that the three methods can successfully solve most of the test problems with NI. However, for Function 3 with 9000 and 90000 variables, the TTPRP and LS algorithms cannot handle the function, but the proposed algorithm can do with NI . To more directly show the methods’ performance, Dolan and Moré [36] proposed a drawing tool that can obtain the performance profiles of methods. Therefore, using the drawing tool, we obtain Figures 1–3, which are related to the NI, NF, and CPU in Tables 1–3. In Figure 1, the ATTPRP algorithm solves all test problems at approximately , while the LS algorithm solves of the test problems at approximately , and the TTPRP algorithm solves at approximately . Thus, we can obtain the result that the ATTPRP algorithm performs slightly better than the other two algorithms. When in Figure 2, the presented algorithm solves all test problems, the TTPRP algorithm solves of all test problems, and the LS algorithm only solves approximately of the test problems. Thus, it is not difficult to see that the ATTPRP algorithm is more competitive than the other two methods. In Figure 3, the curve of the ATTPRP algorithm is above those of the TTPRP and LS algorithms, which indicates that the proposed algorithm is more robust than the other two algorithms in terms of the CPU. In summary, the enhancement of the presented method is noticeable.

Table 1: Test results of the ATTPRP algorithm.

No.
Dim
ATTPRP algorithm

NI/NF
CPU
GN

1
3000
123/124
0.608404
9.97E − 06

9000
88/89
0.733205
9.86E − 06

30000
57/58
1.092007
9.96E − 06

90000
38/39
1.310408
9.85E − 06

2
3000
28/514
0.686404
9.50E − 06

9000
18/386
1.372809
9.22E − 06

30000
15/371
3.432022
9.83E − 06

90000
10/276
3.946825
9.95E − 06

3
3000
14016/20709
107.172687
1.00E − 05

9000
15769/23986
331.158923
9.99E − 06

30000
15609/27514
574.09928
1.00E − 05

90000
17136/35208
1503.022835
1.00E − 05

4
3000
63/558
1.029607
5.24E − 07

9000
106/1096
5.241634
8.39E − 07

30000
195/2337
24.258155
1.20E − 07

90000
336/4521
88.031364
2.21E − 08

5
3000
90/615
0.780005
8.52E − 06

9000
96/759
2.464816
7.76E − 06

30000
130/1194
11.154072
8.62E − 06

90000
176/1984
29.250188
8.82E − 06

6
3000
91/1223
1.918812
8.09E − 06

9000
136/2093
8.845257
9.86E − 06

30000
221/3905
38.282645
9.97E − 06

90000
379/7177
125.814807
9.33E − 06

7
3000
48/335
0.468003
9.66E − 06

9000
71/636
2.480416
8.57E − 06

30000
122/1339
13.135284
9.31E − 06

90000
203/2562
43.63348
8.53E − 06

8
3000
1/2
0.000001
0.00E + 00

9000
1/2
0.0468
0.00E + 00

30000
1/2
0.000001
0.00E + 00

90000
1/2
0.0156
0.00E + 00

9
3000
6014/88059
100.885847
9.93E − 06

9000
6437/103883
312.345202
9.91E − 06

30000
7345/137142
1209.803355
9.96E − 06

90000
8950/196012
2763.386114
9.99E − 06

10
3000
1811/19670
25.693365
9.99E − 06

9000
1947/22554
76.36249
9.82E − 06

30000
2202/28399
261.644877
9.95E − 06

90000
2638/38968
577.983705
1.00E − 05

11
3000
351/4205
5.382034
9.78E − 06

9000
409/5194
16.645307
9.44E − 06

30000
510/7139
64.475213
9.64E − 06

90000
683/10789
161.351834
9.52E − 06

12
3000
184/188
0.156001
9.98E − 06

9000
184/188
0.577204
9.98E − 06

30000
184/188
1.107607
9.98E − 06

90000
184/188
3.07322
9.98E − 06

Table 2: Test results of the TTPRP algorithm.

No.
Dim
TTPRP algorithm

NI/NF
CPU
GN

1
3000
129/130
0.436803
9.97E − 06

9000
89/90
0.873606
9.96E − 06

30000
59/60
1.232408
9.96E − 06

90000
41/42
1.435209
9.73E − 06

2
3000
46/1055
1.435209
9.99E − 06

9000
10/267
0.951606
6.76E − 06

30000
9/278
2.667617
9.83E − 06

90000
8/279
4.040426
9.00E − 06

3
3000
17401/18908
109.481502
9.98E − 06

9000
19999/23227
355.089476
1.37E − 05

30000
19431/26598
612.241525
9.99E − 06

90000
19999/34409
1542.678289
2.56E − 05

4
3000
70/662
1.123207
3.49E − 06

9000
113/1326
6.099639
2.14E − 06

30000
196/2809
28.766584
6.47E − 07

90000
334/5473
101.884253
3.11E − 06

5
3000
54/464
0.514803
3.38E − 06

9000
72/711
2.152814
3.75E − 06

30000
100/1225
10.99807
7.38E − 06

90000
160/2256
31.403001
5.61E − 06

6
3000
84/1417
2.246414
6.94E − 06

9000
127/2473
10.311666
9.41E − 06

30000
211/4669
45.349491
8.60E − 06

90000
346/8524
141.773709
8.53E − 06

7
3000
45/366
0.546003
2.67E − 06

9000
68/729
2.808018
1.23E − 06

30000
117/1558
14.617294
8.34E − 07

90000
195/3039
48.219909
7.08E − 07

8
3000
1/2
0.0624
0.00E + 00

9000
1/2
0.000001
0.00E + 00

30000
1/2
0.0624
0.00E + 00

90000
1/2
0.0624
0.00E + 00

9
3000
6521/123863
139.277693
9.81E − 06

9000
6827/141140
425.679929
9.97E − 06

30000
7711/182289
1559.916399
9.99E − 06

90000
9107/251132
3460.398582
9.85E − 06

10
3000
4904/65319
77.750898
1.00E − 05

9000
5271/71854
229.820673
9.73E − 06

30000
5280/75948
689.150018
9.97E − 06

90000
5655/88105
1269.302137
9.99E − 06

12
3000
300/4529
4.898431
9.88E − 06

9000
387/6163
17.846514
9.61E − 06

30000
472/8397
72.836867
9.83E − 06

90000
620/12490
168.652681
9.27E − 06

13
3000
193/198
0.124801
9.99E − 06

9000
193/198
0.405603
9.99E − 06

30000
193/198
0.826805
9.99E − 06

90000
193/198
2.589617
9.99E − 06

Table 3: Test results of the LS algorithm.

No.
Dim
LS algorithm

NI/NF
CPU
GN

1
3000
174/175
0.982806
9.96E − 06

9000
94/95
0.670804
9.88E − 06

30000
60/61
1.107607
9.93E − 06

90000
41/42
1.248008
9.97E − 06

2
3000
61/1405
1.762811
9.85E − 06

9000
34/916
3.05762
9.98E − 06

30000
15/470
3.946825
9.46E − 06

90000
13/456
6.864044
9.12E − 06

3
3000
19999/21506
125.721206
1.43E − 05

9000
19999/23222
351.267452
1.44E − 05

30000
19999/27153
634.003664
2.24E − 05

90000
19999/34389
1582.942147
1.76E − 05

4
3000
75/666
1.185608
7.17E − 06

9000
118/1331
6.24004
6.87E − 06

30000
202/2813
30.186194
7.82E − 06

90000
338/5467
96.159016
7.65E − 06

5
3000
136/1145
1.404009
8.63E − 06

9000
150/1358
4.305628
9.81E − 06

30000
171/1806
16.801308
5.69E − 06

90000
234/2865
42.08907
9.58E − 06

6
3000
109/1695
2.698817
7.38E − 06

9000
147/2694
11.263272
8.98E − 06

30000
235/4934
47.502305
8.67E − 06

90000
364/8724
147.062143
7.57E − 06

7
3000
69/400
0.530403
6.22E − 06

9000
94/765
2.948419
3.29E − 06

30000
142/1590
15.085297
1.79E − 06

90000
217/3070
50.450723
4.55E − 07

8
3000
1/2
0.0624
0.00E + 00

9000
1/2
0.000001
0.00E + 00

30000
1/2
0.0468
0.00E + 00

90000
1/2
0.0156
0.00E + 00

9
3000
5643/107964
117.484353
9.84E − 06

9000
6131/128776
373.934397
9.86E − 06

30000
6997/169540
1461.323767
9.99E − 06

90000
8449/239554
3276.130201
9.88E − 06

10
3000
4171/55825
65.47362
9.93E − 06

9000
4213/58137
183.113974
1.00E − 05

30000
4616/67344
608.747102
9.88E − 06

90000
4965/79166
1150.008172
9.92E − 06

11
3000
3484/49132
52.073134
9.92E − 06

9000
371/5921
16.504906
9.85E − 06

30000
3625/52579
454.727315
9.93E − 06

90000
3802/57077
801.049535
9.87E − 06

12
3000
194/199
0.109201
9.96E − 06

9000
194/199
0.405603
9.96E − 06

30000
194/199
0.998406
9.96E − 06

90000
194/199
2.776818
9.96E − 06

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

Figure 2: Performance profiles of the methods (NF).

Figure 3: Performance profiles of the methods (CPU).

3.2. Image Restoration Problems

The purpose of this subsection is to recover the original image from an image damaged by impulse noise. It has important practical significance in optimization fields. The selection of parameters is similar to that in the above subsection. The stop condition is or . For the experiments, Cameraman , Barbara , and Man are chosen as the test images. We also perform experiments to compare the ATTPRP algorithm with the TTPRP algorithm, where the step length is generated by Step 2 and Step 3 in the ATTPRP algorithm. More detailed performance results are shown in Figures 4–6. It is not difficult to see that both the ATTPRP and TTPRP algorithms are successful in the image restoration of the three images. The expenditure of the CPU time is listed in Table 4 to compare the ATTPRP algorithm with the TTPRP algorithm.

Figure 4: Restoration of the Cameraman, Barbara, and Man images by using the ATTPRP algorithm and TTPRP algorithm. From left to right: a noisy image with 30% salt-and-pepper noise and the restorations obtained with the ATTPRP algorithm and the TTPRP algorithm by minimizing .

Figure 5: Restoration of the Cameraman, Barbara, and Man images by using the ATTPRP algorithm and TTPRP algorithm. From left to right: a noisy image with 50% salt-and-pepper noise and the restorations obtained with the ATTPRP algorithm and TTPRP algorithm by minimizing .

Figure 6: Restoration of the Cameraman, Barbara, and Man images by using the ATTPRP algorithm and TTPRP algorithm. From left to right: a noisy image with 70% salt-and-pepper noise and the restorations obtained with the ATTPRP algorithm and TTPRP algorithm by minimizing .

Table 4: CPU times of the ATTPRP algorithm and TTPRP algorithm in seconds.

30% noise
Cameraman
Barbara
Man
Total

ATTPRP algorithm
2.184
4.789
20.514
27.487

TTPRP algorithm
2.23
5.179
20.748
28.157

50% noise
Cameraman
Barbara
Man
Total

ATTPRP algorithm
3.276
9.142
35.475
47.893

TTPRP algorithm
3.307
9.204
35.677
48.188

70% noise
Cameraman
Barbara
Man
Total

ATTPRP algorithm
3.619
13.073
58.812
75.504

TTPRP algorithm
3.978
13.4
59.249
76.627

From Figures 1–4, we can obviously note that both algorithms can perfectly restore a noisy image with 30%, 50%, and 70% salt-and-pepper noise. In addition, the results in Table 4 show that the ATTPRP algorithm and the TTPRP algorithm are both successful in restoring these images with an approximate CPU time. The presented algorithm is slightly competitive with the TTPRP algorithm for 30% noise problems, 50% noise problems, and 70% noise problems.

4. Conclusions

In this paper, an accelerated conjugate gradient algorithm that combines the TTPRP method, the acceleration step length, and the hyperplane projection technique is proposed. All search directions generated by using the algorithm automatically have sufficient descent and trust region properties. The global convergence property of the proposed algorithm is established under suitable conditions. The numerical results show that the proposed algorithm is effective. The image restoration problems also demonstrate that the proposed algorithm is successful

For future research, we have some ideas as follows: (i) If the acceleration system is introduced into the quasi-Newton method, does it have some good properties? (ii) Can the acceleration system be introduced into the trust region method to solve unconstrained optimization problems and nonlinear equations? (iii) Can the proposed algorithm be applied to machine learning?

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11661009), the High Level Innovation Teams and Excellent Scholars Program in Guangxi Institutions of Higher Education (Grant No. (2019)52), and the Guangxi Natural Science Key Fund (No. 2017GXNSFDA198046).

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