Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity

Abstract

In this manuscript, we propose an efficient algorithm for solving a class of nonlinear operator equations. The algorithm is an improved version of previously established method. The algorithm’s features are as follows: (i) the search direction is bounded and satisfies the sufficient descent condition; (ii) the global convergence is achieved when the operator is continuous and satisfies a condition weaker than pseudo-monotonicity. Moreover, by comparing it with previously established method the algorithm’s efficiency was shown. The comparison was based on the iteration number required for each algorithm to solve a particular problem and the time taken. Some benchmark test problems, which included monotone and pseudo-monotone problems, were considered for the experiments. Lastly, the algorithm was utilized to solve the logistic regression (prediction) model.

1. Introduction

In the field of computational mathematics, solving nonlinear equations stands as a fundamental challenge with diverse applications cutting across areas such as optimization, physics, engineering, and economics. The investigation for efficient and robust numerical methods to address this challenge has led to the development of numerous algorithms. Among these, the conjugate gradient (CG) method appears to be a powerful tool, particularly due to its convergence properties and effectiveness in solving large-scale nonlinear systems.

Consider finding the solution of

$$F\left(v\right)=0,$$

(1)

where $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous. There are a lot of CG algorithms for solving (1) in the literature, most of which require F to be monotone and Lipschitz continuous (see, for example, [1,2,3,4,5,6,7] and references therein). Recently algorithms that require F to be pseudo-monotone, which is weaker than monotone, were proposed (see [8,9] for more details). In particular, Liu et al. [8] proposed an efficient CG projection algorithm for solving (1). The operator F under consideration was assumed to be pseudo-monotone and continuous, which are weaker than monotonicity and Lipschitz continuity, respectively. In addition, the CG parameter consists of a fixed constant c. After careful observations of the algorithm by Liu et al. [8], we posed the following questions: (i) Can we impose a weaker assumption than pseudo-monotone on F and still achieve convergence? (ii) Can we replace the fixed constant c with an adaptive parameter and achieve a more efficient algorithm?

The answers to the questions above are given in the next section. In what follows, we give a summary of the contributions:

1.

The article proposes an efficient algorithm for solving (1).

2.

The operator F is assumed to satisfy a weaker condition than pseudo-monotonicity.

3.

The proposed algorithm is globally convergent.

4.

On some test problems, which include monotone and pseudo-monotone, the proposed algorithm performs better than that of Liu et al. [8].

This manuscript is organized as follows: Section 2 consists of the proposed algorithm and its convergence analysis. Section 3 captures the numerical experiments, where the performance of the proposed method is observed for some benchmark test problems. In Section 4, we apply the proposed algorithm to solve prediction models. Section 5 has the conclusion of this work.

2. The Proposed Algorithm and Its Convergence

In this section, we start by recalling the search direction of the iterative algorithm proposed by Liu et al. in [8] called Algorithm 2.1. The search direction is defined as

$$d_k=\left\{\begin{array}{c}-F\left(v_k\right)\mathrm{if}k=0\hfill \\ -F\left(v_k\right)+{\beta }_k{d}_{k-1},\mathrm{elsewhere},\hfill \end{array}\right.$$

(2)

where

$${\beta }_k:={\displaystyle \frac{cF{\left(v_k\right)}^T{d}_{k-1}}{d_{k-1}^T{w}_{k-1}}},c\in [0,1),$$

(3)

$$w_{k-1}=y_{k-1}+{\delta }_{k-1}{d}_{k-1},y_{k-1}=F\left(v_k\right)-F\left(v_{k-1}\right),$$

(4)

 and 

$${\delta }_{k-1}=1+max\left\{0,{\displaystyle \frac{-d_{k-1}^T{y}_{k-1}}{\parallel d_{k-1}{\parallel }^2}}\right\}.$$

(5)

As noted in [8], the direction becomes the steepest descent when $c=0$ or $F{\left(v_k\right)}^T{d}_{k-1}=0$.

After careful observation, we realized that replacing the constant c in (3) with an adaptive parameter with the same domain as c can enhance the performance of the algorithm. As such, we propose the following parameter

$${\theta }_k:=1-{\displaystyle \frac{{\left(F{\left(v_k\right)}^T{d}_{k-1}\right)}^2}{\parallel F\left(v_k\right){\parallel }^2{\parallel d_{k-1}\parallel }^2}},$$

(6)

to replace c.

Remark 1. 

Observe that ${\theta }_k\in [0,1)$ since by Cauchy-Schwartz inequality, we have that

$$F{\left(v_k\right)}^T{d}_{k-1}\le \parallel F\left(v_k\right)\parallel \parallel d_{k-1}\parallel .$$

Hence, we use the following for ${\beta }_k$:

$${\beta }_k:={\displaystyle \frac{{\theta }_{k}F{\left(v_k\right)}^T{d}_{k-1}}{d_{k-1}^T{w}_{k-1}}}.$$

(7)

Remark 2. 

Applying the definition of $w_{k-1}$ and ${\delta }_{k-1}$ from (4) and (5), respectively, we have

$$d_{k-1}^T{w}_{k-1}\ge d_{k-1}^T{y}_{k-1}+\parallel d_{k-1}{\parallel }^2-d_{k-1}^T{y}_{k-1}={\parallel d_{k-1}\parallel }^2>0.$$

(8)

Definition 1. 

A function F is pseudo-monotone, if for all $v_1,v_2\in {\mathbb{R}}^n$

$$F{\left(v_1\right)}^T(v_2-v_1)\ge 0\mathit{implies}F{\left(v_2\right)}^T(v_2-v_1)\ge 0,$$

(9)

Before stating the proposed algorithm, we state the following assumptions:

Assumption 1. 

The solution set of (1) is non-empty.

Assumption 2. 

F is continuous and for any $v^{\ast }$ such that $F\left(v^{\ast }\right)=0$ and $v\in R^n$, then

$$F{\left(v\right)}^T(v-v^{\ast })\ge 0.$$

(10)

Remark 3. 

Observe that if F satisfies (9), then F satisfies (10). However, the converse is not true.

By Remark 3, a function that satisfies (10) is more general than a pseudo-monotone function, which is more general than a monotone function.

In what follows, we present the proposed Algorithm 1 for solving (1) under Assumptions 1 and 2.

Algorithm 1: (Modified Algorithm 2.1)

Initialization. $v_0\in {\mathbb{R}}^n$, $\rho \in (0,1)$, $\lambda \in (0,2)$, $\sigma \in (0,1)$, $ϵ>0$. Set $k=0.$ Step 0. Check if $\parallel F\left(v_k\right)\parallel \le ϵ$ and stop. Else, compute $d_k$ by Equations (2) and (7). Step 1. Evaluate $u_k=v_k+{\alpha }_k{d}_k$, where the step size ${\alpha }_k={\rho }^i$ with i being the smallest non-negative integer satisfying $${-F\left(u_k\right)}^T{d}_k\ge \sigma {\alpha }_k\parallel F\left(u_k\right)\parallel \parallel d_k{\parallel }^2.$$ (11) Step 2. Check if $\parallel F\left(u_k\right)\parallel \le ϵ,$ and stop. Else, evaluate $$v_{k+1}=v_k-\lambda {\displaystyle \frac{F{\left(v_k\right)}^T(v_k-u_k)}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right).$$ (12) Step 3. Set $k=k+1$ and repeat the procedure.

We omit the proof of the following Lemma as it is similar to the one in [8].

Lemma 1 

([8]). For all k and ${\theta }_k\in [0,1)$, the search direction defined by Equation (2) satisfies the inequalities

$$d_k^{T}F\left(v_k\right)\le -(1-{\theta }_k){\parallel F\left(v_k\right)\parallel }^2,$$

(13)

and

$$\parallel d_k\parallel \le (1+{\theta }_k)\parallel F\left(v_k\right)\parallel .$$

(14)

Lemma 2. 

Suppose $v^{\ast }\in {\mathbb{R}}^n$ such that $F\left(v^{\ast }\right)=0$ and $\left\{v_k\right\}$ generated by the  Modified Algorithm 2.1, then

$$\parallel v_{k+1}-v^{\ast }{\parallel }^2\le \parallel v_k-v^{\ast }{\parallel }^2-\lambda (2-\lambda ){\sigma }^2{\parallel v_k-u_k\parallel }^4.$$

Proof. 

By (11), and (10),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F{\left(u_k\right)}^T(v_k-v^{\ast })& =F{\left(u_k\right)}^T(v_k-v^{\ast }-u_k+u_k)\hfill \\ \hfill & =F{\left(u_k\right)}^T(v_k-u_k)+F{\left(u_k\right)}^T(u_k-v^{\ast })\hfill \\ \hfill & \ge F{\left(u_k\right)}^T(v_k-u_k)\hfill \\ \hfill & \ge \sigma {\alpha }_k^2\parallel F\left(u_k\right)\parallel \parallel d_k{\parallel }^2\ge 0.\hfill \end{array}\end{array}$$

(15)

Now, by (11), (12), (15) and $\lambda \in (0,2)$, we have

$$\begin{array}{cc}\hfill \parallel v_{k+1}-v^{\ast }{\parallel }^2& ={∥v_k-\lambda {\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k)}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right)-v^{\ast }∥}^2\hfill \\ \hfill & ={∥(v_k-v^{\ast })+(-\lambda {\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k))}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right))∥}^2\hfill \\ \hfill & ={\left((v_k-v^{\ast })+(-\lambda {\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k))}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right))\right)}^T\left((v_k-v^{\ast })+(-\lambda {\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k))}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right))\right)\hfill \\ \hfill & ={(v_k-v^{\ast })}^T(v_k-v^{\ast })-2\lambda {\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k)}{\parallel F\left(u_k\right){\parallel }^2}}F{\left(u_k\right)}^T(v_k-v^{\ast })\hfill \\ \hfill & +{\lambda }^2{\left({\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k))}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right)\right)}^T\left({\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k))}{\parallel F\left(u_k\right){\parallel }^2}}F\left(u_k\right)\right)\hfill \\ \hfill & =\parallel v_k-v^{\ast }{\parallel }^2-2\lambda {\displaystyle \frac{F{\left(u_k\right)}^T(v_k-u_k)}{\parallel F\left(u_k\right){\parallel }^2}}F{\left(u_k\right)}^T(v_k-v^{\ast })+{\lambda }^2{\displaystyle \frac{{\left(F{\left(u_k\right)}^T(v_k-u_k)\right)}^2}{\parallel F\left(u_k\right){\parallel }^2}}\hfill \\ \hfill & \le \parallel v_k-v^{\ast }{\parallel }^2-2\lambda {\displaystyle \frac{{\left(F{\left(u_k\right)}^T(v_k-u_k)\right)}^2}{\parallel F\left(u_k\right){\parallel }^2}}+{\lambda }^2{\displaystyle \frac{{\left(F{\left(u_k\right)}^T(v_k-u_k)\right)}^2}{\parallel F\left(u_k\right){\parallel }^2}}\hfill \\ \hfill & =\parallel v_k-v^{\ast }{\parallel }^2-\lambda (2-\lambda ){\displaystyle \frac{{\left(F{\left(u_k\right)}^T(v_k-u_k)\right)}^2}{\parallel F\left(u_k\right){\parallel }^2}}\hfill \\ \hfill & \le \parallel v_k-v^{\ast }{\parallel }^2-\lambda (2-\lambda ){\sigma }^2{\alpha }_k^4{\parallel d_k\parallel }^4\hfill \\ \hfill & \le \parallel v_k-v^{\ast }{\parallel }^2.\hfill \end{array}$$

(16)

   □

Remark 4. 

${\displaystyle \underset{k\to \infty }{lim}\parallel v_k-v^{\ast }\parallel }$ exists from (16) since $\{\parallel v_k-v^{\ast }\parallel \}$ is non-increasing and hence convergent.

Remark 5. 

From Remark 4, $\{\parallel v_k-v^{\ast }\parallel \}$ is bounded. That is, there exists a positive constant $\overline{M}$ such that for all k,

$$\parallel v_k-v^{\ast }\parallel \le \overline{M}.$$

(17)

Applying the triangle inequality on (17), we have

$$\parallel v_k\parallel \le \overline{M}+\parallel v^{\ast }\parallel :=\tilde{M}.$$

Furthermore, $\left\{F\left(v_k\right)\right\}$ is bounded.

The proof of the following theorem can be found in [8].

Theorem 1 

([8]). If Assumptions 1 and 2 hold and $\left\{v_k\right\}$ is generated by Modified Algorithm 2.1, then

$$\underset{k\to \infty }{lim\; inf}\parallel F\left(v_k\right)\parallel =0.$$

(18)

Corollary 1. 

The sequence $\left\{v_k\right\}$ generated by  Modified Algorithm 2.1converges to some $v^{\ast }$ in the sense that $F\left(v^{\ast }\right)=0.$

Proof. 

The continuity of F implies that there is a limit point $v^{\ast }$ of the sequence $\left\{v_k\right\}$ with $F\left(v^{\ast }\right)=0,$ that is, $v^{\ast }$ is a solution. In addition, because $\parallel v_k-v^{\ast }\parallel$ converges from Remark 4, $v^{\ast }$ is an accumulation point of $\parallel v_k-v^{\ast }\parallel$, then $\left\{v_k\right\}\to v^{\ast }$.    □

3. Numerical Experiments

We consider two experiments in this section. Firstly, we perform experiments on the system of monotone equations since every monotone function is pseudo-monotone. Secondly, we consider pseudo-monotone functions that are not monotone.

3.1. Solving System of Monotone Equations

As every monotone function is pseudo-monotone, in this subsection, we will apply Modified Algorithm 2.1 to solve ten (10) systems of monotone equations given in Appendix A with different dimensions and starting points. To show the impact of the adaptive parameter ${\theta }_k$ in the search direction of Modified Algorithm 2.1, we compared it with Algorithm 2.1 in [8], where the parameter is fixed. We consider the following for the comparison:

• Five dimensions: $n=1000,5000,10,000,50,000,100,000$
• Seven initial points: $v_0={({\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}},\dots ,{\displaystyle \frac{1}{8}})}^T$,  $v_0={\left({\displaystyle \frac{2}{5}},{\displaystyle \frac{2}{5}}\dots ,{\displaystyle \frac{2}{5}}\right)}^T$,  $v_0=({\displaystyle \frac{1}{10}},{\displaystyle \frac{1}{10}},\dots ,{\displaystyle \frac{1}{10}}{)}^T$, $v_0={\left({\displaystyle \frac{1}{100}},{\displaystyle \frac{1}{100}},\dots ,{\displaystyle \frac{1}{100}}\right)}^T$,  $v_0={({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\dots ,{\displaystyle \frac{1}{2}})}^T$,  $v_0={\left({\displaystyle \frac{1}{5}},{\displaystyle \frac{1}{5}},\dots ,{\displaystyle \frac{1}{5}}\right)}^T$, $v_0={({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\dots ,{\displaystyle \frac{1}{4}})}^T$.
• Parameters for Modified Algorithm 2.1: $\rho =0.8,\lambda =1.2,\sigma ={10}^{-4},ϵ={10}^{-6}$.
• The choice of $\lambda$ is by trial and error in the interval $(0,2)$ and we got the best result at 1.7.
• Parameters for Algorithm 2.1: As chosen in [8].
• Stopping condition: when $\parallel F\left(v_k\right)\parallel \le {10}^{-6}$.

The results obtained from the experiment can be found in Appendix B where #ITN, #FEN, CPUTIMS, and Norm represent iteration number, function evaluation number, CPU time in seconds, and norm of the function at the solution, respectively. However, for clarity, we employ the performance profiles of Dolan and Moré [10]. It gives us a graphical interpretation of the percentage success for each algorithm (solver) based on the #ITN, #FEN, and CPUTIMS. The algorithm with the highest percentage on the y-axes of the plot is considered the best (see Figure 1, Figure 2 and Figure 3). MATLAB R2023b was used for the numerical experiments on a personal computer with 2.3 Ghz, 8 GB RAM, using Windows 7 operating system.

From Figure 1, Figure 2 and Figure 3, one can see that the percentage of Modified Algorithm 2.1 is higher than that of Algorithm 2.1 in all the figures. This is associated with an adaptive parameter ${\theta }_k$ in Modified Algorithm 2.1 that replaces the fixed constant in Algorithm 2.1.

3.2. Solving Pseudo-Monotone Equations

As observed in Section 1, Liu et al. [8] proposed an efficient algorithm for solving a system of pseudo-monotone equations. However, they could only provide numerical examples for solving monotone equations. In this subsection, we will provide and solve two systems of equations where F is not monotone but pseudo-monotone, as shown in

Table 1. Results from the experiments on the pseudo-monotone problems P1 and P2 by Modified Algorithm 2.1 and Algorithm 2.1, respectively.

		Modified Algorithm 2.1				Algorithm 2.1
Problem	Initial Point	#ITN	#FEN	CPUTIMS	Norm	#ITN	#FEN	CPUTIMS	Norm
P1	(0,0)	17	17	0.008397	4.48 × ${10}^{-7}$	34	34	0.002663	7.95 × ${10}^{-7}$
	${(1,1)}^T$	18	18	0.006964	7.19 × ${10}^{-7}$	36	36	0.00257	7.75 × ${10}^{-7}$
	${(0.1,0.1)}^T$	17	17	0.004736	7.44 × ${10}^{-7}$	34	34	0.002293	9.13 × ${10}^{-7}$
	${(-1,-1)}^T$	16	16	0.002552	9.75 × ${10}^{-7}$	34	34	0.002786	6.46 × ${10}^{-7}$
	${(-0.1,-0.1)}^T$	16	16	0.004537	8.75 × ${10}^{-7}$	34	34	0.002557	7.00 × ${10}^{-7}$
P2	(0.2,0.2,0.2)	60	66	0.017991	9.63 × ${10}^{-7}$	98	100	0.009262	8.98 × ${10}^{-7}$
	${(0.4,0.4,0.4)}^T$	61	67	0.00344	8.96 × ${10}^{-7}$	102	104	0.005058	9.73 × ${10}^{-7}$
	${(0.6,0.6,0.6)}^T$	55	61	0.003559	9.53 × ${10}^{-7}$	69	71	0.003775	9.38 × ${10}^{-7}$
	${(0.8,0.8,0.8)}^T$	75	81	0.004154	9.09 × ${10}^{-7}$	85	87	0.004471	9.81 × ${10}^{-7}$
	${(1,1,1)}^T$	78	84	0.004725	9.26 × ${10}^{-7}$	56	57	0.0036	9.75 × ${10}^{-7}$

.

Consider the following pseudo-monotone equations defined by:

  P1: [11] $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined as

$$\begin{array}{cc}\hfill F_1\left(v\right)& =(v_1+{(v_2-1)}^2)(1+v_2)\hfill \\ \hfill F_2\left(v\right)& =-v_1^3-v_1{(v_2-1)}^2.\hfill \end{array}$$

P2: [12] $F:{\mathbb{R}}^3\to {\mathbb{R}}^3$ defined as

$$\begin{array}{cc}\hfill F\left(v\right)& =({exp}^{-{\parallel v\parallel }^2}+\mathsf{\Phi })Mv,\mathrm{where},\hfill \\ \hfill M& =\left(\begin{array}{ccc}1& 0& -1\\ 0& 1.5& 0\\ -1& 0& 2\end{array}\right),\mathrm{and}\mathsf{\Phi }=0.2.\hfill \end{array}$$

Table 1 shows that each algorithm successfully found the solution of problems P1 and P2. However, in all instances, Modified Algorithm 2.1 achieved the solution with fewer iterations and function evaluations than Algorithm 2.1. Moreover, the time taken for Algorithm 2.1 to attain a solution was less in six out of the ten instances than in Modified Algorithm 2.1. This may be as a result of the computational cost associated with computing the adaptive parameter in Modified Algorithm 2.1.

4. Solving a Prediction Model

In this section, we will apply Modified Algorithm 2.1 to solve a prediction model called logistic regression. The logistic regression model can be used to predict a categorical dependent variable, which makes it an important tool in machine learning. The logistic regression model [13] is a problem of the form

$$\underset{v\in {\mathbb{R}}^n}{min}f\left(v\right),$$

(19)

where

$$f\left(v\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^{N}log(1+{exp}^{-c_i{l}_i^{T}v})+{\displaystyle \frac{\xi }{2}}{\parallel v\parallel }^2.$$

(20)

$\xi$ is the regularization parameter, ${\displaystyle \frac{1}{N}}{\sum }_{i=1}^{N}log(1+{exp}^{-c_i{l}_i^{T}v})$ is the loss function, $c_i\in \{-1,1\}$ is the binary class and $l_i\in {\mathbb{R}}^n$ is the data variable. f is strongly convex and $v^{\ast }$ is a unique solution of (19) if and only if it solves the nonlinear equation:

$$0=F\left(v\right):={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\displaystyle \frac{-c_i\left({exp}^{-c_i{l}_i^{T}v}\right)l_i}{1+{exp}^{-c_i{l}_i^{T}v}}}+\xi v.$$

(21)

See [14,15] and references therein. Furthermore, Equation (21) is monotone, and therefore the Modified Algorithm 2.1 can solve (21).

As such, Modified Algorithm 2.1 was implemented using MATLAB R2023b to solve (21). Its performance was compared with the following algorithms; Algorithm 2.1 proposed by Liu et al. in [8] and DFMRMIL by Koorapetse et al. [16]. Leveraging the library for support vector machines called LIBSVM (https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ accessed on 1 July 2024) [17], ten test instances were considered (see

Table 2. LIBSVM test instances.

S/N	Dataset	Data Points N	Number of Variables n
1	a1a.t	30,956	123
2	a2a.t	30,296	123
3	a3a.t	29,376	123
4	a4a.t	27,780	123
5	a5a.t	26,147	123
6	a6a.t	21,341	123
7	colon-cancer	62	2000
8	svmguide1.t	4000	4
9	svmguide4.t	312	10
10	usps.t	2007	256

). The starting point is chosen as $v_0=4\times (rand(n,1)-0.5)$ just like in [14] with $\xi =0.1$. The same parameters as in the last section were considered for Modified Algorithm 2.1, Algorithm 2.1, and DFMRMIL; the parameters were chosen as given in [8] and [16], respectively. In addition, each instance was replicated five times and the average values of #ITN, #FEN, CPUTIMS, and Norm were recorded (see

Table 3. Experimental results on the 10 instances in Table 2.

	DFMRMIL	Algorithm 2.1	Modified Algorithm 2.1
S/N	#ITN/#FEN/CPUTIMS/Norm	#ITN/#FEN/CPUTIMS/Norm	#ITN/#FEN/CPUTIMS/Norm
1	715.2/1430.6/7.709/9.92 × ${10}^{-7}$	172.4/345.4/1.881/9.86 × ${10}^{-7}$	116.4/249.6/1.345/9.24 × ${10}^{-7}$
2	715.0/1430.0/7.274/9.94 × ${10}^{-7}$	173.0/346.0/1.700/9.70 × ${10}^{-7}$	109.2/237.6/1.182/9.41 × ${10}^{-7}$
3	715.2/1430.6/7.181/9.94 × ${10}^{-7}$	172.6/345.8/1.746/9.84 × ${10}^{-7}$	103.8/224.8/1.140/9.18 × ${10}^{-7}$
4	714.0/1428.0/6.878/9.95 × ${10}^{-7}$	172.4/345.4/1.669/9.78 × ${10}^{-7}$	110.2/239.2/1.162/8.94 × ${10}^{-7}$
5	713.4/1427.0/6.403/9.93 × ${10}^{-7}$	172.2/344.8/1.552/9.86 × ${10}^{-7}$	113.2/245.2/1.107/8.92 × ${10}^{-7}$
6	711.8/1423.6/5.078/9.94 × ${10}^{-7}$	171.8/344.2/1.213/9.78 × ${10}^{-7}$	110.6/239.2/0.856/9.23 × ${10}^{-7}$
7	1257.0/21529.0/5.968/9.68 × ${10}^{-7}$	788.8/3412.6/0.958/9.64 × ${10}^{-7}$	695.6/3191.4/0.900/9.71 × ${10}^{-7}$
8	548.6/2329.0/0.164/9.95 × ${10}^{-7}$	156.6/399.0/0.027/9.85 × ${10}^{-7}$	249.0/1020.4/0.070/9.38 × ${10}^{-7}$
9	930.4/13502.2/0.158/9.88 × ${10}^{-7}$	652.0/2603.4/0.030/9.71 × ${10}^{-7}$	524.8/2234.4/0.026/9.64 × ${10}^{-7}$
10	881.2/9749.2/69.624/9.95 × ${10}^{-7}$	222.2/615.8/4.378/9.68 × ${10}^{-7}$	483.8/2238.0/15.975/9.41 × ${10}^{-7}$

). It can be seen from Table 3 that the mean values of #ITN, #FEN, CPUTIMS for Modified Algorithm 2.1 are less than those for Algorithm 2.1 and DFMRMIL in $80\%$ of the datasets.

5. Conclusions

An improved projection method based on the CG method for solving continuous nonlinear equations was proposed. The improvement was a result of an adaptive parameter ${\theta }_k$ that replaces the fixed constant c in the CG parameter proposed by [8]. Moreover, the global convergence was established under a weaker assumption than pseudo-monotone on the operator F. Numerical experiments were performed on some benchmark nonlinear equations that involve monotone and pseudo-monotone operators. The results of the experiments indicate that the proposed algorithm is outstanding. This is surely a sign that replacing c with ${\theta }_k$ was a better choice. Furthermore, the algorithm was employed to solve a prediction model and the results were also satisfactory. Finally, based on the reported results, we can conclude by recommending an adaptive parameter over a fixed constant.