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

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. .is 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.


Introduction
Consider the following nonlinear equation: where S is a closed and convex subset. e function F: R n ⟶ R n is monotone and continuously differentiable which satisfies is problem of monotone equations has all kinds of applications, such as compressive sensing [1] and chemical equilibrium problems [2]. e conjugate gradient algorithm generates the iteration point via where α k is the step size generated from a proper line search and d k is a search direction. ere are many methods to find d k , such as the Newton method [3], quasi-Newton method [4], and conjugate gradient method [5][6][7][8][9][10][11][12][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][15][16][17][18]. e search direction of the conjugate gradient method is usually defined as where F k � F(x k ) and β k is a parameter. Diverse β k means a diverse CG method. ere 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]. e parameter we mentioned is as follows: where y k− 1 � F k − F k− 1 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. e main contributions of the hybrid CG method are as follows: (i) e given direction is designed as a convex combination of PRP and FR methods (ii) e given direction has the sufficient descent property (iii) e given direction has the trust region property (iv) e 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. e conclusion is reported in the last section.

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 t > 0 is a constant. Next, we utilized the global convergence of β MPRP k and the excellent numerical behavior of β MFR k by defining a new parameter called β PRPFR k . e new parameter is defined as where e definition of s k− 1 is by Li and Fukushima [4], and c k is proposed in [26]. We now proposed a hybrid gradient search direction as where β PRPFR k is defined in (13).

Remark 1.
By the definition of s k− 1 and y k− 1 , we obtain that erefore, we have Remark 2. By the definition of β PRPFR k and c k , we have Now, we introduce the hyperplane projection method by Solodov and Svaiter [22]. We first give the definition of the projection operator P S (·):

2
Mathematical Problems in Engineering ere is a fundamental property, that is, e hyperplane projection method is as follows: let x k be the current iteration point and According to the monotonicity of F(x), we have if x * is a solution. en, the hyperplane strictly separates the current iteration x k from the solution of problem (1). e next iterate can be computed by projecting on it. at 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 α k in a multiplicative manner to improve the reduction of the function values along the iterations. e step size is defined as follows: If ϕ k > 0, let α k � α k . Motivated by the above discussions, we proposed a hybrid conjugate gradient algorithm in Algorithm 1.

Convergence Analysis
We will establish the convergence of Algorithm 1 in this section. First, we give the following assumptions.
Assumption 2. e solution set of problem (1) is nonempty. From Assumption 1, there exists a positive constant ω such that (11)- (14), and then d k satisfies the sufficient descent condition. at is,

Lemma 1. Let d k be defined by
□ Lemma 2. From d k defined in (11)- (14), d k satisfies the trust region property independent of the line search. at is, Proof. According to equality (28) and Cauchy-Schwartz inequality, we have en, the proof is complete.
□ Lemma 3. If x k and z k can be generated by the PRPRFR algorithm, then the step size α k satisfies Proof. From the line search (25), supposing α k ≠ κ, then α k ′ � α k ρ − 1 does not satisfy the line search (25). at is, Using the Lipschitz continuous of F(x) and (28), we have erefore, e proof is complete.
□ Lemma 4 (see [22]). Suppose that x * ∈ R n satisfies F(x * ) � 0. Let x k be generated by the PRPFR algorithm. en, Lemma 5. Let x k be generated by the PRPFR algorithm, and then Proof. Lemma 4 indicates that x k 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 x k be generated by the PRPFR algorithm. en, Proof. We assume (41) does not hold, and then there exists According to (21)- (26) and (42), we get 4 Mathematical Problems in Engineering e last inequality implies that ‖d k ‖ is bounded. By (30), we have where M � (1 + (2/t))ω. Multiplying both sides of (33) with ‖d k ‖, we obtain that

Numerical Experiments
In this section, we report some numerical experiments to investigate the computational efficiency of the proposed hybrid conjugate gradient algorithm. e 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.

Normal Nonlinear Equations.
In this section, we test some normal nonlinear equations. e 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]. e columns of Tables 2-4 have the following meaning.
According to Tables 2-4, it is evident that the three methods can solve most of the test problems with NI ≥ 20000. 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 NI ≥ 20000. 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 τ > 1.2, 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 τ � 4.4, while the FR algorithm solves 90% of the test problems at approximately τ � 4.6, and the modified PRP algorithm solves 75% of the test problems at approximately τ � 5. In Figure 3, the PRPFR algorithm solves 94% of the problems at approximately τ � 3.3. e FR algorithm and the modified PRP algorithm solve 77% and 58% of the test problems at approximately τ � 3.3 and τ � 4, 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.
erefore, we conclude that the PRPFR algorithm is competitive to the FR algorithm and the modified PRP algorithm.

Image Restoration Problems.
Image restoration aims to recover the original image from an image damaged by impulse noise. ese problems are significant in optimization fields. e stop rule is e experiments choose Lena (512 × 512), Barbara (512 × 512), and Man (1024 × 1024) as the test images. Meanwhile, we compare the PRPFR algorithm experiments' performance with that of the modified PRP algorithm, where the step size α k is generated by Step 2 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    Figures 4-6, we can easily notice that both algorithms are successful for restoring these noisy images with 20%, 45%, and 70% 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 20% noise problems, 45% noise problems, and 70% noise problems.

Conclusion
In this paper, a new hybrid conjugate gradient algorithm that combines the PRP and FR methods is proposed, while using the projection technique. e direction d k has the sufficient descent and trust region properties automatically. Global convergence of the proposed algorithm is established under appropriate assumptions. e 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
ere are no potential conflicts of interest.