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## Machine Learning and its Applications in Image Restoration

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Research Article | Open Access

Volume 2020 |Article ID 6391321 | https://doi.org/10.1155/2020/6391321

Yingjie Zhou, Yulun Wu, Xiangrong Li, "A New Hybrid PRPFR Conjugate Gradient Method for Solving Nonlinear Monotone Equations and Image Restoration Problems", Mathematical Problems in Engineering, vol. 2020, Article ID 6391321, 13 pages, 2020. https://doi.org/10.1155/2020/6391321

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

Guest Editor: Wenjie Liu
Accepted25 Aug 2020
Published25 Sep 2020

#### 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 viawhere 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 [513]. 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 [1418]. The search direction of the conjugate gradient method is usually defined aswhere 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 aswhere 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 aswhere , , .

The definition of is by Li and Fukushima [4], and is proposed in [26]. We now proposed a hybrid gradient search direction aswhere is defined in (13).

Remark 1. By the definition of and , we obtain thatTherefore, we have

Remark 2. By the definition of and , we haveNow, 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 haveif is a solution. Then, the hyperplanestrictly 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 thatStep 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 haveFurthermore, since and , thenThen, 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 haveTherefore,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 andFrom (22) and (25), we have thatFrom 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) yieldsAccording to (21)–(26) and (42), we getThe fourth inequality holds since and . The last inequality implies that is bounded. By (30), we havewhere . Multiplying both sides of (33) with , we obtain thatThis 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 24 have the following meaning.

 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
 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.
 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
 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 24, 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 24 and Figures 13, 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.

 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 46. 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.

From Figures 46, 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).

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