A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations

Xiaowei Fang; Xiaowei Fang

doi:10.3934/math.2025528

AIMS Mathematics

2025, Volume 10, Issue 5: 11656-11675. doi: 10.3934/math.2025528

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Research article

A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations

Xiaowei Fang ^,

Department of Mathematics, Huzhou University, Huzhou Zhejiang, 313000, China

Received: 16 March 2025 Revised: 06 May 2025 Accepted: 12 May 2025 Published: 21 May 2025
MSC : 90C56, 90C30

In this paper, we introduce a derivative-free RMIL conjugate gradient method designed for nonlinear systems of monotone equations with convex constraints. The proposed method represents a refinement of the RMIL method through its integration with projection techniques and a derivative-free line search strategy. When certain reasonable assumptions are satisfied, we can prove the global convergence of the proposed method. Numerical experiments demonstrate that the method is highly effective. Moreover, we employ the method to solve sparse signal restoration problems.

Keywords:

Citation: Xiaowei Fang. A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations[J]. AIMS Mathematics, 2025, 10(5): 11656-11675. doi: 10.3934/math.2025528

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Abstract

1. Introduction

This paper aims at finding solutions to the following constrained nonlinear monotone equations:

$\begin{equation} F(x) = 0,\quad x \in \Omega, \end{equation}$

(1.1)

in which $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is monotone and continuously differentiable, and $\Omega \subseteq \mathbb{R}^n$ is a nonempty closed convex set. The property of monotonicity for $F$ is defined as

$\begin{equation} (F(x) - F(y))^T (x - y) \geq 0, \quad \forall x,y\in \mathbb{R}^n. \end{equation}$

(1.2)

Nonlinear monotone equations emerge from different problems and are widely applied in interdisciplinary domains. For example, the automatic control systems ^[1], the compressive sensing^[2,3,4], and the optimal power flow ^[5].

There are numerous iterative algorithms for solving (1.1). For instance, the Newton algorithm^[6], the quasi-Newton algorithm^[7], the Gauss-Newton algorithm^[8], and the BFGS algorithm^[9]. These algorithms exhibit good convergence properties from suitable initial guesses. Nevertheless, they are unsuitable for handling large-scale nonlinear equations since the computation of the Jacobian matrix or its approximation is required in every iteration.

Recently, some optimization methods that build on the projection and proximal point method presented by Solodov and Svaiter^[10] have been extended to solve (1.1). These include conjugate gradient methods originally applied to solve large-scale unconstrained optimization problems due to the low memory and simple structures. For example, Li and Li ^[11] described a category of derivative-free approaches for systems of monotone equations. Cheng ^[12] formulated a PRP-type method using the line search strategy, and Awwal et al.^[13] proposed a DFP-type derivative-free method. Ahookhosh et al.^[14] developed two new derivative-free approaches for systems of nonlinear monotone equations. According to the RMIL conjugate gradient method presented by Rivaie et al ^[15], Fang^[16,17] put forward a set of adjusted derivative-free gradient-type approaches for solving nonlinear equations. Depending on the hybrid idea and a new adaptive line search strategy, Liu et al. in ^[18] proposed a three-term algorithm to solve the nonlinear monotone equations under convex constraints. According to the spectral secant equation, Zhang et al.^[19] developed a three-term projection method for the purpose of dealing with nonlinear monotone equations. Moreover, they utilized this method to solve the problem in signal processing. Abdullahi et al.^[20,21] described a modified self-adaptive algorithm and a three-term projection algorithm for solving systems of monotone nonlinear equations. Furthermore, they employ these algorithms in signal and image recovery problems. Aji et al. ^[22] proposed a novel spectral algorithm using an inertial technique to solve a system of monotone equations and for its applications.

Motivated by the aforementioned works, we give a new conjugate gradient projection algorithm for nonlinear monotone equation (1). The structure of this paper is as follows. Section 2 is dedicated to proposing the new method. In Section 3, we conduct the convergence analysis of the newly proposed method. Section 4 showcases the numerical results obtained from solving monotone nonlinear equations. Subsequently, in Section 5, we apply the proposed method to address the signal recovery problem. Finally, Section 6 presents the conclusions. The symbol $\|\cdot\|$ stands for Euclidean norm, and $\|\cdot\|_1$ stands for $l_1$ norm.

2. Algorithm

In this part, we initially review the conjugate gradient method that is employed to solve the following unconstrained optimization problems:

$\begin{equation} \mathop{\min} f(x), \end{equation}$

(2.1)

in which $f$ is a smooth function mapping from $\mathbb{R}^n$ to $\mathbb{R}$ , and the gradient of $f(x_k)$ is symbolized as $g_k$ . The conjugate gradient method has the following iterative formula:

$\begin{equation} x_{k+1} = x_k + \alpha_k d_k, \end{equation}$

(2.2)

in which the step length $\alpha_k$ is given through some line search techniques, and the search direction $d_k$ is derived from

$\begin{equation} d_k = \left\{ \begin{array}{ll} -g_k & \text{if} \ k = 0\\ -g_k + \beta_k d_{k-1} & \text{if} \ k \geq 1 \end{array} \right. . \end{equation}$

(2.3)

In recent times, Rivaie et al. ^[15] proposed the RMIL conjugate gradient method (RMIL, named after its developers Rivaie, Mustafa, Ismail, and Leong), where the $\beta_k$ is computed by

$\begin{equation} \beta_k = \frac{g_k^T (g_k - g_{k-1})}{||d_{k-1}||^2}. \end{equation}$

(2.4)

Results from numerical experiments reveal that the method is highly efficient.

Motivated by the results of the RMIL method, we develop a derivative-free approach aimed at solving nonlinear systems of monotone equations. For the sake of simplicity, we substitute $g_k$ in equations (2.3) and (2.4) with $F_k$ , where $F_k = F(x_k)$ . Moreover, we change the coefficient of $g_k$ from -1 to $-\theta_k$ when $k\geq 1$ . In this way, it can ensure that the search direction generated by the algorithm satisfies the property of sufficient descent, and it can effectively improve the computational efficiency of the algorithm. Consequently, the search direction $d_k$ is given by

$\begin{equation} d_k = \left\{ \begin{array}{ll} - F_k & \text{if} \ k = 0\\ -\theta_k F_k + \beta_k d_{k-1} & \text{if} \ k \geq 1 \end{array} \right. , \end{equation}$

(2.5)

where $\theta_k = \beta_k\frac{F_k^T d_{k-1}}{\|F_k\|^2} + 1$ , $\beta_k = \frac{F_k^T y_{k-1}}{\|d_{k-1}\|^2}$ , $F_k = F(x_k)$ , $y_{k-1} = F_k-F_{k-1}$ . From the definitions of $d_k$ , it is not difficult to see that for $k \in \mathbb{N}$ , we have

$\begin{equation} \begin{split} F_k^T d_k& = -F_k^T(\beta_k\frac{F_k^T d_{k-1}}{\|F_k\|^2} + 1 )F_k + F_k^T \beta_k d_{k-1}\\ & = - \beta_k\frac{F_k^T d_{k-1}}{\|F_k\|^2} \|F_k\|^2 - \|F_k\|^2 + \beta_k F_k^T d_{k-1}\\ & = -\|F_k\|^2 .\\ \end{split} \end{equation}$

(2.6)

When $k = 0$ , we get $F_0^T d_0 = -\|F_0\|^2$ .

Next we state the projection operator $P_{\Omega}[.]$ , which is described as follows:

$\begin{equation} P_{\Omega}[x] = \arg \min \{ \|x-y\| \; | \; y \in \Omega \},\quad\forall x \in \mathbb{R}^n. \end{equation}$

(2.7)

This operation refers to projecting the vector $x$ onto the closed convex set $\Omega$ . By doing so, it is guaranteed that the subsequent iterative point produced by our algorithm remains within the domain $\Omega$ . The projection operator is known to possess the non-expansive property, namely

$\begin{equation} \| P_{\Omega}[x] - P_{\Omega}[y]\| \leq \|x-y\| ,\quad\forall x, y \in \mathbb{R}^n. \end{equation}$

(2.8)

From here on, our attention is directed towards the method for solving the monotone equations (1.1). We adopt the projection procedure presented in ^[10] and obtain a point

$\begin{equation} z_k = x_k + \alpha_k d_k, \end{equation}$

(2.9)

with the result that

$\begin{equation} F(z_k)^T (x_k - z_k) > 0, \end{equation}$

(2.10)

where $\alpha_k$ is obtained using a suitable line search technique.

Meanwhile, for any $x^*$ satisfying $F(x^*) = 0$ , owing to the monotonic property of $F$ , we have

$\begin{equation} F(z_k)^T(x^* - z_k) = -(F(x^*)-F(z_k))^T (x^* - z_k) \leq 0. \end{equation}$

(2.11)

Incorporating the expression (2.10), we have the conclusion that the hyperplane

$\begin{equation} H_k = \{ x \in \mathbb{R}^n | F(z_k)^T (x - z_k) = 0\} \end{equation}$

(2.12)

strictly divides the solutions of the equations (1.1) and $x_k$ . Consequently, Solodov and Svaiter ^[10] calculate the next iterate $x_{k+1}$ through the projection of $x_k$ onto the hyperplane $H_k$ . The $x_{k+1}$ is defined as

$\begin{equation} x_{k+1} = x_k - \frac{F(z_k)^T (x_k - z_k)}{\|F(z_k)\|^2}F(z_k). \end{equation}$

(2.13)

Now, we give the steps of our algorithm.

Algorithm 2.1. :

$\mathbf{Step\ 0 :}$ Choose $\sigma > 0, s > 0, \epsilon > 0, 1 > \rho > 0, 2 > \gamma > 0.$ and the starting point $x_{0}\in\mathbb{R}^n$ . Set $k = 0$ .

$\mathbf{Step\ 1 :}$ If $\|F(x_{k})\| < \epsilon$ , stop. Otherwise, move to step 2.

$\mathbf{Step\ 2 :}$ Define the search direction $d_k$ through (2.5).

$\mathbf{Step\ 3 : }$ Determine the steplength $\alpha_k = \max \{s\rho^i | i = 0, 1, 2, \cdots\}$ satisfies

$\begin{equation} -F(x_k + \alpha_k d_k)^T d_k \geq \sigma \alpha_k \|F(x_k + \alpha_k d_k)\|\|d_k\|^2, \end{equation}$

(2.14)

then set $z_k = x_k + \alpha_k d_k$ and move to step 4.

$\mathbf{Step\ 4 : }$ If $z_k \in \Omega$ and $\|F(z_k)\| < \epsilon$ , set $x_{k+1} = z_k$ , stop. Else compute

$\begin{equation} x_{k+1} = P_{\Omega}\Big{[}x_k - \gamma\frac{F(z_k)^T (x_k - z_k)}{\|F(z_k)\|^2}F(z_k)\Big{]} \end{equation}$

(2.15)

and set $k: = k+1$ , move to step 1.

3. Convergence analysis

In this section, for the purpose of attaining the global convergence of Algorithm 2.1, the following assumptions are necessary.

Assumption 3.1. (1) The set of solutions for problem (1.1) is nonempty.

(2) The mapping $F(\cdot)$ is Lipschitz continuous on $\mathbb{R}^n$ , that is

$\begin{equation} \|F(x) - F(y)\| \leq L \|x - y \|, \quad \forall x,y \in \mathbb{R}^n, \end{equation}$

(3.1)

in which $L$ represents a positive constant.

From Assumption 3.1, it can be inferred that

$\begin{equation} \|F(x)\| \leq \kappa \quad \forall x \in \mathbb{R}^n, \end{equation}$

(3.2)

in which $\kappa$ is defined as a positive constant.

Lemma 3.1. Assume that Assumption 3.1 holds and the sequences $\{x_k\}, \{\alpha_k\}, \{d_k\}$ are yielded in accordance with Algorithm 2.1. Then, for all $x^* \in \Omega$ that satisfies $F({x^*}) = 0$ , and given that $\gamma \in (0, 2)$ , we obtain

$\begin{equation} \|x_{k+1} - x^*\|^2 - \|x_k - x^*\|^2 \leq -\gamma(2-\gamma)\sigma^2\|\alpha_k d_k\|^4. \end{equation}$

(3.3)

Furthermore, it holds that

$\begin{equation} \lim\limits_{k\rightarrow +\infty}\|\alpha_k d_k\| = 0. \end{equation}$

(3.4)

Proof. Suppose $F(x^*) = 0$ and $x^* \in \Omega$ ; due to the monotonicity of $F$ , we obtain

$\begin{equation} F(z_k)^T (z_k - x^*) \geq 0. \end{equation}$

(3.5)

Combining with (3.5), we get

$\begin{equation} F(z_k)^T(x_k-x^*) = F(z_k)^T(x_k-z_k) + F(z_k)^T(z_k-x^*) \geq F(z_k)^T(x_k-z_k). \end{equation}$

(3.6)

Thus, from (2.8), (2.10), (2.14), (2.15), and (3.6), we have

$\begin{equation} \begin{split} \|x_{k+1}-x^*\|^2 - \|x_k-x^*\|^2& = \| P_{\Omega}[x_k - \gamma\frac{F(z_k)^T(x_k-z_k)}{\|F(z_k)\|^2}F(z_k)]-x^*\|^2 - \|x_k - x^*\|^2\\ & \leq \| x_k - \gamma\frac{F(z_k)^T(x_k-z_k)}{\|F(z_k)\|^2}F(z_k)-x^*\|^2 - \|x_k - x^*\|^2\\ & = -2\gamma\frac{F(z_k)^T(x_k-z_k)}{\|F(z_k)\|^2}F(z_k)^T(x_k-x^*) + \gamma^2\frac{(F(z_k)^T(x_k-z_k))^2}{\|F(z_k)\|^2}\\ & \leq -2\gamma\frac{F(z_k)^T(x_k-z_k)}{\|F(z_k)\|^2}F(z_k)^T(x_k-z_k) + \gamma^2\frac{(F(z_k)^T(x_k-z_k))^2}{\|F(z_k)\|^2}\\ & = -\frac{\gamma(2-\gamma)(F(z_k)^T(x_k-z_k))^2}{\|F(z_k)\|^2}\\ & \leq -\gamma(2-\gamma)\sigma^2\|\alpha_k d_k\|^4.\\ \end{split} \end{equation}$

(3.7)

Then, we can deduce that

$\begin{equation} \gamma(2-\gamma)\sigma^2\sum\limits_{i = 0}^{k}\|\alpha_i d_i\|^4 \leq \|x_0 - x^*\|^2 - \|x_{k+1} - x^*\|^2 \leq \|x_0 - x^*\|^2, \end{equation}$

(3.8)

which together with $\gamma \in (0, 2)$ , means that

$\begin{equation} \lim\limits_{k\rightarrow +\infty}\|\alpha_k d_k \| = 0. \end{equation}$

(3.9)

□

Lemma 3.2. Assume that Assumption 3.1 holds and sequences $\{x_k\}, \{d_k\}$ are yielded in accordance with Algorithm 2.1.Then we have

$\begin{equation} \|d_k\| \leq \kappa (1 + 2\gamma L s), \end{equation}$

(3.10)

in which $\kappa > 0, s > 0, L > 0, 2 > \gamma > 0$ .

Proof. From (2.8) and step 4 of Algorithm 2.1, we obtain

$\begin{equation} \begin{split} \|x_k - x_{k-1}\|& = \|P_{\Omega}\Big{[}x_{k-1} - \gamma\frac{F(z_{k-1})^T (x_{k-1} - z_{k-1})}{\|F(z_{k-1})\|^2}F(z_{k-1})\Big{]} - x_{k-1} \|\\ & \leq \|x_{k-1} - \gamma\frac{F(z_{k-1})^T (x_{k-1} - z_{k-1})}{\|F(z_{k-1})\|^2}F(z_{k-1}) - x_{k-1} \|\\ & = \| \gamma\frac{F(z_{k-1})^T (x_{k-1} - z_{k-1})}{\|F(z_{k-1})\|^2} F(z_{k-1}) \|\\ & \leq \gamma \| x_{k-1} - z_{k-1} \|\\ \end{split} \end{equation}$

(3.11)

For $k \in \mathbb{N}$ , using (2.5), (3.1), (3.2), (3.11), and step 3 of Algorithm 2.1, we have

$\begin{equation} \begin{split} \|d_k\|& = \| -( \beta_k\frac{F_k^T d_{k-1}}{\|F_k\|^2} + 1)F_k + \beta_k d_{k-1}\|\\ & \leq \|F_k\| + 2\|\beta_k d_{k-1}\|\\ & \leq \|F_k\| + 2\frac{\|F_k\|\|d_{k-1}\|\|y_{k-1}\|}{\|d_{k-1}\|^2}\\ & = \|F_k\| + 2\frac{\|F_k\|\|F_k-F_{k-1}\|}{\|d_{k-1}\|}\\ & \leq \|F_k\| + 2L\frac{\|F_k\|\|x_k-x_{k-1}\|}{\|d_{k-1}\|}\\ & \leq \|F_k\| + 2\gamma L\frac{\|F_k\|\|x_{k-1}-z_{k-1}\|}{\|d_{k-1}\|}\\ & = \|F_k\| + 2\gamma L\alpha_{k-1}\frac{\|F_k\|\|d_{k-1}\|}{\|d_{k-1}\|}\\ & = \|F_k\| (1 + 2\gamma L\alpha_{k-1})\\ & \leq \kappa(1 + 2\gamma L s ).\\ \end{split} \end{equation}$

(3.12)

From (2.5) and (3.2), we get

$\begin{equation} \|d_0\| = \|-F_0\| \leq \kappa. \end{equation}$

(3.13)

Combining with (3.7), we yield (3.10). □

Lemma 3.3. Assume that Assumption 3.1 holds and the sequences $\{x_k\}$ , $\{z_k\}$ , and $\{\alpha_k\}$ are yielded in accordance with Algorithm 2.1; then we get

$\begin{equation} \alpha_k \geq \min \left \{s, \frac{\|F_k\|^2}{\rho^{-1} \kappa^2 \Big(1 + 2\gamma Ls\Big)^2 \Big(L + \sigma\kappa + \sigma\kappa\rho^{-1}Ls\big(1 + 2\gamma Ls \big)\Big)} \right \}. \end{equation}$

(3.14)

Proof. In the case where $\alpha_k \neq s$ , relying on the line search technique of Algorithm 2.1, it follows that $\rho^{-1}\alpha_{k}$ fails to satisfy (2.14), namely

$\begin{equation} -F(x_{k} + \rho^{-1}\alpha_{k} d_{k})^T d_{k} < \sigma \rho^{-1}\alpha_{k} \|F(x_{k} + \rho^{-1}\alpha_{k} d_{k})\| \|d_{k}\|^2. \end{equation}$

(3.15)

Combining with (3.1), (3.2) and (3.10), we obtain

$\begin{equation} \begin{split} \|F(x_{k} + \rho^{-1}\alpha_{k}d_{k})\|& = \|F(x_{k} + \rho^{-1}\alpha_{k}d_{k}) - F(x_{k})\| + \|F(x_{k})\|\\ & \leq L \rho^{-1}\alpha_k \|d_{k}\| + \kappa\\ & \leq \kappa\Big(1 + \rho^{-1}Ls(1+ 2\gamma Ls ) \Big).\\ \end{split} \end{equation}$

(3.16)

It follows from (2.6), (3.1), (3.10), (3.15), and (3.16) that we have

$\begin{equation} \begin{split} \|F_k\|^2& = -F_{k}^{T}d_{k}\\ & = [F(x_{k} + \rho^{-1}\alpha_{k}d_{k}) - F(x_{k})]^T d_{k} - [F(x_{k} + \rho^{-1}\alpha_{k}d_{k})]^T d_{k} \\ & \leq L\rho^{-1}\alpha_k\|d_{k}\|^2 + \sigma\rho^{-1}\alpha_k (L\rho^{-1}\alpha_k\|d_k\| + \kappa) \|d_k\|^2 \\ & = \rho^{-1}\alpha_k\|d_{k}\|^2 \Big(L + \sigma (L\rho^{-1}\alpha_k\|d_k\|+\kappa)\Big)\\ & \leq \rho^{-1}\alpha_{k} \kappa^2 \Big(1 + 2\gamma L s\Big)^2 \Big(L + \sigma\kappa + \sigma\kappa\rho^{-1}Ls\big(1 + 2\gamma Ls \big)\Big) .\\ \end{split} \end{equation}$

(3.17)

The above inequality shows that

$\begin{equation} \begin{split} \alpha_{k}& \geq \frac{\|F_k\|^2}{\rho^{-1} \kappa^2 \Big(1 + 2\gamma L s\Big)^2 \Big(L + \sigma\kappa + \sigma\kappa\rho^{-1}Ls\big(1 + 2\gamma Ls \big)\Big)}.\\ \end{split} \end{equation}$

(3.18)

This implies (3.14). □

Theorem 3.4. Assume that Assumption 3.1 holds and the sequences $\{x_k\}$ and $\{ F_k \}$ are yielded in accordance with Algorithm 2.1. Then we obtain

$\begin{equation} \liminf\limits_{k\rightarrow \infty}\|F_k\| = 0. \end{equation}$

(3.19)

Proof. Assume that (3.19) is false. Then, there is a positive constant $k$ for which

$\begin{equation} \|F_k\| \geq \xi. \end{equation}$

(3.20)

Based on (2.6) and (3.20), we obtain

$\begin{equation} \|d_k\| = \frac{\|F_k\|\|d_k\|}{\|F_k\|} \geq \frac{\|F_k^T d_k\|}{\|F_k\|}\geq \|F_k\| \geq \xi. \end{equation}$

(3.21)

Combining with Lemma 3.3, we obtain

$\begin{equation} \alpha_k \|d_k\| \geq \min \left \{s\xi, \frac{\xi^3} {\rho^{-1} \kappa^2 \Big(1 + 2\gamma Ls\Big)^2 \Big(L + \sigma\kappa + \sigma\kappa\rho^{-1}Ls\big(1 + 2\gamma Ls \big)\Big)} \right \} > 0. \end{equation}$

(3.22)

which contradicts the conclusion (3.4) of Lemma 3.1, then (3.19) holds. □

4. Numerical experiments

In the present section, we conduct a series of numerical experiments in order to assess the performance of Algorithm 2.1. We make a comparison between it and the three-term projection method put forward by Zhang et al.^[19], the three-term conjugate method proposed by Gao et al.^[23], and the PRP-like method presented by Abubakar et al.^[24]. We conduct our tests using Matlab R2018a on a personal computer equipped with 32 GB of RAM and an 11th Gen Intel(R) Core(TM) i7-11700K processor.

We test the algorithms on ten problems that have been widely used in the literature for dimensions 50000 and 200000. All test problems are initialized with the following eight starting points: $x_1 = (10, 10, \cdots, 10)^T$ , $x_2 = (-10, -10, \cdots, -10)^T$ , $x_3 = (-1, -1, \cdots, -1)^T$ , $x_4 = (0.1, 0.1, \cdots, 0.1)^T$ , $x_5 = (\frac{1}{2}, \frac{1}{2^2}, \cdots, \frac{1}{2^n})^T$ , $x_6 = (1, \frac{1}{2}, \cdots, \frac{1}{n})^T$ , $x_7 = (\frac{1}{n}, \frac{2}{n}, \cdots, 1)^T$ , $x_8 = (\frac{n-1}{n}, \frac{n-2}{n}, \cdots, 0)^T$ , which are derived from ^[17,19,23]. With respect to every algorithm and each test problem, , the program concludes when any of the three conditions described below is achieved: (1) $\|F(x_k)\|\leq 10^{-6}$ , (2) $\|F(z_k)\|\leq 10^{-6}$ , (3) The algorithm still fails to converge after 1000 iterations. The parameters of methods introduced by Zhang, Gao and Abubakar are set as described in their respective articles. Regarding Algorithm 2.1, we define $\sigma = 10^{-4}, s = 1, \epsilon = 10^{-5}, \rho = 0.55, \gamma = 1.2.$ At present, we are going to list out the test problems, where the mapping $F$ has the following definition: $F(x) = (f_1(x), f_2(x), \cdots, f_n(x))^T.$

Test Problem 1 This problem sourced from ^[23] is

$\begin{equation} \begin{split} f_i(x) & = x_i - sin(|x_i|-1), i = 1,2,\cdots,n,\\ \Omega & = \{ x \in \mathbb{R}^n | x_i \geq -1, \sum\limits_{i = 1}^n x_i \leq n, i = 1,2,\cdots,n \} . \nonumber \end{split} \end{equation}$

Test Problem 2 This problem sourced from ^[23] is

$\begin{equation} \begin{split} f_i(x) & = e^{x_i} - 1, i = 1,2,\cdots,n,\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

Test Problem 3 This problem sourced from ^[23] is

$\begin{equation} \begin{split} f_i(x) & = 2x_i - sin(x_i), i = 1,2,\cdots,n,\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

Test Problem 4 This problem sourced from ^[19] is

$\begin{equation} \begin{split} f_i(x) & = x_i - 3 x_i \Big(\frac{sin(x_i)}{3} - 0.66\Big) + 2, i = 1,2,\cdots,n,\\ \Omega & = \{ x \in \mathbb{R}^n | x_i \geq -5, i = 1,2,\cdots,n \} . \nonumber \end{split} \end{equation}$

Test Problem 5 This problem sourced from ^[19] is

$\begin{equation} \begin{split} f_i(x) & = (e^{x_i})^2 + 3 sin(x_i)cos(x_i) - 1, i = 1,2,\cdots,n,\\ \Omega & = \{ x \in \mathbb{R}^n | x_i \geq -5, i = 1,2,\cdots,n\} . \nonumber \end{split} \end{equation}$

Test Problem 6 This problem sourced from ^[19] is

$\begin{equation} \begin{split} f_1(x) & = 4x_1(x_1^2 + x_2^2) - 4,\\ f_i(x) & = 4x_i(x_{i-1}^2 + x_i^2) + 4x_i(x_i^2 + x_{i+1}^2) - 4, i = 2,3,\cdots,n-1,\\ f_n(x) & = 4x_n(x_{n-1}^2 + x_n^2),\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

Test Problem 7 This problem sourced from ^[12] is

$\begin{equation} \begin{split} f_1(x) & = 2.5 x_1 + x_2 - 1,\\ f_i(x) & = x_{i-1} + 2.5x_i + x_{i+1}, i = 2,3,\cdots,n-1,\\ f_n(x) & = x_{n-1} + 2.5 x_n - 1),\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

Test Problem 8 This problem sourced from ^[25] is

$\begin{equation} \begin{split} f_1(x) & = cos(x_{1}) - 9 + 3x_1 + 8exp(x_{2}),\\ f_i(x) & = cos(x_{i}) - 9 + 3x_i + 8exp(x_{i-1}), i = 2,3,\cdots,n,\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

Test Problem 9 This problem sourced from ^[25] is

$\begin{equation} \begin{split} f_1(x) & = \frac{1}{(x_1 + 1)^2} - exp(x_i),\\ f_i(x) & = \frac{1}{(x_i + 1)^2} + cos(x_{i+1}) - 2exp(x_i), i = 2,3,\cdots,n-1,\\ f_n(x) & = \frac{1}{(x_n + 1)^2} - exp(x_n),\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

Test Problem 10 This problem sourced from ^[25] is

$\begin{equation} \begin{split} f_i(x) & = 2x_i - sin|x_i|, i = 1,2,\cdots,n,\\ \Omega & = \mathbb{R}^n_+ . \nonumber \end{split} \end{equation}$

In Tables 1, 2, and 3, we present the detailed outcomes using the format of "Niter/ Nfun/Time". Here, "Niter" represents the number of iterations, "Nfun" denotes the number of function value calculations, and "Time" stands for the CPU time counted in seconds. "Dim" indicates the dimension of test problems, and "Sta" represents the starting points. Through a comprehensive analysis of Tables 1, 2 and 3, it can be clearly observed that, for the given problems, Algorithm 2.1 registered the fewest values for Niter and Nfun in a large number of instances.

Table 1. The results of four algorithms for Test Problems 1, 2, 3, and 4.

Problem	Dim	Sta	Zhang	Gao	Abubakar	Algorithm 2.1
			Niter/Nfun/Time	Niter/Nfun/Time	Niter/Nfun/Time	Niter/Nfun/Time
TP1	50000	$x_1$	18 / 53 / 0.043	12 / 34 / 0.031	10 / 39 / 0.025	18 / 53 / 0.047
		$x_2$	18 / 54 / 0.038	11 / 33 / 0.021	10 / 39 / 0.020	18 / 54 / 0.040
		$x_3$	17 / 52 / 0.035	10 / 30 / 0.024	9 / 38 / 0.022	17 / 52 / 0.033
		$x_4$	14 / 42 / 0.030	11 / 33 / 0.025	10 / 40 / 0.024	14 / 42 / 0.031
		$x_5$	17 / 51 / 0.034	11 / 33 / 0.021	36 / 179 / 0.099	17 / 51 / 0.035
		$x_6$	17 / 51 / 0.036	11 / 33 / 0.021	39 / 194 / 0.105	18 / 54 / 0.033
		$x_7$	21 / 58 / 0.043	14 / 39 / 0.030	48 / 228 / 0.129	22 / 61 / 0.042
		$x_8$	21 / 58 / 0.041	14 / 39 / 0.027	48 / 228 / 0.140	22 / 61 / 0.045
	200000	$x_1$	19 / 56 / 0.191	13 / 37 / 0.137	11 / 43 / 0.123	19 / 56 / 0.171
		$x_2$	19 / 57 / 0.195	12 / 36 / 0.143	11 / 43 / 0.122	19 / 57 / 0.193
		$x_3$	18 / 55 / 0.189	11 / 33 / 0.117	10 / 42 / 0.123	18 / 55 / 0.166
		$x_4$	15 / 45 / 0.161	12 / 36 / 0.136	10 / 40 / 0.126	15 / 45 / 0.142
		$x_5$	17 / 51 / 0.186	11 / 33 / 0.111	39 / 192 / 0.516	18 / 54 / 0.203
		$x_6$	18 / 54 / 0.180	11 / 33 / 0.114	39 / 194 / 0.527	18 / 54 / 0.173
		$x_7$	21 / 58 / 0.199	14 / 39 / 0.141	50 / 238 / 0.670	21 / 60 / 0.192
		$x_8$	21 / 58 / 0.222	14 / 39 / 0.142	50 / 238 / 0.663	21 / 60 / 0.197
TP2	50000	$x_1$	1 / 15 / 0.009	15 / 59 / 0.046	1 / 37 / 0.017	1 / 15 / 0.007
		$x_2$	1 / 2 / 0.002	1 / 2 / 0.002	1 / 2 / 0.002	1 / 2 / 0.002
		$x_3$	1 / 2 / 0.002	1 / 2 / 0.002	1 / 2 / 0.001	1 / 2 / 0.001
		$x_4$	16 / 33 / 0.021	11 / 23 / 0.018	1 / 3 / 0.001	16 / 33 / 0.020
		$x_5$	13 / 27 / 0.012	10 / 22 / 0.010	31 / 64 / 0.028	13 / 27 / 0.013
		$x_6$	15 / 31 / 0.021	11 / 25 / 0.015	33 / 71 / 0.041	14 / 29 / 0.016
		$x_7$	19 / 39 / 0.026	14 / 31 / 0.020	55 / 116 / 0.082	19 / 39 / 0.022
		$x_8$	19 / 39 / 0.030	14 / 31 / 0.021	55 / 116 / 0.075	19 / 39 / 0.024
	200000	$x_1$	1 / 15 / 0.045	16 / 61 / 0.220	1 / 37 / 0.085	1 / 15 / 0.045
		$x_2$	1 / 2 / 0.009	1 / 2 / 0.007	1 / 2 / 0.009	1 / 2 / 0.007
		$x_3$	1 / 2 / 0.007	1 / 2 / 0.006	1 / 2 / 0.006	1 / 2 / 0.006
		$x_4$	17 / 35 / 0.131	12 / 25 / 0.113	1 / 3 / 0.007	17 / 34 / 0.124
		$x_5$	13 / 27 / 0.095	10 / 22 / 0.069	31 / 64 / 0.176	13 / 27 / 0.075
		$x_6$	15 / 31 / 0.112	11 / 25 / 0.096	33 / 71 / 0.238	14 / 29 / 0.109
		$x_7$	20 / 41 / 0.167	15 / 33 / 0.121	57 / 120 / 0.421	20 / 41 / 0.150
		$x_8$	20 / 41 / 0.166	15 / 33 / 0.120	57 / 120 / 0.413	20 / 41 / 0.144
TP3	50000	$x_1$	2 / 6 / 0.005	14 / 31 / 0.022	1 / 6 / 0.007	2 / 6 / 0.007
		$x_2$	1 / 4 / 0.005	1 / 3 / 0.003	7 / 18 / 0.012	1 / 4 / 0.004
		$x_3$	1 / 3 / 0.002	1 / 3 / 0.002	7 / 15 / 0.010	1 / 3 / 0.002
		$x_4$	16 / 33 / 0.022	12 / 25 / 0.017	6 / 13 / 0.007	16 / 32 / 0.017
		$x_5$	13 / 27 / 0.015	9 / 19 / 0.009	20 / 41 / 0.018	13 / 27 / 0.011
		$x_6$	14 / 29 / 0.016	10 / 21 / 0.011	37 / 75 / 0.035	14 / 29 / 0.013
		$x_7$	18 / 37 / 0.024	13 / 27 / 0.015	47 / 95 / 0.059	18 / 37 / 0.025
		$x_8$	18 / 37 / 0.024	13 / 27 / 0.014	47 / 95 / 0.059	18 / 37 / 0.021
	200000	$x_1$	2 / 6 / 0.030	15 / 33 / 0.148	1 / 6 / 0.029	2 / 6 / 0.031
		$x_2$	1 / 4 / 0.019	1 / 3 / 0.013	7 / 18 / 0.065	1 / 4 / 0.019
		$x_3$	1 / 3 / 0.008	1 / 3 / 0.010	7 / 15 / 0.047	1 / 3 / 0.010
		$x_4$	17 / 35 / 0.131	12 / 25 / 0.099	6 / 13 / 0.036	15 / 30 / 0.091
		$x_5$	13 / 27 / 0.080	9 / 19 / 0.057	20 / 41 / 0.119	13 / 27 / 0.068
		$x_6$	14 / 29 / 0.101	10 / 21 / 0.068	37 / 75 / 0.221	14 / 29 / 0.088
		$x_7$	18 / 37 / 0.140	14 / 29 / 0.111	49 / 99 / 0.300	18 / 37 / 0.109
		$x_8$	18 / 37 / 0.126	14 / 29 / 0.115	49 / 99 / 0.330	18 / 37 / 0.117
TP4	50000	$x_1$	14 / 56 / 0.044	27 / 131 / 0.107	6 / 48 / 0.032	14 / 56 / 0.046
		$x_2$	14 / 56 / 0.045	27 / 132 / 0.082	6 / 44 / 0.031	14 / 56 / 0.041
		$x_3$	13 / 53 / 0.038	27 / 134 / 0.093	5 / 41 / 0.021	13 / 53 / 0.035
		$x_4$	14 / 57 / 0.040	21 / 103 / 0.069	5 / 40 / 0.023	14 / 57 / 0.034
		$x_5$	23 / 93 / 0.062	29 / 144 / 0.100	45 / 360 / 0.197	19 / 77 / 0.050
		$x_6$	23 / 93 / 0.055	29 / 144 / 0.105	52 / 416 / 0.233	20 / 81 / 0.049
		$x_7$	30 / 120 / 0.090	30 / 148 / 0.093	70 / 556 / 0.317	23 / 92 / 0.056
		$x_8$	30 / 120 / 0.077	30 / 148 / 0.099	70 / 556 / 0.336	23 / 92 / 0.067
	200000	$x_1$	15 / 60 / 0.198	28 / 136 / 0.472	6 / 48 / 0.156	15 / 60 / 0.196
		$x_2$	15 / 60 / 0.195	28 / 137 / 0.406	6 / 44 / 0.132	15 / 60 / 0.199
		$x_3$	13 / 53 / 0.164	28 / 139 / 0.416	5 / 41 / 0.098	13 / 53 / 0.150
		$x_4$	14 / 57 / 0.183	22 / 108 / 0.314	5 / 40 / 0.100	14 / 57 / 0.156
		$x_5$	23 / 93 / 0.279	30 / 149 / 0.463	45 / 360 / 0.838	20 / 81 / 0.218
		$x_6$	23 / 93 / 0.264	30 / 149 / 0.473	53 / 422 / 0.986	20 / 81 / 0.220
		$x_7$	31 / 124 / 0.368	31 / 153 / 0.506	72 / 572 / 1.386	23 / 92 / 0.262
		$x_8$	31 / 124 / 0.374	31 / 153 / 0.469	72 / 572 / 1.395	23 / 92 / 0.257

| Show Table

DownLoad: CSV

Table 2. The results of four algorithms for Test Problems 5, 6, 7, and 8.

Problem	Dim	Sta	Zhang	Gao	Abubakar	Algorithm 2.1
			Niter/Nfun/Time	Niter/Nfun/Time	Niter/Nfun/Time	Niter/Nfun/Time
TP5	50000	$x_1$	9 / 45 / 0.197	27 / 157 / 0.758	9 / 106 / 0.478	9 / 56 / 0.215
		$x_2$	9 / 24 / 0.040	21 / 85 / 0.111	9 / 47 / 0.066	9 / 24 / 0.040
		$x_3$	5 / 20 / 0.022	14 / 67 / 0.079	6 / 51 / 0.045	5 / 20 / 0.025
		$x_4$	4 / 17 / 0.021	13 / 64 / 0.067	5 / 46 / 0.045	4 / 17 / 0.020
		$x_5$	33 / 142 / 0.098	12 / 59 / 0.041	48 / 419 / 0.261	11 / 45 / 0.028
		$x_6$	34 / 147 / 0.135	14 / 72 / 0.068	48 / 436 / 0.362	13 / 52 / 0.058
		$x_7$	38 / 167 / 0.190	17 / 87 / 0.095	fail / fail / fail	15 / 62 / 0.073
		$x_8$	37 / 164 / 0.168	17 / 87 / 0.106	fail / fail / fail	15 / 62 / 0.063
	200000	$x_1$	9 / 45 / 0.802	29 / 165 / 3.370	9 / 106 / 1.911	9 / 56 / 0.875
		$x_2$	9 / 24 / 0.179	21 / 85 / 0.456	9 / 47 / 0.256	9 / 24 / 0.174
		$x_3$	5 / 20 / 0.091	14 / 67 / 0.313	6 / 51 / 0.176	5 / 20 / 0.089
		$x_4$	4 / 17 / 0.070	13 / 64 / 0.279	5 / 46 / 0.151	4 / 17 / 0.067
		$x_5$	33 / 142 / 0.462	12 / 59 / 0.203	48 / 419 / 1.064	11 / 45 / 0.131
		$x_6$	32 / 139 / 0.521	14 / 72 / 0.265	48 / 436 / 1.342	13 / 52 / 0.196
		$x_7$	34 / 150 / 0.656	17 / 87 / 0.397	fail / fail / fail	15 / 62 / 0.266
		$x_8$	43 / 174 / 0.859	17 / 87 / 0.451	fail / fail / fail	15 / 62 / 0.250
TP6	50000	$x_1$	41 / 269 / 0.102	32 / 287 / 0.103	69 / 1169 / 0.378	37 / 239 / 0.085
		$x_2$	140 / 1211 / 0.420	320 / 4034 / 1.358	fail / fail / fail	144 / 1415 / 0.479
		$x_3$	45 / 283 / 0.104	61 / 471 / 0.165	54 / 809 / 0.258	31 / 188 / 0.069
		$x_4$	51 / 313 / 0.117	388 / 2730 / 1.005	177 / 2499 / 0.813	30 / 180 / 0.063
		$x_5$	34 / 208 / 0.075	27 / 208 / 0.076	50 / 746 / 0.245	31 / 188 / 0.069
		$x_6$	44 / 268 / 0.097	27 / 210 / 0.075	55 / 820 / 0.264	31 / 189 / 0.067
		$x_7$	49 / 308 / 0.113	495 / 3482 / 1.295	68 / 1070 / 0.344	39 / 244 / 0.086
		$x_8$	53 / 336 / 0.122	31 / 244 / 0.088	75 / 1193 / 0.374	34 / 216 / 0.077
	200000	$x_1$	40 / 265 / 0.572	47 / 448 / 0.916	67 / 1123 / 2.037	36 / 235 / 0.477
		$x_2$	180 / 1562 / 3.159	fail / fail / fail	50 / 903 / 1.714	221 / 2338 / 4.424
		$x_3$	42 / 262 / 0.588	59 / 453 / 0.967	53 / 794 / 1.399	32 / 194 / 0.403
		$x_4$	47 / 292 / 0.635	404 / 2835 / 6.352	99 / 1401 / 2.509	32 / 193 / 0.397
		$x_5$	53 / 328 / 0.715	30 / 232 / 0.504	52 / 779 / 1.373	31 / 189 / 0.398
		$x_6$	42 / 263 / 0.581	28 / 218 / 0.469	55 / 824 / 1.472	35 / 213 / 0.448
		$x_7$	66 / 414 / 0.907	399 / 2816 / 6.304	75 / 1202 / 2.128	38 / 239 / 0.483
		$x_8$	59 / 378 / 0.828	32 / 252 / 0.543	76 / 1211 / 2.237	37 / 234 / 0.508
TP7	50000	$x_1$	62 / 217 / 0.078	112 / 487 / 0.172	46 / 372 / 0.114	66 / 239 / 0.085
		$x_2$	76 / 257 / 0.092	101 / 432 / 0.150	59 / 481 / 0.141	71 / 255 / 0.091
		$x_3$	69 / 237 / 0.089	139 / 563 / 0.202	47 / 383 / 0.115	64 / 231 / 0.081
		$x_4$	69 / 233 / 0.087	136 / 548 / 0.203	46 / 369 / 0.119	61 / 219 / 0.086
		$x_5$	64 / 217 / 0.082	100 / 432 / 0.156	44 / 356 / 0.103	60 / 216 / 0.073
		$x_6$	62 / 212 / 0.078	94 / 394 / 0.143	46 / 375 / 0.111	58 / 209 / 0.074
		$x_7$	63 / 216 / 0.078	93 / 406 / 0.148	47 / 384 / 0.116	62 / 224 / 0.081
		$x_8$	63 / 216 / 0.078	93 / 406 / 0.144	47 / 382 / 0.111	62 / 224 / 0.081
	200000	$x_1$	61 / 212 / 0.520	114 / 501 / 1.400	42 / 344 / 0.617	69 / 248 / 0.565
		$x_2$	76 / 262 / 0.668	101 / 444 / 1.087	60 / 490 / 0.908	70 / 252 / 0.557
		$x_3$	70 / 243 / 0.618	102 / 445 / 1.098	53 / 427 / 0.835	65 / 234 / 0.559
		$x_4$	71 / 241 / 0.672	97 / 421 / 1.077	46 / 370 / 0.692	65 / 233 / 0.511
		$x_5$	68 / 233 / 0.619	92 / 399 / 0.966	46 / 368 / 0.664	62 / 223 / 0.486
		$x_6$	56 / 194 / 0.508	127 / 513 / 1.289	44 / 355 / 0.644	59 / 213 / 0.472
		$x_7$	63 / 219 / 0.551	95 / 417 / 1.014	45 / 363 / 0.666	66 / 237 / 0.559
		$x_8$	63 / 219 / 0.530	95 / 417 / 1.010	45 / 363 / 0.669	66 / 237 / 0.538
TP8	50000	$x_1$	21 / 140 / 0.149	28 / 215 / 0.232	1 / 46 / 0.077	21 / 140 / 0.149
		$x_2$	1 / 5 / 0.008	1 / 4 / 0.007	2 / 21 / 0.019	1 / 5 / 0.008
		$x_3$	1 / 6 / 0.006	1 / 5 / 0.006	2 / 24 / 0.023	1 / 6 / 0.007
		$x_4$	18 / 109 / 0.091	25 / 173 / 0.163	1 / 13 / 0.011	18 / 109 / 0.097
		$x_5$	373 / 2929 / 1.437	26 / 90 / 0.053	fail / fail / fail	12 / 58 / 0.033
		$x_6$	146 / 1338 / 1.183	8 / 36 / 0.035	fail / fail / fail	8 / 34 / 0.033
		$x_7$	284 / 1640 / 1.546	48 / 222 / 0.150	65 / 936 / 0.759	56 / 299 / 0.273
		$x_8$	300 / 1714 / 1.616	32 / 156 / 0.112	32 / 984 / 3.637	45 / 243 / 0.224
	200000	$x_1$	22 / 146 / 0.601	29 / 222 / 1.012	1 / 46 / 0.308	22 / 146 / 0.596
		$x_2$	1 / 5 / 0.033	1 / 4 / 0.028	2 / 21 / 0.089	1 / 5 / 0.032
		$x_3$	1 / 6 / 0.037	1 / 5 / 0.026	2 / 24 / 0.099	1 / 6 / 0.029
		$x_4$	19 / 115 / 0.385	25 / 173 / 0.650	1 / 13 / 0.047	19 / 115 / 0.373
		$x_5$	373 / 2929 / 6.855	26 / 90 / 0.313	fail / fail / fail	12 / 58 / 0.166
		$x_6$	fail / fail / fail	8 / 36 / 0.152	fail / fail / fail	8 / 34 / 0.155
		$x_7$	318 / 1853 / 8.556	67 / 298 / 1.056	65 / 1010 / 4.233	62 / 334 / 1.378
		$x_8$	305 / 2005 / 9.320	43 / 188 / 0.723	34 / 584 / 5.046	49 / 261 / 1.107

| Show Table

DownLoad: CSV

Table 3. The results of four algorithms for Test Problems 9 and 10.

Problem	Dim	Sta	Zhang	Gao	Abubakar	Algorithm 2.1
			Niter/Nfun/Time	Niter/Nfun/Time	Niter/Nfun/Time	Niter/Nfun/Time
TP9	50000	$x_1$	1 / 2 / 0.004	1 / 2 / 0.004	1 / 2 / 0.004	1 / 2 / 0.004
		$x_2$	1 / 2 / 0.004	1 / 2 / 0.005	1 / 2 / 0.004	1 / 2 / 0.005
		$x_3$	1 / 2 / 0.008	1 / 2 / 0.007	1 / 2 / 0.006	1 / 2 / 0.007
		$x_4$	3 / 4 / 0.016	4 / 8 / 0.027	4 / 5 / 0.019	3 / 4 / 0.013
		$x_5$	3 / 4 / 0.005	3 / 4 / 0.005	3 / 4 / 0.005	3 / 4 / 0.004
		$x_6$	4 / 5 / 0.015	3 / 4 / 0.011	3 / 4 / 0.014	4 / 5 / 0.015
		$x_7$	2 / 3 / 0.012	2 / 3 / 0.009	2 / 3 / 0.006	2 / 3 / 0.011
		$x_8$	2 / 3 / 0.014	2 / 3 / 0.006	2 / 3 / 0.005	2 / 3 / 0.013
	200000	$x_1$	1 / 2 / 0.017	1 / 2 / 0.017	1 / 2 / 0.017	1 / 2 / 0.017
		$x_2$	1 / 2 / 0.019	1 / 2 / 0.018	1 / 2 / 0.017	1 / 2 / 0.018
		$x_3$	1 / 2 / 0.025	1 / 2 / 0.024	1 / 2 / 0.024	1 / 2 / 0.028
		$x_4$	3 / 4 / 0.072	4 / 8 / 0.087	4 / 5 / 0.069	3 / 4 / 0.039
		$x_5$	3 / 4 / 0.022	3 / 4 / 0.023	3 / 4 / 0.023	3 / 4 / 0.021
		$x_6$	4 / 5 / 0.062	3 / 4 / 0.053	3 / 4 / 0.049	4 / 5 / 0.063
		$x_7$	2 / 3 / 0.053	2 / 3 / 0.028	2 / 3 / 0.029	2 / 3 / 0.044
		$x_8$	2 / 3 / 0.050	2 / 3 / 0.034	2 / 3 / 0.028	2 / 3 / 0.051
TP10	50000	$x_1$	2 / 6 / 0.006	14 / 31 / 0.022	1 / 6 / 0.005	2 / 6 / 0.007
		$x_2$	1 / 4 / 0.004	1 / 3 / 0.002	2 / 8 / 0.007	1 / 4 / 0.003
		$x_3$	16 / 34 / 0.020	1 / 4 / 0.003	7 / 19 / 0.009	16 / 34 / 0.022
		$x_4$	16 / 33 / 0.017	12 / 25 / 0.016	6 / 13 / 0.008	16 / 32 / 0.022
		$x_5$	13 / 27 / 0.013	9 / 19 / 0.010	20 / 42 / 0.019	13 / 27 / 0.010
		$x_6$	14 / 29 / 0.016	10 / 21 / 0.010	24 / 49 / 0.023	14 / 29 / 0.012
		$x_7$	18 / 37 / 0.024	13 / 27 / 0.014	29 / 60 / 0.031	18 / 37 / 0.019
		$x_8$	18 / 37 / 0.020	13 / 27 / 0.015	29 / 60 / 0.036	18 / 37 / 0.020
	200000	$x_1$	2 / 6 / 0.024	15 / 33 / 0.129	1 / 6 / 0.017	2 / 6 / 0.025
		$x_2$	1 / 4 / 0.013	1 / 3 / 0.011	2 / 8 / 0.030	1 / 4 / 0.015
		$x_3$	17 / 36 / 0.127	1 / 4 / 0.013	7 / 19 / 0.061	13 / 27 / 0.097
		$x_4$	17 / 35 / 0.128	12 / 25 / 0.098	6 / 13 / 0.038	15 / 30 / 0.093
		$x_5$	13 / 27 / 0.087	9 / 19 / 0.067	20 / 42 / 0.117	13 / 27 / 0.074
		$x_6$	14 / 29 / 0.103	10 / 21 / 0.074	24 / 49 / 0.170	14 / 29 / 0.093
		$x_7$	18 / 37 / 0.141	14 / 29 / 0.103	31 / 64 / 0.191	18 / 37 / 0.112
		$x_8$	18 / 37 / 0.130	14 / 29 / 0.107	31 / 64 / 0.210	18 / 37 / 0.118

| Show Table

DownLoad: CSV

In an effort to comprehensively compare all algorithms, we utilize the performance profiles that were proposed by Dolan and Moré ^[26]. The performance profiles concerning the number of iterations are shown in Figure 1. The performance profiles for the number of function evaluations are displayed in Figure 2, while Figure 3 reveals the performance profiles of the CPU time. It is readily observable that the Algorithm 2.1 introduced in this paper outperforms the algorithms presented by Zhang, Gao, and Abubakar in terms of efficiency.

Figure 1. Performance profiles for the number of iterations.

DownLoad: Full-Size Img PowerPoint

Figure 2. Performance profiles for the number of function evaluations.

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Figure 3. Performance profiles for the CPU time.

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5. Applications in sparse signal restoration

In this section, we make use of Algorithm 2.1 to tackle sparse signal restoration problems and compare its performance with that of Zhang^[19], Gao^[23], and Abubakar^[24]. We begin by recalling the compressed sensing model. This model is utilized to retrieve sparse solutions or signals from underdetermined linear systems represented as $Ax = b$ , where $b \in \mathbb{R}^m$ represents the data obtained from observations and $A \in \mathbb{R}^{m \times n} (m \ll n)$ denotes a linear mapping. Specifically, we can achieve sparse signal restoration through the solution of the following $l_1$ -norm regularized optimization problem

$\begin{equation} \min\limits_{x \in \mathbb{R}^n}\quad \frac{1}{2}\|Ax-b\|_2^2 + \tau \|x\|_1, \end{equation}$

(5.1)

where $\tau$ acts as a constant that reconciles data fitting and sparsity. The vector $x \in \mathbb{R}^n$ can be decomposed as $x = u-v, u \geq 0, v\geq 0,$ where $u \in \mathbb{R}^n, v \in \mathbb{R}^n$ , and $u_i = \max\{ x_i, 0\}, v_i = \max\{-x_i, 0\}$ for $i = 1, 2, \cdots, n.$ Therefore, we can replace problem (5.1) with the next programming formulation:

$\begin{equation} \min\limits_{u,v \geq 0} \frac{1}{2}\|A(u-v) - b\|_2^2 + \tau e_n^T u + \tau e_n^T v, \end{equation}$

(5.2)

in which $e_n = (1, 1, \cdots, 1)^T \in \mathbb{R}^n$ .

Further, we obtain its standard form

$\begin{equation} \min\limits_{z \geq 0} \frac{1}{2}z^T H z + c^T z, \end{equation}$

(5.3)

in which

$\begin{equation} z = \left[ \begin{array}{c} u \\ v \end{array} \right], \ c = \left[ \begin{array}{c} \tau e_n - A^T b \\ \tau e_n + A^T b \end{array} \right],\ H = \left[ \begin{array}{cc} A^T A & - A^T A \\ -A^T A & A^T A \end{array} \right]. \end{equation}$

(5.4)

Evidently, matrix $H$ has semi-positive definiteness; thus, Eq (5.3) is a convex quadratic problem. Based on first-order optimality conditions for (5.3), z qualifies as a minimizer if and only if it satisfies the subsequent equations:

$\begin{equation} F(z) = \min\{z,Hz+c \} = 0 ,z\geq 0, \end{equation}$

(5.5)

in which $F(z)$ is monotone and continuous. Henceforth, our proposed method may be employed for resolving (5.5).

During these experiments, the primary objective is to reconstruct a one-dimensional sparse signal with a length of $n$ from $m (m \ll n)$ observations. The performance of signal restoration is evaluated using the mean squared error (MSE), defined as

$\begin{equation} \text{MSE} = \frac{1}{n}\|x - x^*\|^2, \end{equation}$

(5.6)

in which $x^*$ represents the recovered signal and $x$ indicates the initial signal. Let $f(x) = \frac{1}{2}\|Ax-b\|_2^2 + \tau \|x\|_1$ serves as the auxiliary function. The iterative sequence commences with the initial measurement signal $x_0 = A^T b$ and comes to an end when

$\begin{equation} \frac{\|f(x_k) - f(x_{k-1})\|}{\|f(x_{k-1})\|} < 10^{-5}. \end{equation}$

(5.7)

During our experiments, we define $r$ as the quantity of nonzero entries in $x$ . For a specified set of values of $m, n$ , and $r$ , we generate the subsequent test data using MATLAB:

$\begin{equation} \begin{split} x & = zeros(n,1);\\ q & = randperm(n);\\ x&(q(1:r)) = randn(r,1);\\ b & = orth(randn(m,n)')'*x+0.001* randn(m,1);\\ x_0 & = A'*b.\\ \end{split} \end{equation}$

(5.8)

The numerical outcomes presented in Figure 4 contain the original sparse signal, the measurement data, and the reconstructed signals yielded by four algorithms. Clearly, all four algorithms reconstruct the original signal from the measurements with near perfection. However, Algorithm 2.1 outperforms the other algorithms in terms of MSE, the number of iterations, and CPU time. Figure 5 depicts four graphs to visually assess the performance of the four algorithms. These graphs display the convergence characteristics of the algorithms, depicting the changes in the values of the merit function and the MSE as the iterations progress and as the CPU time is consumed. Table 4 displays the numerical outcomes obtained from the experiment on ten diverse noise samples. Evidently, in a majority of cases, Algorithm 2.1 outperforms other algorithms. It achieves a lower MSE, demands a smaller number of iterations, and consumes less time on the CPU.

Figure 4. From top to bottom: the original signal, the measurement, and the reconstructed signals by four algorithms.

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Figure 5. The upper graphs show MSE against iterations and CPU time, and the lower graphs show objective function value against iterations and CPU time.

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Table 4. The results of four algorithms in ten experiments on the signal recovery problem.

	Zhang	Gao	Abubakar	Algorithm 2.1
	Niter/Time/MSE	Niter/Time/MSE	Niter/Time/MSE	Niter/Time/MSE
	87 / 4.47 / 2.07E-05	116 / 6.20 / 2.39E-05	85 / 4.80 / 1.26E-05	70 / 3.73 / 1.93E-05
	144 / 7.28 / 1.21E-05	150 / 7.81 / 4.06E-05	103 / 5.75 / 1.41E-05	89 / 4.78 / 1.20E-05
	120 / 6.18 / 1.26E-05	132 / 6.92 / 5.45E-05	112 / 6.21 / 1.61E-05	66 / 3.51 / 1.34E-05
	135 / 6.85 / 1.77E-05	124 / 6.47 / 9.44E-05	139 / 7.63 / 1.88E-05	95 / 4.99 / 1.05E-05
	146 / 7.40 / 1.72E-05	136 / 7.14 / 7.16E-05	115 / 6.32 / 2.32E-05	91 / 4.74 / 1.62E-05
	115 / 5.90 / 1.17E-05	127 / 6.62 / 4.50E-05	89 / 4.94 / 1.66E-05	70 / 3.66 / 1.75E-05
	153 / 7.77 / 1.25E-05	161 / 8.31 / 3.99E-05	157 / 8.51 / 1.49E-05	105 / 5.44 / 1.16E-05
	104 / 5.38 / 1.21E-05	113 / 6.10 / 4.62E-05	103 / 5.73 / 1.30E-05	82 / 4.42 / 1.15E-05
	132 / 6.71 / 1.32E-05	132 / 7.01 / 4.64E-05	95 / 5.34 / 1.34E-05	85 / 4.48 / 1.30E-05
	164 / 8.26 / 1.69E-05	160 / 8.22 / 7.25E-05	113 / 6.23 / 2.17E-05	108 / 5.58 / 1.74E-05
Average	130 / 6.62 / 1.47E-05	135 / 7.08 / 5.35E-05	111 / 6.15 / 1.64E-05	86 / 4.53 / 1.42E-05

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6. Conclusions

This paper presents a derivative-free conjugate gradient method for the purpose of solving large-scale nonlinear systems of monotone equations that are bounded by convex constraints. The proposed approach is a novel expansion of the RMIL conjugate gradient method. It incorporates projection techniques, which play a crucial role in ensuring that the iterates remain within the feasible region defined by the convex constraints. This not only helps in maintaining the validity of the solution process but also improves the efficiency of the algorithm by avoiding unnecessary computations outside the constraint set. Additionally, the method features a line search strategy that operates without using derivatives. Under certain reasonable assumptions, we prove the global convergence property of the method. To further demonstrate the practical performance of our proposed method, we carry out extensive numerical experiments. The experimental results illustrate that the method we proposed has great promise. It outperforms several existing methods in terms of computational efficiency and accuracy when solving large-scale nonlinear systems of monotone equations. Furthermore, our numerical experiments on sparse signal restoration problems further validate the effectiveness and practicality of the method.

Acknowledgments

This research was supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY23A010007 and the Huzhou Natural Science Foundation under Grant No. 2023YZ29.

Conflict of interest

The author declares to have no competing interests.

Use of Generative-AI tools declaration

The author declares they have not used Artificial Intelligence (AI) tools in the creation of this article.

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A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations

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1. Introduction

2. Algorithm

3. Convergence analysis

4. Numerical experiments

5. Applications in sparse signal restoration

6. Conclusions

Acknowledgments

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A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations

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1. Introduction

2. Algorithm

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4. Numerical experiments

5. Applications in sparse signal restoration

6. Conclusions

Acknowledgments

Conflict of interest

Use of Generative-AI tools declaration

References

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