Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 6391321, 13 pages
https://doi.org/10.1155/2020/6391321

Research Article

A New Hybrid PRPFR Conjugate Gradient Method for Solving Nonlinear Monotone Equations and Image Restoration Problems

Yingjie Zhou, Yulun Wu, and Xiangrong Li

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

Correspondence should be addressed to Xiangrong Li; xrli68@163.com

Received 24 July 2020; Accepted 25 August 2020; Published 25 September 2020

Guest Editor: Wenjie Liu

Copyright © 2020 Yingjie Zhou 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.

A new hybrid PRPFR conjugate gradient method is presented in this paper, which is designed such that it owns sufficient descent property and trust region property. This method can be considered as a convex combination of the PRP method and the FR method while using the hyperplane projection technique. Under accelerated step length, the global convergence property is gained with some appropriate assumptions. Comparing with other methods, the numerical experiments show that the PRPFR method is more competitive for solving nonlinear equations and image restoration problems.

1. Introduction

Consider the following nonlinear equation:

where S is a closed and convex subset. The function is monotone and continuously differentiable which satisfies

This problem of monotone equations has all kinds of applications, such as compressive sensing [1] and chemical equilibrium problems [2]. The conjugate gradient algorithm generates the iteration point via

where is the step size generated from a proper line search and is a search direction. There are many methods to find , such as the Newton method [3], quasi-Newton method [4], and conjugate gradient method [5–13]. As we all know, the Newton method, the quasi-Newton method, and their related methods are very popular due to their local superlinear convergence property. But, it is expensive for them to compute the Jacobian matrix or the approximate Jacobian matrix in per iteration while the dimensions are very large.

Due to the simplicity, less storage, efficiency, and nice convergence property, the conjugate gradient method becomes more and more popular for solving nonlinear equations [14–18]. The search direction of the conjugate gradient method is usually defined as

where and is a parameter. Diverse means a diverse CG method. There are some famous CG methods such as the FR method [7], the PRP method [10], the DY method [5], the LS method [9], the HS method [8], and the CD method [6]. The parameter we mentioned is as follows:

where and means the Euclidian norm.

In 1990, the first hybrid conjugate gradient method was proposed by Touati-Ahmed and Storey [19], and this method has global convergence while using the strong Wolfe line search. Dai and Yuan [20] proposed a CG method for unconstrained optimization and proved the global convergence of these hybrid computational schemes. Dong and Jiao [21] proposed a convex combination of the PRP and DY methods for solving nonlinear equations. Furthermore, the projection technique proposed by Solodov and Svaiter [22] motivated many scholars to further develop methods for solving (1) [23, 24].

Inspired by the convex combination proposed in [25], we proposed a convex combination of a modified PRP and a modified FR method with trust region property which [25] was not present before. We also try to make the algorithm inherit the convergence of the PRP method and excellent performance of the FR method.

The main contributions of the hybrid CG method are as follows:

(i)The given direction is designed as a convex combination of PRP and FR methods(ii)The given direction has the sufficient descent property(iii)The given direction has the trust region property(iv)The global convergence of the presented algorithm is proved(v)Numerical experiments show that the algorithm is more competitive for nonlinear equations and image restoration problems

In Section 2, we introduce the motivation and the PRPFR algorithm. In Section 3, we prove the sufficient property and the trust region property and give the convergence analysis. Section 4 shows the numerical experiment results. The conclusion is reported in the last section.

2. Algorithm

Motivated by the convex combination proposed in [25], we proposed a method which is a convex combination of a modified PRP method and a modified FR method for solving problem (1) and image restoration problems. We have recalled the classic PRP and classic FR conjugate gradient parameters in (5) and (6). For global convergence and good numerical performance, we defined the modified parameters as

where is a constant.

Next, we utilized the global convergence of and the excellent numerical behavior of by defining a new parameter called . The new parameter is defined as

where , , .

The definition of is by Li and Fukushima [4], and is proposed in [26]. We now proposed a hybrid gradient search direction as

where is defined in (13).

Remark 1. By the definition of and , we obtain that

Therefore, we have

Remark 2. By the definition of and , we have

Now, we introduce the hyperplane projection method by Solodov and Svaiter [22]. We first give the definition of the projection operator :

There is a fundamental property, that is,

The hyperplane projection method is as follows: let be the current iteration point and be obtained by line search direction such that . According to the monotonicity of , we have

if is a solution. Then, the hyperplane

strictly separates the current iteration from the solution of problem (1). The next iterate can be computed by projecting on it. That is,

As for the step size, we will use an appropriate line search to make performance better. Andrei [27] presented an acceleration scheme that generates the step size in a multiplicative manner to improve the reduction of the function values along the iterations. The step size is defined as follows:

If , let .
Motivated by the above discussions, we proposed a hybrid conjugate gradient algorithm in Algorithm 1.

Algorithm 1. PRPFR conjugate gradient algorithm.

 Step 0: given an initial point and constant , , , , and , let . Step 1: if , stop. Otherwise, compute by (11)–(14). Step 2: choosing such that

 Step 3: computing through (24), if , then by (23). Step 4: setting , if , stop. Otherwise, compute the next iterate using (22). Step 5: let , go to Step 2.

3. Convergence Analysis

We will establish the convergence of Algorithm 1 in this section. First, we give the following assumptions.

Assumption 1. The function is Lipschitz continuous; that is, there exists a positive constant L such that

Assumption 2. The solution set of problem (1) is nonempty.
From Assumption 1, there exists a positive constant such that

Lemma 1. Let be defined by (11)–(14), and then satisfies the sufficient descent condition. That is,

Proof. If , we have . If , by (14), we have

Lemma 2. From defined in (11)–(14), satisfies the trust region property independent of the line search. That is,

Proof. According to equality (28) and Cauchy–Schwartz inequality, we have

Furthermore, since and , then

Then, the proof is complete.

Lemma 3. If and can be generated by the PRPRFR algorithm, then the step size satisfies

Proof. From the line search (25), supposing , then does not satisfy the line search (25). That is,

Using the Lipschitz continuous of and (28), we have

Therefore,

The proof is complete.

Lemma 4. (see [22]). Suppose that satisfies . Let be generated by the PRPFR algorithm. Then,

Lemma 5. Let be generated by the PRPFR algorithm, and then

Proof. Lemma 4 indicates that is bounded and

From (22) and (25), we have that

From the abovementioned, the proof is complete.
Now, we establish the global convergence of the PRPFR algorithm.

Theorem 1. Let be generated by the PRPFR algorithm. Then,

Proof. We assume (41) does not hold, and then there exists such that , . This together with (30) yields

According to (21)–(26) and (42), we get

The fourth inequality holds since and . The last inequality implies that is bounded. By (30), we have

where . Multiplying both sides of (33) with , we obtain that

This inequality mentioned above contradicts (38); therefore, (41) holds.

4. Numerical Experiments

In this section, we report some numerical experiments to investigate the computational efficiency of the proposed hybrid conjugate gradient algorithm. The numerical experiments will be divided into two sections including normal nonlinear equations and image restoration problems. All tests in this section are written in MATLAB 2018a, run on a PC with AMD Ryzen 7 4800 U with Radeon Graphics 1.80 Hz, 16 GB of SDRAM memory, and Windows 10 system.

4.1. Normal Nonlinear Equations

In this section, we test some normal nonlinear equations. The concrete test problems come from [28] and are listed in Table 1. To compare the numerical performance of the PRPFR algorithm, we also do the experiments with the modified PRP algorithm [29] and the FR algorithm [7]. The columns of Tables 2–4 have the following meaning.

Table 1: Test problems.

No.
Test problems

1
Exponential function 2

2
Trigonometric function

3
Singular function

4
Logarithmic function

5
Broyden tridiagonal function

6
Trigexp function

7
Strictly convex function 1

8
Variable dimensioned function

9
Tridiagonal system

10
Five-diagonal system

11
Extended Freudenstein and Roth function (n is even)

12
Discrete boundary value problem

Table 2: Test results of the PRPFR algorithm.

No.
Dim
PRPFR algorithm

NI/NF
CPU
GN

1
3000
13/135
0.03125
8.02E − 06

9000
17/205
1.71875
9.48E − 06

30000
61/802
13.046875
9.73E − 06

90000
6/91
2.421875
7.17E − 06

2
3000
14/39
0.0625
9.54E − 06

9000
14/39
0.828125
5.50E − 06

30000
13/36
0.984375
6.01E − 06

90000
12/33
1.21875
6.18E − 06

3
3000
19999/20253
33.25
1.46E − 05

9000
19501/19813
425.3125
9.99E − 06

30000
17291/17984
783.796875
1.00E − 05

90000
19518/21490
1765.453125
1.00E − 05

4
3000
24/49
0.0625
7.65E − 06

9000
25/51
0.515625
6.59E − 06

30000
26/53
1.0625
6.00E − 06

90000
27/55
2.734375
5.20E − 06

5
3000
49/192
0.078125
6.04E − 06

9000
60/229
1.140625
9.16E − 06

30000
49/190
3.5
4.52E − 06

90000
46/179
5.25
5.04E − 06

6
3000
28/147
0.109375
7.39E − 06

9000
29/151
1.1875
7.66E − 06

30000
29/154
3.09375
7.03E − 06

90000
33/170
5.171875
3.56E − 06

7
3000
22/46
0.046875
5.78E − 06

9000
23/48
0.234375
5.00E − 06

30000
23/48
1.203125
9.14E − 06

90000
24/50
2.234375
7.91E − 06

8
3000
1/3
0.0625
3.19E − 08

9000
1/3
0
0.00E + 00

30000
1/3
0.03125
0.00E + 00

90000
1/3
0.078125
0.00E + 00

9
3000
626/3337
1.96875
9.89E − 06

9000
1696/10439
71.9375
9.91E − 06

30000
1410/8756
150.765625
9.98E − 06

90000
1374/8779
236.90625
9.99E − 06

10
3000
917/5266
1.53125
9.96E − 06

9000
664/3474
24.890625
9.92E − 06

30000
480/3047
49.15625
9.87E − 06

90000
759/4313
119.015625
9.87E − 06

11
3000
305/1898
0.625
9.96E − 06

9000
276/1714
11.078125
9.69E − 06

30000
333/2069
35.9375
9.91E − 06

90000
283/1759
44.671875
8.69E − 06

12
3000
12/37
0.046875
5.29E − 06

9000
11/34
0.953125
5.99E − 06

30000
10/31
0.75
6.52E − 06

90000
9/28
1.5
7.52E − 06

: the serial number of the problem. : the dimensions of x. : the function evaluation numbers. : the iteration numbers. : the calculation time in seconds. : the final function norm evaluations when the program is stopped.

 : the parameters are chosen as , , , , and . : when or is satisfied, we stop the process.

Table 3: Test results of the modified PRP algorithm.

No.
Dim
Modified PRP algorithm

NI/NF
CPU
GN

1
3000
38/389
0.234375
9.84E − 06

9000
21/250
1.6875
9.08E − 06

30000
20/271
4.203125
9.70E − 06

90000
10/155
3.78125
8.09E − 06

2
3000
18/47
0.0625
9.87E − 06

9000
18/47
0.6875
5.67E − 06

30000
17/44
0.90625
6.32E − 06

90000
15/39
1.703125
7.76E − 06

3
3000
9999/10404
10.984375
2.83E − 05

9000
9999/10873
190.140625
5.01E − 05

30000
9999/11895
435.296875
3.15E − 05

90000
9999/13913
839.734375
4.85E − 05

4
3000
42/194
0.109375
5.69E − 09

9000
66/352
3
3.37E − 06

30000
121/748
13.28125
4.61E − 07

90000
218/1538
41.4375
7.05E − 08

5
3000
67/280
0.1875
5.90E − 06

9000
90/393
3.171875
8.05E − 06

30000
111/541
9.453125
7.88E − 06

90000
139/787
24.28125
6.16E − 06

6
3000
75/517
0.265625
6.67E − 06

9000
107/840
7.203125
3.88E − 06

30000
165/1487
26.96875
7.27E − 06

90000
252/2537
77.25
8.93E − 06

7
3000
39/138
0.046875
6.65E − 06

9000
56/245
2.84375
6.87E − 06

30000
89/477
7.53125
5.70E − 06

90000
139/869
25.09375
6.00E − 06

8
3000
1/2
0.03125
0.00E + 00

9000
1/2
0
0.00E + 00

30000
1/2
0.0625
0.00E + 00

90000
1/2
0.15625
0.00E + 00

9
3000
5544/43870
12.765625
9.85E − 06

9000
5832/49309
347.203125
9.84E − 06

30000
6424/60507
960.546875
9.68E − 06

90000
7445/80662
1983.578125
9.83E − 06

10
3000
768/4997
1.515625
9.84E − 06

9000
881/6108
44.8125
9.28E − 06

30000
1043/8118
121.359375
9.89E − 06

90000
1343/11835
301.625
8.92E − 06

11
3000
512/3345
1.84375
9.85E − 06

9000
545/3646
23.953125
9.95E − 06

30000
1122/7498
119.3125
9.90E − 06

90000
1229/8722
223.140625
9.54E − 06

12
3000
16/48
0.078125
7.22E − 06

9000
13/39
0.5625
8.96E − 06

30000
11/33
1.390625
8.41E − 06

90000
10/30
1.671875
7.92E − 06

Table 4: Test results of the FR algorithm.

No.
Dim
FR algorithm

NI/NF
CPU
GN

1
3000
29/371
0.203125
9.84E − 06

9000
21/294
2.25
9.38E − 06

30000
12/183
3.984375
9.11E − 06

90000
5/80
2.09375
6.44E − 06

2
3000
13/50
0.078125
6.48E − 06

9000
12/46
0.484375
7.47E − 06

30000
11/42
0.984375
8.14E − 06

90000
10/38
1.484375
9.72E − 06

3
3000
19999/266078
194
1.12E − 03

9000
19999/265865
2918.28125
1.20E − 03

30000
19999/265318
6049.65625
7.84E − 04

90000
19999/266012
8251.859375
3.84E − 04

4
3000
24/72
0.046875
5.59E − 06

9000
24/72
0.59375
9.65E − 06

30000
25/75
1.28125
8.80E − 06

90000
26/78
3.203125
7.62E − 06

5
3000
62/381
0.21875
7.73E − 06

9000
69/462
3
9.20E − 06

30000
101/767
12.609375
9.75E − 06

90000
128/1215
26.90625
8.22E − 06

6
3000
26/165
0.109375
9.06E − 06

9000
21/127
0.90625
6.10E − 06

30000
22/134
1.9375
7.51E − 06

90000
22/134
3.359375
5.31E − 06

7
3000
23/70
0.0625
5.30E − 06

9000
23/70
0.40625
9.17E − 06

30000
24/73
1.578125
8.37E − 06

90000
25/76
1.90625
7.25E − 06

8
3000
1/3
0
3.19E − 08

9000
1/3
0.078125
0.00E + 00

30000
1/3
0.140625
0.00E + 00

90000
1/3
0.03125
0.00E + 00

9
3000
1248/15399
4.46875
9.91E − 06

9000
1544/19579
125.3125
9.87E − 06

30000
1175/14881
203.828125
9.96E − 06

90000
1027/13000
263.96875
9.91E − 06

10
3000
763/8892
2.640625
9.90E − 06

9000
765/8912
53.84375
9.92E − 06

30000
806/9498
127.421875
9.65E − 06

90000
1572/20089
423.4375
9.97E − 06

11
3000
373/4072
1.078125
9.98E − 06

9000
387/4226
24.71875
9.49E − 06

30000
402/4393
57.828125
9.77E − 06

90000
415/4537
97.296875
9.44E − 06

12
3000
10/36
0.0625
7.27E − 06

9000
9/31
0.21875
5.83E − 06

30000
6/19
0.234375
7.66E − 06

90000
6/19
0.765625
3.95E − 06

According to Tables 2–4, it is evident that the three methods can solve most of the test problems with . However, the FR method cannot handle the function 3 with 9000 dimensions, 30000 dimensions, and 90000 dimensions, while the PRPFR can solve the problem with . For more directly knowing the performance of this method, Dolan and Moré [30] introduced a technique to compare the performance of different algorithms. In Figure 1, when , the PRPFR algorithm is obviously better than the FR algorithm and the modified PRP algorithm. In Figure 2, the PRPFR algorithm solves all the problems at approximately , while the FR algorithm solves of the test problems at approximately , and the modified PRP algorithm solves of the test problems at approximately . In Figure 3, the PRPFR algorithm solves of the problems at approximately . The FR algorithm and the modified PRP algorithm solve and of the test problems at approximately and , respectively. From Tables 2–4 and Figures 1–3, it is obvious that the performance of the PRPFR algorithm is better than that of the FR algorithm and the modified PRP algorithm for most problems. Therefore, we conclude that the PRPFR algorithm is competitive to the FR algorithm and the modified PRP algorithm.

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

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

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

Table 5: CPU times of the PRPFR algorithm and the modified PRP algorithm in seconds.

20% noise
Lena
Barbara
Man
Total

PRPFR algorithm
2.719
3.578
12.844
19.141

Modified PRP algorithm
3.281
4.563
13.125
20.969

45% noise
Lena
Barbara
Man
Total

PRPFR algorithm
5.672
7.172
29.969
42.813

Modified PRP algorithm
6.156
7.797
31.609
45.563

70% noise
Lena
Barbara
Man
Total

PRPFR algorithm
14.063
17.391
61.906
93.359

Modified PRP algorithm
17.078
18.797
62.453
98.328

4.2. Image Restoration Problems

Image restoration aims to recover the original image from an image damaged by impulse noise. These problems are significant in optimization fields. The stop rule is or . The experiments choose Lena (), Barbara (), and Man () as the test images. Meanwhile, we compare the PRPFR algorithm experiments’ performance with that of the modified PRP algorithm, where the step size is generated by Step 2 and Step 3 in the PRPFR algorithm. The detailed performance results are shown in Figures 4–6. It can be observed that both the PRPFR algorithm and the modified PRP algorithm are able to restore the blur image of these three images.

Figure 4: From left to right: a noisy image with salt-and-pepper noise and the restorations obtained with the PRPFR algorithm and the modified PRP algorithm.

Figure 5: From left to right: a noisy image with salt-and-pepper noise and the restorations obtained with the PRPFR algorithm and the modified PRP algorithm.

Figure 6: From left to right: a noisy image with salt-and-pepper noise and the restorations obtained with the PRPFR algorithm and the modified PRP algorithm.

From Figures 4–6, we can easily notice that both algorithms are successful for restoring these noisy images with , , and noise. According to the results in Table 5, we can draw the conclusion that the PRPFR algorithm is more effective than the modified PRP algorithm for noise problems, noise problems, and noise problems.

5. Conclusion

In this paper, a new hybrid conjugate gradient algorithm that combines the PRP and FR methods is proposed, while using the projection technique. The direction has the sufficient descent and trust region properties automatically. Global convergence of the proposed algorithm is established under appropriate assumptions. The numerical experiments show that the proposed algorithm is competitive and efficient for solving nonlinear equations and image restoration problems.

For further research, we have some thinking as follows: (i) If the convex combination is applied to the quasi-Newton method, can it have better properties? (ii) Under other line search techniques, can this conjugate gradient method have global convergence? (iii) Can the proposed algorithm be applied to compressive sensing?

Data Availability

All data are included in the paper.

Conflicts of Interest

There are no potential conflicts of interest.

Acknowledgments

The authors want to thank the support of the funds. This work was supported by the High Level Innovation Teams and Excellent Scholars Program in Guangxi Institutions of Higher Education (Grant no. [2019]32), the National Natural Science Foundation of China (Grant no. 11661009), and the Guangxi Natural Science Key Foundation (no. 2017GXNSFDA198046).

References

Y. Xiao and H. Zhu, “A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing,” Journal of Mathematical Analysis and Applications, vol. 405, no. 1, pp. 310–319, 2013.

K. Meintjes and A. P. Morgan, “Chemical equilibrium systems as numerical test problems,” ACM Transactions on Mathematical Software, vol. 16, no. 2, pp. 143–151, 1990.

P. N. Brown and Y. Saad, “Convergence theory of nonlinear Newton-Krylov algorithms,” SIAM Journal on Optimization, vol. 4, no. 2, pp. 297–330, 1994.

D.-H. Li and M. Fukushima, “A modified BFGS method and its global convergence in nonconvex minimization,” Journal of Computational and Applied Mathematics, vol. 129, no. 1-2, pp. 15–35, 2001.

Y. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM Journal on Optimization, vol. 10, pp. 177–182, 2000.

R. Fletcher, “Unconstrained optimization,” Pratical Method of Optimization, vol. 1, 1987.

R. Fletcher and C. Reeves, “Function minimization by conjugate gradients,” The Computer Journal, vol. 7, no. 2, pp. 149–154, 1964.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, no. 6, pp. 409–432, 1952.

Y. Liu and C. Storey, “Efficient generalized conjugate gradient algorithms, part 1: theory,” Journal of Optimization Theory and Applications, vol. 69, no. 1, pp. 129–137, 1991.

E. Polak and G. Ribière, “Note sur la convergence de méthodes de directions conjuguées,” Revue française d'informatique et de recherche opérationnelle. Série rouge, vol. 3, no. 16, pp. 35–43, 1969.

G. Yuan, J. Lu, and Z. Wang, “The PRP conjugate gradient algorithm with a modified WWP line search and its application in the image restoration problems,” Applied Numerical Mathematics, vol. 152, pp. 1–11, 2020.

G. Yuan, X. Wang, and Z. Sheng, “Family weak conjugate gradient algorithms and their convergence analysis for nonconvex functions,” Numerical Algorithms, vol. 84, no. 3, pp. 935–956, 2020.

G. Yuan, Z. Wei, and Y. Yang, “The global convergence of the Polak-Ribière-Polyak conjugate gradient algorithm under inexact line search for nonconvex functions,” Journal of Computational and Applied Mathematics, vol. 362, pp. 262–275, 2019.

A. B. Abubakar and P. Kumam, “A descent Dai-Liao conjugate gradient method for nonlinear equations,” Numerical Algorithms, vol. 81, no. 1, pp. 197–210, 2019.

A. B. Abubakar and P. Kumam, “An improved three-term derivative-free method for solving nonlinear equations,” Computational and Applied Mathematics, vol. 37, no. 5, pp. 6760–6773, 2018.

Z. Dai and H. Zhu, “A modified Hestenes-Stiefel-type derivative-free method for large-scale nonlinear monotone equations,” Mathematics, vol. 8, no. 2, p. 168, 2020.

G. Yuan, T. Li, and W. Hu, “A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems,” Applied Numerical Mathematics, vol. 147, pp. 129–141, 2020.

G. Yuan and M. Zhang, “A three-terms Polak-Ribière-Polyak conjugate gradient algorithm for large-scale nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 286, pp. 186–195, 2015.

D. Touati-Ahmed and C. Storey, “Efficient hybrid conjugate gradient techniques,” Journal of Optimization Theory and Applications, vol. 64, no. 2, pp. 379–397, 1990.

Y. Dai and Y. Yuan, “An efficient hybrid conjugate gradient method for unconstrained optimization,” Annals of Operations Research, vol. 103, pp. 33–47, 2001.

J. Dong and B. Jiao, “A new hybrid PRP-DY conjugate gradient method,” 2010 Third International Joint Conference on Computational Science and Optimization, vol. 2, pp. 70–74, 2010.

M. V. Solodov and B. F. Svaiter, “A globally convergent inexact Newton method for systems of monotone equations,” Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, vol. 22, no. 1998, pp. 355–369, 1969.

X. Y. Wang, S. J. Li, and X. P. Kou, “A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints,” Calcolo, vol. 53, no. 2, pp. 133–145, 2016.

C. Wang, Y. Wang, and C. Xu, “A projection method for a system of nonlinear monotone equations with convex constraints,” Mathematical Methods of Operations Research, vol. 66, no. 1, pp. 33–46, 2007.

K. Zahra and A. Ali, “A new hybrid conjugate gradient method for large-scale unconstrained optimization problem with non-convex objective function,” Computational and Applied Mathematics, vol. 38, p. 186, 2019.

E. G. Birgin and J. Martnez, “A spectral conjugate gradient method for unconstrained optimization,” Applied Mathematics and Optimization, vol. 43, no. 2, pp. 117–128, 2001.

N. Andrei, “Another conjugate gradient algorithm with guaranteed descent and conjugacy conditions for large-scale unconstrained optimization,” Journal of Optimization Theory and Applications, vol. 159, no. 1, pp. 159–182, 2013.

G. Yuan, Z. Wei, and S. Lu, “Limited memory BFGS method with backtracking for symmetric nonlinear equations,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 367–377, 2011.

L. Zhang, W. Zhou, and D.-H. Li, “A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence,” IMA Journal of Numerical Analysis, vol. 26, no. 4, pp. 629–640, 2006.

E. D. Dolan and J. J. Moré, “Benchmarking optimization software with performance profiles,” Mathematical Programming, vol. 91, no. 2, pp. 201–213, 2002.