Support vector machine optimization via an improved elephant herding algorithm for motor energy efficiency rating

Xinrui Ren; Jianbo Yu; Zhaomin Lv; Xinrui Ren; Jianbo Yu; Zhaomin Lv

doi:10.3934/mbe.2022557

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 11957-11982. doi: 10.3934/mbe.2022557

Previous Article Next Article

Research article Special Issues

Support vector machine optimization via an improved elephant herding algorithm for motor energy efficiency rating

1.
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
State Key Laboratory of ASIC and System, School of Microelectronics, Fudan University, Shanghai 200433, China
3.
School of Urban Railway Transportation, Shanghai University of Engineering Science, Shanghai 201620, China
Academic editor: Hamid Reza Karimi

Received: 23 June 2022 Revised: 18 July 2022 Accepted: 19 July 2022 Published: 17 August 2022

Accurate evaluation of motor energy efficiency under off-condition operation can provide an important basis for an energy-saving upgrade of the motor and the elimination of backward motors. By considering the power quality, motor characteristics and load characteristics, a motor energy efficiency evaluation system with seven indexes and 10 grades was constructed. An improved elephant herding optimization method combined with a support vector machine rating model is proposed, it achieved an accuracy higher than 98%. Considering the slow convergence speed and low convergence precision of the standard elephant herding optimization (EHO) method, it is easy to fall into the local optimum problem. To improve population initialization, chaotic mapping and adversarial learning were used to achieve EHO with population diversity and global search capability. Group learning and elite retention have been added to improve the local development ability of the algorithm. The improved EHO has been compared with other intelligent optimization algorithms by using 12 benchmark functions, and the results show that the improved algorithm has better optimization performance.

Keywords:

Citation: Xinrui Ren, Jianbo Yu, Zhaomin Lv. Support vector machine optimization via an improved elephant herding algorithm for motor energy efficiency rating[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 11957-11982. doi: 10.3934/mbe.2022557

Related Papers:

[1]	Xiaowei Fang . A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations. AIMS Mathematics, 2025, 10(5): 11656-11675. doi: 10.3934/math.2025528
[2]	Sani Aji, Poom Kumam, Aliyu Muhammed Awwal, Mahmoud Muhammad Yahaya, Kanokwan Sitthithakerngkiet . An efficient DY-type spectral conjugate gradient method for system of nonlinear monotone equations with application in signal recovery. AIMS Mathematics, 2021, 6(8): 8078-8106. doi: 10.3934/math.2021469
[3]	Austine Efut Ofem, Jacob Ashiwere Abuchu, Godwin Chidi Ugwunnadi, Hossam A. Nabwey, Abubakar Adamu, Ojen Kumar Narain . Double inertial steps extragadient-type methods for solving optimal control and image restoration problems. AIMS Mathematics, 2024, 9(5): 12870-12905. doi: 10.3934/math.2024629
[4]	Ting Lin, Hong Zhang, Chaofan Xie . A modulus-based modified multivariate spectral gradient projection method for solving the horizontal linear complementarity problem. AIMS Mathematics, 2025, 10(2): 3251-3268. doi: 10.3934/math.2025151
[5]	Habibu Abdullahi, A. K. Awasthi, Mohammed Yusuf Waziri, Issam A. R. Moghrabi, Abubakar Sani Halilu, Kabiru Ahmed, Sulaiman M. Ibrahim, Yau Balarabe Musa, Elissa M. Nadia . An improved convex constrained conjugate gradient descent method for nonlinear monotone equations with signal recovery applications. AIMS Mathematics, 2025, 10(4): 7941-7969. doi: 10.3934/math.2025365
[6]	Younes Talaei, Sanda Micula, Hasan Hosseinzadeh, Samad Noeiaghdam . A novel algorithm to solve nonlinear fractional quadratic integral equations. AIMS Mathematics, 2022, 7(7): 13237-13257. doi: 10.3934/math.2022730
[7]	Ziqi Zhu, Kaiye Zheng, Shenghua Wang . A new double inertial subgradient extragradient method for solving a non-monotone variational inequality problem in Hilbert space. AIMS Mathematics, 2024, 9(8): 20956-20975. doi: 10.3934/math.20241020
[8]	Yu Zhang, Xiaojun Ma . An accelerated conjugate method for split variational inclusion problems with applications. AIMS Mathematics, 2025, 10(5): 11465-11487. doi: 10.3934/math.2025522
[9]	Aliyu Muhammed Awwal, Poom Kumam, Kanokwan Sitthithakerngkiet, Abubakar Muhammad Bakoji, Abubakar S. Halilu, Ibrahim M. Sulaiman . Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application. AIMS Mathematics, 2021, 6(8): 8792-8814. doi: 10.3934/math.2021510
[10]	Yali Zhao, Qixin Dong, Xiaoqing Huang . A self-adaptive viscosity-type inertial algorithm for common solutions of generalized split variational inclusion and paramonotone equilibrium problem. AIMS Mathematics, 2025, 10(2): 4504-4523. doi: 10.3934/math.2025208

Abstract

1. Introduction

Spectral gradient methods are among the widely known first-order methods for unconstrained optimization problems $\min_{m \in \mathbb{R}^n} f(m),$ where $f:\mathbb{R}^n\to \mathbb{R}$ is a smooth nonlinear function that is bounded below. These methods generate a sequence of approximations using the iterative formula:

$\begin{equation} m_{k+1} = m_k + \Lambda_k t_k, \end{equation}$

(1.1)

where $\Lambda_k$ is called the step size and is obtained using some line search procedures, and $t_k$ is called a search direction, defined as

$\begin{equation} {t_k}: = \begin{cases} -g_k, & {\rm{if}}\; k = 0 , \\ -\overline{\gamma}_k g_{k}, & {\rm{if}}\; k\ge 1 , \end{cases} \end{equation}$

(1.2)

where $g_k: = \nabla f(m_k),$ and the coefficient $\overline{\gamma}_k$ is a scalar known as the spectral parameter, which differentiates any two spectral gradient methods (see Barzilai and Borwein (BB) ^[1]). The classical forms of the parameter $\overline{\gamma}_k$ given in ^[1] are

$\begin{align} \overline{\gamma}_k^{long} = \frac{\|m_{k+1}-m_{k}\|^2}{\langle g(m_{k+1})-g(m_k), m_{k+1}-m_{k}\rangle}, \end{align}$

(1.3)

$\begin{align} \overline{\gamma}_k^{short} = \frac{\langle g(m_{k+1})-g(m_k), m_{k+1}-m_{k}\rangle}{\|g(m_{k+1})-g(m_k)\|^2}. \end{align}$

(1.4)

Among the advantages of the spectral gradient method is its simplicity in implementation and low storage requirement; thus, it is suitable for large-scale problems. As a result, researchers have extended the spectral gradient method to solve systems of nonlinear equations (see, ^[2,3,4]). A system of nonlinear equations involves finding a vector $m$ in a nonempty, closed and convex set $C \subset \mathbb{R}^n$ such that

$\begin{equation} \Gamma(m) = 0, \end{equation}$

(1.5)

where $\Gamma:\mathbb{R}^n\to \mathbb{R}^n$ is continuous. When the mapping $\Gamma$ is monotone, i.e.,

$\begin{equation*} \langle \Gamma(m)-\Gamma(\overline{m}), m-\overline{m} \rangle \geq 0, \quad \forall m, \overline{m}\in \mathbb{R}^n, \end{equation*}$

Problem (1.5) becomes a system of nonlinear monotone equations. Many problems arising from science and engineering can be translated into the form of problem (1.5). For example, applications of system of nonlinear equations (1.5) have appeared in different fields ^{[5,6,7,8,9,10]}. In recent years, algorithms for solving systems of nonlinear monotone equations have been used in signal and image recovery, (see ^{[11,12,13,14,15,16,17]}).

Motivated by the hyperplane projection strategy of Solodov and Svaiter ^[18], spectral gradient-like methods for solving (1.5) have gained more attention. Zhang and Zhou ^[19] developed a spectral gradient projection method for unconstrained nonlinear equations based on the modified version of the $\overline{\gamma}_k^{long}$ (1.3). Later on, Yu et al.^[20] extended the work in ^[19] to solve convex constrained nonlinear equations. In addition, Yu et al. ^[21] proposed another spectral method for solving a system of nonlinear equations. The search direction in their work uses a convex combination of the modified $\overline{\gamma}_k^{long}$ (1.3) and $\overline{\gamma}_k^{short}$ (1.4). Their work revealed that combining the modified BB parameters gives better numerical performance than deploying them separately. This means that the efficiency of the convex combination of the $\overline{\gamma}_k^{long}$ (1.3) and $\overline{\gamma}_k^{short}$ (1.4) can be further explored.

Nowadays, there is a growing interest in incorporating the inertial technique into algorithms for solving systems of nonlinear equations (see, for example, ^[22,23]). The inertial step defined as $i_{k+1}: = m_{k+1}+\alpha_k(m_{k+1}-m_{k}), \; \alpha_k\in(0, 1),$ was proposed by Polyak ^[24] in order to speed up the performance of iterative algorithms. It can be observed that the inertial technique is using two previous iterates to compute the current iterate. This has also been shown to accelerate the iteration process of algorithms for solving nonlinear problems such as the proximal point method ^[25,26] and auxiliary problem principle ^[27].

Motivated by the contributions of the above-mentioned literature, this work seeks to explore the effect of the inertial technique on the convex combination of the $\overline{\gamma}_k^{long}$ and $\overline{\gamma}_k^{short}$ based on some modifications. This idea can be viewed as the modification of the work of Yu et al. ^[21]. Numerical experiments conducted in Section 3 reveal some level of improvement in numerical performance. Some of the notable contributions of this work include the following:

● A new spectral method for solving a system of nonlinear monotone equations based on the inertial technique is proposed.

● This work generalizes some existing algorithms in the literature.

● The global convergence of the proposed method is discussed under standard conditions.

● To depict the efficiency of the new method, a numerical experiment on a collection of test problems in comparison with some existing methods is presented.

● Subsequently, the proposed algorithm is applied to problems arising from robotic motion control.

In the remaining part of this work, the next section gives some definitions, details of the algorithms and global convergence. Section three gives some numerical experiments, while the application part is in Section four. In the final section, some concluding remarks are given.

2. Proposed algorithm and convergence analysis

We begin this section by recalling the definition of the projection operator as follows:

Definition 2.1. Suppose $C\subset \mathbb{R}^n$ is a convex, nonempty and closed set. Then, any point $m\in \mathbb{R}^n$ can be projected onto the set $C$ using

$\begin{equation} P_{C}(m) = \arg \min \{\|m-\overline{m}\|\ : \overline{m}\in C\}. \end{equation}$

(2.1)

The relation (2.1) satisfies the following useful property:

$\begin{equation} \|P_{C}(m)-\overline{m}\|\leq \|m-\overline{m}\|, \quad \forall \overline{m}\in C. \end{equation}$

(2.2)

Definition 2.2. Any vector-valued map that satisfies

$\|\Gamma(m)-\Gamma(\overline{m})\|\leq L\|m-\overline{m}\|, \; \forall m, \overline{m} \in \mathbb{R}^n, \; L > 0,$

is said to be Lipschitz continuous.

In what follows, we present the proposed algorithm and subsequently give some remarks.

Algorithm 1: Derivative-Free Spectral Method with Inertia (iSDFM).
Input: Choose $m_{-1}, \; m_0\in C,$ $0 < \eta < 2,$ $\kappa > 0,$ $\sigma, \mu, \varsigma, \alpha_k \in(0, 1),$ $r > 0$ and $Tol > 0.$
Step 0: Set $k=0,$ compute $t_0:=-\Gamma(m_0)$ and $i_0:=m_0+\alpha_0(m_0-m_{-1}).$
Step 1: If $\\|\Gamma(m_k)\\|\le Tol$ , then terminate; else, continue with Step 2.
Step 2: Set
$\begin{equation} p_k:=m_k+\Lambda_kt_k, \; \; \; \Lambda_k= \kappa\varsigma^{j}, \; \; \; \; \; \; \; (2.3)\end{equation}$
where $j$ is the least non–negative integer such that
$\begin{equation} - \langle \Gamma(m_k+\kappa \varsigma^j t_k), t_k\rangle \geq \sigma \kappa \varsigma^{j}\\|t_k\\|^2 \min \{1, \; \\|\Gamma(m_k + \kappa \varsigma^{j} t_k)\\|^{\frac{1}{c}}\}, \; c \geq1.\; \; \; \; \; \; (2.4) \end{equation}$
Step 3: If $\\|\Gamma(p_k)\\|=0$ , stop. Else, compute
$\begin{equation} m_{k+1}:=P_{C} \left[m_k-\eta \frac{\langle\Gamma(p_k), m_k-p_k\rangle}{\\|\Gamma(p_k)\\|^2} \Gamma(p_k)\right].\; \; \; \; \; \; \; (2.5) \end{equation}$
Step 4: Set $k:=k+1$ and redo the task from step 1, where the inertial step is updated as $i_{k+1}:=m_{k+1}+\alpha_k(m_{k+1}-m_{k})$ with the search direction as
$\begin{equation} t_k:= -\gamma_k \Gamma(m_k), \; \; \; \; \; \; (2.6)\end{equation}$
where
$\begin{align} & \gamma_k = (1-\theta_k) \overline{\beta}_k +\theta_k\widehat{\beta}_k, \; \; \; \; \; (2.7) \end{align}$
$\begin{align} & \overline{\beta}_k= \frac{\\|i_{k+1}-i_k\\|^2}{\langle i_{k+1}-i_k, \; \Gamma(i_{k+1})-\Gamma(i_k)+r(i_{k+1}-i_k)\rangle}, \; \; \; \; \; (2.8)\end{align}$
$\begin{align} & \widehat{\beta}_k = \frac{\langle i_{k+1}-i_k, \; \Gamma(i_{k+1})-\Gamma(i_k)+r(i_{k+1}-i_k)\rangle}{\\|\Gamma(i_{k+1})-\Gamma(i_k)+r(i_{k+1}-i_k)\\|^2}, \; \; \; \; \; (2.9)\end{align}$
$\begin{align} & \theta_k = 1 - \mu \cdot \frac{\langle\Gamma(m_k), i_{k+1}-i_k\rangle^2}{\overline{\Gamma(m_k)}^2{ \\|i_{k+1}-i_k\\|^2}}, \; \; \; \; \; (2.10)\end{align}$
$\begin{align} & \overline{\Gamma(m_k)} = \max \{\\|\Gamma(m_{k-1})\\|, \\|\Gamma(m_k)\\|\}. \; \; \; \; \; (2.11)\end{align}$

| Show Table

DownLoad: CSV

Remark 2.3. The choice of the $\theta_k$ defined by (2.10) is prompted by the work of Awwal et al. ^[28]. Now, since $\max \{\|\Gamma(m_{k-1})\|, \|\Gamma(m_k)\|\}\geq \|\Gamma(m_k)\|,$ then by the Cauchy-Schwarz inequality we have

$\begin{equation} 0 \leq \frac{\langle \Gamma(m_k), \; i_{k+1}-i_k\rangle^2}{\|\overline{\Gamma(m_k)}\|^2\|i_{k+1}-i_k\|^2} \leq \frac{\|\Gamma(m_k)\|^2\|i_{k+1}-i_k\|^2}{\|\Gamma(m_k)\|^2\|i_{k+1}-i_k\|^2} = 1, \end{equation}$

(2.12)

and by the fact that $\mu\in(0, 1),$ we get

$\begin{equation} 0 \leq \theta_k \leq 1. \end{equation}$

(2.13)

Remark 2.4. We note that if for all $k$ , $i_k = m_k$ (2.8) and (2.9) and $\theta_k\in[0, 1],$ then the search direction (2.6) reduces to that of Yu et al. ^[21]. Moreover, if for all $k$ , $i_k = m_k$ in (2.8) and $\theta_k = 0,$ then the search direction (2.6) reduces to that of Yu et al.^[20]. Furthermore, the new search direction (2.6) also reduces to DAIS1 and DAIS2 of Awwal et al. in ^[22] if $\theta_k = 1$ and $\theta_k = 0,$ respectively. Hence, our proposed work is regarded as an extension of the algorithms proposed in ^[21,22].

To show the global convergence of the proposed algorithms, we assumed the following:

( $A_{1}$ ) The mapping $\Gamma$ is monotone.

( $A_{2}$ ) The mapping $\Gamma$ is Lipschitz continuous.

( $A_{3}$ ) The solution set of problem (1.5) is nonempty.

Lemma 2.5. Suppose that the assumptions ( $A_{1}$ )–( $A_{3}$ ) hold, and that $\{m_k\}$ and $\{i_k\}$ are produced by Algorithm 1. Then,

(ⅰ) $\lim\limits_{k \to \infty }\|m_{k}-\widehat{m}\|$ exists.

(ⅱ) $\{m_k\}$ , $\{i_k\}$ and $\|\Gamma(m_k)\|$ are bounded.

(ⅲ) The search direction is bounded, i.e.,

$\begin{equation} \|t_k\| \leq c_1, \; c_1 > 0. \end{equation}$

(2.14)

(ⅳ) The search direction satisfies

$\begin{equation} \langle \Gamma(m_k), t_k\rangle \leq -c_2\|\Gamma(m_k)\|^2, \; c_2 > 0. \end{equation}$

(2.15)

(ⅴ)

$\begin{equation} \lim \limits_{k\to \infty}\Lambda_k\|t_k\| = 0. \end{equation}$

(2.16)

Proof. Let $\widehat{m}$ be a solution of problem (1.5), and by the assumption ( $A_{1}$ ), we have

$\begin{align} \langle \Gamma(p_k), \; m_k-\widehat{m} \rangle& = \langle \Gamma(p_k), \; m_k-p_k+p_k-\widehat{m} \rangle \\ & = \langle \Gamma(p_k), \; m_k-p_k\rangle + \langle \Gamma(p_k)-\Gamma(\widehat{m}), \; p_k-\widehat{m} \rangle\\ &\geq\langle \Gamma(p_k), \; m_k-p_k\rangle. \end{align}$

(2.17)

Now, since $0 < \eta < 2,$ from (2.2), (2.5) and (2.17) we have

$\begin{array}{l} \|m_{k+1}-\widehat{m}\|^2 = \left\|P_C \left[m_k-\eta \frac{\langle \Gamma(p_k), \; m_k-p_k \rangle}{\|\Gamma(p_k)\|^2}\Gamma(p_k)\right]-\widehat{m}\right\|^2\\ \leq \left\|m_k-\widehat{m}-\eta\frac{\langle \Gamma(p_k), \; m_k-p_k \rangle}{\|\Gamma(p_k)\|^2}\Gamma(p_k)\right\|^2\\ = \|m_k-\widehat{m}\|^2-2\eta\frac{\langle \Gamma(p_k), \; m_k-p_k \rangle}{\|\Gamma(p_k)\|^2}\langle \Gamma(p_k), \; m_k-\widehat{m}\rangle \\+\eta^2\frac{\langle \Gamma(p_k), \; m_k-p_k \rangle^2}{\|\Gamma(p_k)\|^2}\\ \leq \|m_k-\widehat{m}\|^2-2\eta\frac{\langle \Gamma(p_k), \; m_k-p_k \rangle}{\|\Gamma(p_k)\|^2}\langle \Gamma(p_k), \; m_k-p_k\rangle \\+\eta^2\frac{\langle \Gamma(p_k), \; m_k-p_k \rangle^2}{\|\Gamma(p_k)\|^2}\\ = \|m_k-\widehat{m}\|^2-\eta(2-\eta)\frac{\langle \Gamma(p_k), \; m_k-p_k \rangle^2}{\|\Gamma(p_k)\|^2}\\ \leq \|m_k-\widehat{m}\|^2. \end{array}$

(2.18)

This means that $\|m_k-\widehat{m}\|\leq \|m_{k-1}-\widehat{m}\|\leq \cdots\leq \|m_0-\widehat{m}\|,$ and thus $\lim\limits_{k\to\infty}\|m_k-\widehat{m}\|$ exists.

(ⅱ) Since $\lim\limits_{k\to\infty}\|m_k-\widehat{m}\|$ exists, $\{m_k\}$ is bounded. Combined with the fact that $0 < \alpha_k < 1, \; \forall k,$ this gives the boundedness of $\{i_k\}.$

In addition, since the mapping $\Gamma$ is Lipschitz continuous and $\{m_k\}$ is bounded, we can find a positive constant $c_3 > 0$ such that

$\begin{equation} \|\Gamma(m_k)\|\leq c_3. \end{equation}$

(2.19)

(ⅲ) To prove that the search direction defined by (2.6) is bounded, we need to show that the parameter $\gamma_k$ (2.7) is bounded.

From assumption ( $A_{1}$ ), the mapping $\Gamma$ is monotone. This gives us $\langle \Gamma(i_{k+1}) - \Gamma(i_k), \; i_{k+1}-i_k \rangle \geq 0,$ and thus

$\begin{align} \langle \Gamma(i_{k+1}) - \Gamma(i_k)+ r(i_{k+1}-i_k), \; i_{k+1}-i_k \rangle& = \langle \Gamma(i_{k+1}) - \Gamma(i_k), \; i_{k+1}-i_k \rangle +r\|i_{k+1}-i_k\|^2\\ &\geq r\|i_{k+1}-i_k\|^2. \end{align}$

(2.20)

Moreover, using the assumption that the mapping $\Gamma$ is Lipschitz continuous, together with the Cauchy Schwarz inequality, we have

(2.21)

Thus, (2.20) and (2.21) imply

$\begin{equation} r\|i_{k+1}-i_k\|^2 \leq \langle \Gamma(i_{k+1}) - \Gamma(i_k)+ r(i_{k+1}-i_k), \; i_{k+1}-i_k \rangle \leq (L+r)\|i_{k+1}-i_k\|^2, \end{equation}$

(2.22)

and therefore

$\begin{equation} \frac{1}{L+r} \leq \overline{\beta}_k \leq \frac{1}{r}. \end{equation}$

(2.23)

On the other hand,

$\begin{align} & \| \Gamma(i_{k+1}) - \Gamma(i_k)+ r(i_{k+1}-i_k)\|^2\\ & = \langle \Gamma(i_{k+1}) - \Gamma(i_k)+ r(i_{k+1}-i_k), \Gamma(i_{k+1}) - \Gamma(i_k) + r(i_{k+1}-i_k)\rangle\\ & = \|\Gamma(i_{k+1}) - \Gamma(i_k)\|^2 +2r\langle \Gamma(i_{k+1}) - \Gamma(i_k), i_{k+1}-i_k\rangle + r^2\|i_{k+1}-i_k\|^2\\ & \geq \|\Gamma(i_{k+1}) - \Gamma(i_k)\|^2 + r^2\|i_{k+1}-i_k\|^2\\ & \geq r^2\|i_{k+1}-i_k\|^2. \end{align}$

(2.24)

Using Lipschitz continuity, we have

$\begin{equation} \| \Gamma(i_{k+1}) - \Gamma(i_k)+ r(i_{k+1}-i_k)\| \leq(L+r)\|i_{k+1}-i_k\|. \end{equation}$

(2.25)

Combining (2.24) and (2.25) gives

$\begin{equation} r^2\|i_{k+1}-i_k\|^2 \leq \| \Gamma(i_{k+1}) - \Gamma(i_k)+ r(i_{k+1}-i_k)\|^2 \leq(L+r)^2\|i_{k+1}-i_k\|^2. \end{equation}$

(2.26)

Using (2.22) and (2.26), we have

$\begin{equation} \frac{r}{(L+r)^2} \leq \widehat{\beta}_k\leq \frac{(L+r)}{r^2}. \end{equation}$

(2.27)

Therefore, setting $M = \frac{1}{r} + \frac{L+r}{r^2}$ yields

$\begin{equation} \overline{\beta}_k + \widehat{\beta}_k \leq M. \end{equation}$

(2.28)

Since $\forall k,$ $\theta_k\leq 1$ (see Remark 1),

$\begin{equation} \gamma_k = (1-\theta_k) \overline{\beta}_k + \theta_k\widehat{\beta}_k\leq \overline{\beta}_k + \widehat{\beta}_k\leq M. \end{equation}$

(2.29)

Combining this with (2.19) gives (2.14) with $c_1 = Mc_3.$

(ⅳ) By the definition of $\theta_k,$ we have three possibilities, and thus the parameter $\gamma_k$ may take any of the three different following forms: $\gamma_k = \overline{\beta}_k,$ $\gamma_k = \widehat{\beta}_k$ and $\gamma_k = (1-\theta_k) \overline{\beta}_k +\theta_k\widehat{\beta}_k$ for $\theta_k = 0,$ $\theta_k = 1$ and $\theta_k\in(0, 1),$ respectively. Therefore, we divide this proof into three cases:

Case Ⅰ: If $\theta = 0, \; \forall k,$ then the search direction (2.6) reduces to $t_k = -\overline{\beta}_k\Gamma_k(m_k),$ and therefore, using (2.23) gives

$\begin{equation} \langle\Gamma(m_k), t_k\rangle = -\overline{\beta}_k\|\Gamma(m_k)\|^2\leq -\frac{1}{L+r}\|\Gamma(m_k)\|^2. \end{equation}$

(2.30)

Case Ⅱ: If $\theta = 1, \; \forall k,$ then the search direction (2.6) becomes $t_k = -\widehat{\beta}_k\Gamma_k(m_k),$ and thus, using (2.27) yields

$\begin{equation} \langle\Gamma(m_k), t_k\rangle = -\widehat{\beta}_k\|\Gamma(m_k)\|^2\leq -\frac{r}{(L+r)^2}\|\Gamma(m_k)\|^2. \end{equation}$

(2.31)

Case Ⅲ: If $0 < \theta < 1, \; \forall k,$ then we can find some constant $c_4 > 0$ such that $\theta_k > c_4$ and $(1-\theta_k) > 0.$ Therefore, from (2.7) and (2.27), we have

$\begin{equation} \gamma_k \geq \theta_k\widehat{\beta}_k \geq c_4\frac{r}{(L+r)^2}: = c_5. \end{equation}$

(2.32)

Thus, from the search direction (2.6) and (2.32), it holds that

$\begin{equation} \langle\Gamma(m_k), t_k\rangle = -\gamma_k\|\Gamma(m_k)\|^2\leq -c_5\|\Gamma(m_k)\|^2. \end{equation}$

(2.33)

Hence, from the three cases above, we see that (2.15) holds.

(ⅴ) Using the boundedness of $\{m_k\}$ , (2.14) and the definition of $p_k$ in (2.3), $\{p_k\}$ is bounded. Also, using the assumption that $\Gamma$ is Lipschitz continuous, we get

$\begin{equation} \|\Gamma(p_k)\|\leq n_1, \; n_1 > 0. \end{equation}$

(2.34)

Since $\min\left\{1, \|\Gamma(m_k + \Lambda_k t_k)\|^{\frac{1}{c}} \right\} \leq 1,$ squaring from both sides of (2.4) yields

$\begin{equation} \sigma^2\Lambda_k^4\|t_k\|^4\leq \langle \Gamma(p_k), \; \Lambda_kt_k\rangle^2. \end{equation}$

(2.35)

Furthermore, since $0 < \eta < 2,$ from (2.18) we obtain

$\begin{equation} \langle \Gamma(p_k), \; m_k-p_k \rangle^2 \leq \frac{\|\Gamma(p_k)\|^2(\|m_k-\widehat{m}\|^2-\|m_{k+1}-\widehat{m}\|^2)}{\eta(2-\eta)}. \end{equation}$

(2.36)

This together with (2.35) gives

$\begin{equation} \sigma^2\Lambda_k^4\|t_k\|^4 \leq \frac{\|\Gamma(p_k)\|^2(\|m_k-\widehat{m}\|^2-\|m_{k+1}-\widehat{m}\|^2)}{\eta(2-\eta)}. \end{equation}$

(2.37)

Recall that $\|\Gamma(p_k)\|$ is bounded by $n_1$ (see (2.34)), and we obtain

$\begin{equation} \sigma^2\Lambda_k^4\|t_k\|^4\leq \frac{\|\Gamma(p_k)\|^2(\|m_k-\widehat{m}\|^2-\|m_{k+1}-\widehat{m}\|^2)}{\eta(2-\eta)}\leq \frac{n_1^2(\|m_k-\widehat{m}\|^2-\|m_{k+1}-\widehat{m}\|^2)}{\eta(2-\eta)} . \end{equation}$

(2.38)

Since $\lim\limits_{k\to \infty}\|m_{k}-\widehat{m}\|$ exists, taking the limit as $k \to \infty$ on both sides of (2.38) gives

$\begin{equation*} \sigma^2\lim\limits _{k\to \infty}\Lambda_k^4\|t_k\|^4 = 0, \end{equation*}$

which implies (2.16).

Lemma 2.6. Suppose that the Assumption ( $A_{2}$ ) holds. Let the sequences $\{m_k\}$ and $\{p_k\}$ be generated by Algorithm 1. Then,

$\begin{equation} \Lambda_k\geq\max\left\{\kappa, \frac{c_2\varsigma\|\Gamma(m_k)\|^2}{(L+\sigma)\|t_k\|^2}\right\}. \end{equation}$

(2.39)

Proof. From (2.4), if $\Lambda_k\neq \kappa$ , then $\tilde{\Lambda}_{k} = \Lambda_k\varsigma^{-1}$ violates (2.4), that is,

$- \langle \Gamma(m_k+\tilde{\Lambda}_{k} t_k), t_k\rangle < \sigma\|t_k\|^2\tilde{\Lambda}_{k}\min\{ 1, \|\Gamma(m_k+\varsigma^jt_k)\|^\frac{1}{c}\}.$

We know that $\min \{ 1, \|\Gamma(m_k+\varsigma^jt_k)\|^\frac{1}{c}\} \leq 1.$ Thus, from (2.15) and Assumption ( $A_{2}$ ), we get

$\begin{equation} \nonumber \begin{aligned} c_2\|\Gamma(m_k)\|^2 & \leq -\Gamma(m_{k})^T t_k\\ & = (\Gamma(m_k+\tilde{\Lambda}_{k} t_k)-\Gamma(m_k))^T t_k-\langle \Gamma(m_k+\tilde{\Lambda}_{k} t_k), t_k\rangle\\ & \leq \|\Gamma(m_k+\tilde{\Lambda}_{k}t_k) - \Gamma(m_k) \|\|t_k\| -\langle \Gamma(m_k+\tilde{\Lambda}_{k} t_k), t_k\rangle\\ & \leq L\| m_k+\tilde{\Lambda}_{k}t_k - m_k\|\|t_k\| + \sigma \tilde{\Lambda}_{k}\|t_k\|^2\min\{ 1, \|\Gamma(m_k+\varsigma^jt_k)\|^\frac{1}{c}\}\\ & \leq L\| m_k+\tilde{\Lambda}_{k}t_k - m_k\|\|t_k\| + \sigma \tilde{\Lambda}_{k}\|t_k\|^2\\ &\leq \tilde{\Lambda}_{k}L\|t_k\|^2+\sigma\tilde{\Lambda}_{k}\|t_k\|^2\\ &\leq \tilde{\Lambda}_{k}(L + \sigma)\|t_k\|^2. \end{aligned} \end{equation}$

Therefore,

$\begin{equation} \tilde{\Lambda}_{k} \geq \frac{c_2\|\Gamma(m_k)\|^2}{(L + \sigma)\|t_k\|^2}. \end{equation}$

(2.40)

Substituting $\tilde{\Lambda}_{k} = \Lambda_k \varsigma^{-1}$ in (2.40) and solving for $\Lambda_k,$ we get

$\begin{equation} \Lambda_k\geq\frac{c_2 \varsigma\|\Gamma(m_k)\|^2}{(L + \sigma)\|t_k\|^2}. \end{equation}$

(2.41)

Thus, we have

$\begin{equation*} \Lambda_k\geq\max\left\{\kappa, \frac{c_2\varsigma\|\Gamma(m_k)\|^2}{(L+\sigma)\|t_k\|^2}\right\}. \end{equation*}$

Theorem 2.7. If the Assumptions $(A_{1}-A_{3})$ hold, and the sequence $\{m_k\}$ is produced by Algorithm 1, then

$\begin{equation} \liminf\limits_{k\to \infty}\|\Gamma(m_k)\| = 0. \end{equation}$

(2.42)

Proof. We show the proof by contradiction. Suppose (2.42) does not hold, and then there exists $s > 0$ such that $\forall k\geq0,$

$\begin{equation} \|\Gamma(m_k)\|\geq s. \end{equation}$

(2.43)

From Eqs (2.15) and (2.43), we get $\forall k\geq0,$

$\begin{equation} \|t_k\|\geq sc_2. \end{equation}$

(2.44)

Multiplying $\|t_k\|$ on both sides of (2.39), and using (2.14) and (2.43), we obtain

$\begin{equation} \begin{aligned} \Lambda_k\|t_k\| &\geq\max\left\{\kappa, \frac{c_2\varsigma\|\Gamma(m_k)\|^2}{(L+\sigma)\|t_k\|^2}\right\} \|t_k\|\\ &\geq\max\left\{\kappa, \frac{c_2\varsigma c_3^2}{(L+\sigma)c_2^2}\right\} c_2s\\ & = \max\left\{\kappa c_2s, \frac{\varsigma c_3^2 s}{(L+\sigma)}\right\}. \end{aligned} \end{equation}$

(2.45)

Taking the limit as $k\to \infty$ on both sides gives

$\begin{equation} \lim\limits _{k\to \infty}\Lambda_k\|t_k\| > 0, \end{equation}$

(2.46)

which contradicts (2.16). Hence, $\liminf\limits_{k\to \infty}\|\Gamma(m_k)\| = 0.$

3. Numerical experiments

In this section, we present the numerical experiments performed by solving a set of test problems taken from the literature. To depict the efficiency of the iSDFM algorithm, we perform numerical comparison with two other existing methods. The first one is the DAIS1 algorithm proposed in ^[22], which is an inertial-based algorithm for solving a system of nonlinear equations. The second algorithm is the MSGPALG proposed by Yu et al. in ^[21] based on the convex combination of the modified BB long and short parameters. As noted in Remark 2.4, both DAIS1 and MSGPALG can be viewed as special cases of the proposed iSDFM algorithm. We test the performances of all these three algorithms on seven test problems with eight different initial points (see ) and five dimensions (1000, 5000, 10000, 50000, and 100000), thus making the total number of the test problems 280. These three algorithms are coded on MATLAB R2019b which runs on a PC of corei3-4005U processor with 4 GB RAM and 1.70 GHz CPU. The choice of parameters for MSGPALG and DAIS1 are maintained as reported in their respective references [,]. In the iSDFM algorithm, we choose $\varsigma = 0.47,$ $\alpha_k = 1/(k+1)^2,$ $\eta = 1.79,$ $\mu = 0.5,$ $\sigma = 0.01,$ $r = 0.001,$ $c = 2$ and $\kappa = 1$ . The stopping criterion is set to be $\|\Gamma(m_k)\| < 10^{-6}.$

Table 1. Initial guesses used for the problems.

Initial guess	Value
$m_1$	$(1, 1, 1, \ldots, 1)^{T}$
$m_2$	$(0.1, 0.1, 0.1, \cdots, 0.1)^{T}$
$m_3$	$(\frac{1}{2}, \frac{1}{2^2}, \frac{1}{2^3}, \ldots, \frac{1}{2^n})^{T}$
$m_4$	$(1-\frac{1}{n}, 1-\frac{2}{n}, 1-\frac{3}{n}, \ldots, 0)^T$
$m_5$	$(0, \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n-1}{n})^{T}$
$m_6$	$(1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n})^{T}$
$m_7$	$(\frac{n-1}{n}, \frac{n-2}{n}, \frac{n-3}{n}, \ldots, 0)^{T}$
$m_8$	$(\frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \ldots, 0)^{T}$

| Show Table

DownLoad: CSV

We consider the following test problems, where $\Gamma (m) = (g_1(m), g_2(m), \ldots, g_n(m))^T$ :

Problem 1. ^[29]:

$\begin{equation*} \begin{aligned} g_1(m)& = e^{m_1}-1, \; \; g_j(m)& = e^ {m_j}+{m_j}-1, \; {\rm{for }}\;j = 2, ..., n, \;{\rm{ and }}\; & C = \mathbb{R}_{+}^n. \end{aligned} \end{equation*}$

Problem 2. ^[28] Modified logarithmic function:

$\begin{equation*} \begin{aligned} g_j(m)& = \ln(m_j+1)-\frac{m_j}{n}, \; {\rm{for}} \; \; {j = 1, 2, 3, ..., n, }\\ {\rm{and }}\; & C = \{m\in \mathbb{R}^n: \sum _{j = 1}^{n}m_j\leq n, \; m_j > -1, \; j = 1, 2, \ldots , n\}. \end{aligned} \end{equation*}$

Problem 3. ^[30] Nonsmooth function:

$\begin{equation*} \begin{aligned} g_j(m)& = 2m_j-\sin |m_j|, \; j = 1, 2, 3, ..., n, \\ {\rm{and }}\; C & = \{m\in \mathbb{R}^n: \sum _{j = 1}^{n}m_j\le n, \; m_j \ge 0, \; j = 1, 2, \ldots , n\}. \end{aligned} \end{equation*}$

Problem 4. ^[31] Strictly convex function:

$\begin{equation*} \begin{aligned} g_j(m)& = e^{m_j}-1, \; {\rm{for }}\;j = 1, 2, ..., n, \; \; {\rm{and }}\; & C = \mathbb{R}_{+}^n. \end{aligned} \end{equation*}$

Problem 5. ^[20] Nonsmooth function:

$\begin{equation*} \begin{aligned} g_j(m)& = m_j-\sin |m_j-1|, \; j = 1, 2, 3, ..., n, \\ {\rm{and }} C & = \{m\in \mathbb{R}^n: \sum _{j = 1}^{n}m_i\le n, \; m_j \ge -1, \; j = 1, 2, \ldots , n\}. \end{aligned} \end{equation*}$

Problem 6.^[29]:

$\begin{equation*} \begin{aligned} g_j(m)& = e^{{m_j}^2}+1.5\sin(2m_j)-1, \; {\rm{for }}j = 1, 2, ..., n, \; \; {\rm{and }} & C = \mathbb{R}_{+}^n. \end{aligned} \end{equation*}$

Problem 7. ^[32]:

$\begin{equation*} \begin{aligned} g_1(m)& = \frac{5}{2}m_1+m_2-1, \\ g_j(m)& = m_{j-1}+\frac{5}{2}m_j+m_{j+1}-1, \; {\rm{for }}j = 2, 3, ..., n-1, \\ g_n(m)& = m_{n-1}+\frac{5}{2}m_n-1 \;{\rm{ and }}\; C = \mathbb{R}_{+}^n. \end{aligned} \end{equation*}$

Based on these settings, the results of the experiments are tabulated in Tables 2–8 with ITER, FVAL and TIME denoting the number of iterations, number of function evaluations and CPU time, respectively. Based on these metrics, it can be observed that the performances of the three algorithms varies in terms of the ITER, FVAL and TIME. However, taking the whole results of the experiment into consideration, it can be seen that the proposed iSDFM algorithm outperformed the DAIS1 and the MSGPALG algorithms in most instances. By outperforming we mean the new iSDFM recorded the least ITER, FVAL and TIME in most cases of the experiment.

Table 2. Numerical results of the three algorithms on Problem 1.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	1	2	0.0747	0.00E+00	5	7	0.1166	9.24E-08	8	9	0.1666	2.76E-07
	$m_2$	1	2	0.0054	0.00E+00	5	7	0.0519	5.76E-08	7	8	0.0395	1.06E-07
	$m_3$	2	3	0.0186	9.93E-16	10	11	0.0344	8.05E-08	10	11	0.0240	2.07E-07
	$m_4$	2	3	0.0052	0.00E+00	6	8	0.0253	2.30E-07	8	9	0.0112	2.52E-07
	$m_5$	2	3	0.0078	0.00E+00	6	8	0.0385	1.10E-07	8	9	0.0213	2.46E-07
	$m_6$	3	4	0.0064	0.00E+00	9	11	0.0278	7.28E-08	9	10	0.0057	1.98E-07
	$m_7$	2	3	0.0064	0.00E+00	6	8	0.0118	2.30E-07	8	9	0.0094	2.52E-07
	$m_8$	2	3	0.0041	0.00E+00	6	8	0.0101	1.11E-07	8	9	0.0100	2.47E-07
5000	$m_1$	1	2	0.1996	0.00E+00	5	7	0.2617	2.37E-07	8	9	0.0676	6.05E-07
	$m_2$	1	2	0.0113	0.00E+00	4	6	0.0242	8.65E-08	7	8	0.0182	1.65E-07
	$m_3$	2	3	0.0138	9.93E-16	10	11	0.0423	8.05E-08	10	11	0.0224	2.07E-07
	$m_4$	2	3	0.0145	0.00E+00	6	8	1.1522	2.91E-07	8	9	0.1283	5.54E-07
	$m_5$	2	3	0.0556	0.00E+00	6	8	0.0308	2.55E-07	8	9	0.0286	5.52E-07
	$m_6$	3	4	0.1059	0.00E+00	9	11	0.0446	7.33E-08	9	10	0.0253	1.95E-07
	$m_7$	2	3	0.2689	0.00E+00	6	8	0.0281	2.91E-07	8	9	0.0196	5.54E-07
	$m_8$	2	3	0.0683	0.00E+00	6	8	0.0291	2.55E-07	8	9	1.4239	5.52E-07
10000	$m_1$	1	2	0.0335	0.00E+00	5	7	0.9254	3.37E-07	8	9	0.0582	8.53E-07
	$m_2$	1	2	0.0139	0.00E+00	4	6	0.0311	1.06E-07	7	8	0.3954	2.21E-07
	$m_3$	2	3	1.7858	9.93E-16	10	11	0.1457	8.05E-08	10	11	0.1475	2.07E-07
	$m_4$	2	3	0.0597	0.00E+00	6	8	0.0676	3.85E-07	8	9	0.0401	7.82E-07
	$m_5$	2	3	0.0287	0.00E+00	6	8	0.0427	3.60E-07	8	9	0.0347	7.81E-07
	$m_6$	3	4	0.0428	0.00E+00	9	11	0.4798	7.33E-08	9	10	0.0475	1.94E-07
	$m_7$	2	3	0.0353	0.00E+00	6	8	0.0518	3.85E-07	8	9	0.1343	7.82E-07
	$m_8$	2	3	0.5467	0.00E+00	6	8	0.1948	3.61E-07	8	9	2.2142	7.81E-07
50000	$m_1$	1	2	0.1128	0.00E+00	5	7	0.2563	7.55E-07	9	10	0.2087	9.07E-08
	$m_2$	1	2	0.5707	0.00E+00	4	6	2.7546	1.93E-07	7	8	0.1446	4.74E-07
	$m_3$	2	3	0.1359	9.93E-16	10	11	1.7252	8.05E-08	10	11	0.1304	2.07E-07
	$m_4$	2	3	0.1764	0.00E+00	6	8	0.1702	8.17E-07	9	10	0.1625	8.32E-08
	$m_5$	2	3	0.0968	0.00E+00	6	8	0.5717	8.07E-07	9	10	1.3341	8.31E-08
	$m_6$	3	4	1.1641	0.00E+00	9	11	0.4882	7.34E-08	9	10	0.1229	1.94E-07
	$m_7$	2	3	0.1440	0.00E+00	6	8	0.2247	8.17E-07	9	10	0.1264	8.32E-08
	$m_8$	2	3	0.0931	0.00E+00	6	8	1.1037	8.07E-07	9	10	0.1459	8.31E-08
100000	$m_1$	1	2	0.1883	0.00E+00	6	8	0.4077	5.32E-09	9	10	0.9471	1.28E-07
	$m_2$	1	2	0.6308	0.00E+00	4	6	0.2373	2.65E-07	7	8	0.2009	6.66E-07
	$m_3$	2	3	0.2199	9.93E-16	10	11	4.2777	8.05E-08	10	11	0.2548	2.07E-07
	$m_4$	2	3	0.1781	0.00E+00	7	9	0.4334	5.71E-09	9	10	0.2946	1.18E-07
	$m_5$	2	3	0.8688	0.00E+00	7	9	1.8487	5.68E-09	9	10	0.2558	1.18E-07
	$m_6$	3	4	0.1510	0.00E+00	9	11	0.6243	7.34E-08	9	10	2.5348	1.94E-07
	$m_7$	2	3	0.3463	0.00E+00	7	9	0.3045	5.71E-09	9	10	0.2682	1.18E-07
	$m_8$	2	3	0.3872	0.00E+00	7	9	3.9198	5.68E-09	9	10	0.3627	1.18E-07

| Show Table

DownLoad: CSV

Table 3. Numerical results of the three algorithms on Problem 2.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	4	6	0.0300	2.13E-07	8	10	0.0332	6.00E-08	9	10	0.0283	3.85E-07
	$m_2$	19	21	0.0497	9.06E-07	4	6	0.0119	1.25E-07	6	7	0.0145	5.14E-07
	$m_3$	7	9	0.0239	4.09E-07	7	9	0.0146	2.78E-08	7	8	0.0096	5.83E-07
	$m_4$	7	9	0.0172	9.09E-07	8	10	0.0195	2.49E-07	9	10	0.0096	4.50E-07
	$m_5$	7	9	0.0333	9.09E-07	8	10	0.0177	2.49E-07	9	10	0.0115	4.50E-07
	$m_6$	8	10	0.0192	4.95E-08	8	10	0.0181	1.95E-08	7	8	0.0120	7.79E-07
	$m_7$	7	9	0.0145	9.09E-07	8	10	0.0181	2.49E-07	9	10	0.0185	4.50E-07
	$m_8$	7	9	0.0165	8.88E-07	8	10	0.0203	2.50E-07	9	10	0.0107	4.54E-07
5000	$m_1$	4	6	0.4667	9.03E-07	8	10	1.2040	1.34E-07	9	10	0.7974	8.59E-07
	$m_2$	23	25	3.0068	7.92E-07	4	6	0.8070	2.75E-07	7	8	0.4587	1.03E-07
	$m_3$	7	9	0.1027	4.14E-07	7	9	0.0807	2.75E-08	7	8	0.1973	5.62E-07
	$m_4$	10	12	0.0980	8.56E-07	8	10	2.7056	5.64E-07	10	11	0.1100	9.17E-08
	$m_5$	10	12	0.0644	8.56E-07	8	10	0.4899	5.64E-07	10	11	0.1570	9.17E-08
	$m_6$	8	10	0.0563	1.62E-07	8	10	0.0748	2.00E-08	7	8	0.0203	7.77E-07
	$m_7$	10	12	1.7635	8.56E-07	8	10	0.5620	5.64E-07	10	11	1.6032	9.17E-08
	$m_8$	10	12	2.0989	8.51E-07	8	10	0.3175	5.64E-07	10	11	0.0398	9.18E-08
10000	$m_1$	6	8	0.0674	8.99E-07	8	10	3.0765	1.90E-07	10	11	0.0635	1.10E-07
	$m_2$	24	26	4.3594	8.95E-07	4	6	0.8040	3.88E-07	7	8	0.4238	1.45E-07
	$m_3$	7	9	0.0568	4.14E-07	7	9	0.1828	2.75E-08	7	8	0.1624	5.59E-07
	$m_4$	11	13	0.1394	9.00E-07	8	10	0.2587	7.98E-07	10	11	1.5964	1.30E-07
	$m_5$	11	13	1.8668	9.00E-07	8	10	0.1017	7.98E-07	10	11	0.7227	1.30E-07
	$m_6$	8	10	0.0638	1.75E-07	8	10	0.1647	2.02E-08	7	8	0.0366	7.77E-07
	$m_7$	11	13	0.1010	9.00E-07	8	10	2.4034	7.98E-07	10	11	0.1130	1.30E-07
	$m_8$	11	13	0.7089	8.97E-07	8	10	0.2677	7.99E-07	10	11	0.0738	1.30E-07
50000	$m_1$	10	12	0.3468	8.09E-07	8	10	0.6028	4.24E-07	10	11	4.0162	2.47E-07
	$m_2$	27	29	2.2526	9.72E-07	4	6	2.6259	8.66E-07	7	8	0.3940	3.23E-07
	$m_3$	7	9	0.2112	4.15E-07	7	9	0.5050	2.75E-08	7	8	0.6980	5.57E-07
	$m_4$	14	16	1.2757	9.82E-07	9	11	1.2621	1.77E-08	10	11	0.2290	2.90E-07
	$m_5$	14	16	0.4675	9.82E-07	9	11	4.5622	1.77E-08	10	11	0.5405	2.90E-07
	$m_6$	8	10	1.9899	1.84E-07	8	10	1.0180	2.04E-08	7	8	0.3657	7.77E-07
	$m_7$	14	16	0.4672	9.82E-07	9	11	1.4871	1.77E-08	10	11	2.3145	2.90E-07
	$m_8$	14	16	1.4642	9.82E-07	9	11	0.8096	1.77E-08	10	11	1.8791	2.90E-07
100000	$m_1$	11	13	4.5167	9.03E-07	8	10	1.1287	5.99E-07	10	11	1.8709	3.49E-07
	$m_2$	29	31	2.7946	8.55E-07	5	7	0.3214	1.21E-08	7	8	0.3972	4.56E-07
	$m_3$	7	9	1.2771	4.15E-07	7	9	1.3914	2.75E-08	7	8	0.5613	5.57E-07
	$m_4$	16	18	2.8902	8.59E-07	9	11	0.8460	2.50E-08	10	11	1.2346	4.10E-07
	$m_5$	16	18	2.5125	8.59E-07	9	11	2.1172	2.50E-08	10	11	0.5491	4.10E-07
	$m_6$	8	10	1.8713	1.85E-07	8	10	0.5126	2.04E-08	7	8	0.4579	7.77E-07
	$m_7$	16	18	2.3005	8.59E-07	9	11	1.5732	2.50E-08	10	11	1.2220	4.10E-07
	$m_8$	16	18	2.3307	8.58E-07	9	11	1.8067	2.50E-08	10	11	0.5572	4.10E-07

| Show Table

DownLoad: CSV

Table 4. Numerical results of the three algorithms on Problem 3.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	2	3	0.0489	0.00E+00	7	9	0.0312	2.58E-08	9	10	0.0271	2.19E-07
	$m_2$	2	3	0.0036	0.00E+00	5	7	0.0089	8.35E-07	7	8	0.0061	9.15E-07
	$m_3$	2	3	0.0068	0.00E+00	6	8	0.0109	1.27E-08	7	8	0.0068	2.49E-07
	$m_4$	2	3	0.0039	0.00E+00	6	8	0.0106	7.79E-07	9	10	0.0130	1.09E-07
	$m_5$	2	3	0.0046	0.00E+00	6	8	0.0139	7.79E-07	9	10	0.0106	1.09E-07
	$m_6$	2	3	0.0046	0.00E+00	6	8	0.0091	5.53E-08	8	9	0.0098	1.00E-07
	$m_7$	2	3	0.0047	0.00E+00	6	8	0.0115	7.79E-07	9	10	0.0087	1.09E-07
	$m_8$	2	3	0.0041	0.00E+00	6	8	0.0122	7.78E-07	9	10	0.0075	1.10E-07
5000	$m_1$	2	3	0.6745	0.00E+00	7	9	0.0553	5.78E-08	9	10	0.5417	4.89E-07
	$m_2$	2	3	0.0132	0.00E+00	6	8	0.0326	1.85E-08	8	9	0.0784	1.86E-07
	$m_3$	2	3	0.0482	0.00E+00	6	8	0.0719	1.27E-08	7	8	1.2708	2.49E-07
	$m_4$	2	3	0.0132	0.00E+00	8	10	0.9730	5.57E-07	9	10	1.0767	2.45E-07
	$m_5$	2	3	0.0159	0.00E+00	8	10	0.2707	5.57E-07	9	10	0.0275	2.45E-07
	$m_6$	2	3	0.4898	0.00E+00	6	8	0.0905	5.53E-08	8	9	0.2965	1.00E-07
	$m_7$	2	3	0.0259	0.00E+00	8	10	0.2239	5.57E-07	9	10	0.0474	2.45E-07
	$m_8$	2	3	0.0089	0.00E+00	8	10	0.0748	5.57E-07	9	10	0.0289	2.45E-07
10000	$m_1$	2	3	0.6468	0.00E+00	7	9	0.0893	8.17E-08	9	10	1.1107	6.91E-07
	$m_2$	2	3	0.0174	0.00E+00	6	8	2.0953	2.61E-08	8	9	0.0422	2.63E-07
	$m_3$	2	3	0.0241	0.00E+00	6	8	0.1933	1.27E-08	7	8	0.0554	2.49E-07
	$m_4$	2	3	0.0180	0.00E+00	8	10	0.5469	7.88E-07	9	10	0.7111	3.46E-07
	$m_5$	2	3	1.5830	0.00E+00	8	10	2.3199	7.88E-07	9	10	0.0512	3.46E-07
	$m_6$	2	3	0.0905	0.00E+00	6	8	0.1884	5.53E-08	8	9	1.9455	1.00E-07
	$m_7$	2	3	0.0225	0.00E+00	8	10	0.0539	7.88E-07	9	10	0.0370	3.46E-07
	$m_8$	2	3	0.4293	0.00E+00	8	10	0.5833	7.88E-07	9	10	0.2404	3.47E-07
50000	$m_1$	2	3	0.1251	0.00E+00	7	9	0.2378	1.83E-07	10	11	0.1597	1.41E-07
	$m_2$	2	3	0.2099	0.00E+00	6	8	0.3338	5.84E-08	8	9	0.3668	5.88E-07
	$m_3$	2	3	0.2718	0.00E+00	6	8	0.1467	1.27E-08	7	8	0.1518	2.49E-07
	$m_4$	2	3	0.0892	0.00E+00	9	11	1.8976	1.74E-08	9	10	3.4018	7.75E-07
	$m_5$	2	3	0.0738	0.00E+00	9	11	0.8494	1.74E-08	9	10	0.5303	7.75E-07
	$m_6$	2	3	0.4336	0.00E+00	6	8	0.2686	5.53E-08	8	9	0.3395	1.00E-07
	$m_7$	2	3	0.1016	0.00E+00	9	11	3.6617	1.74E-08	9	10	0.1285	7.75E-07
	$m_8$	2	3	0.4282	0.00E+00	9	11	0.2094	1.74E-08	9	10	0.3406	7.75E-07
100000	$m_1$	2	3	0.1608	0.00E+00	7	9	0.3985	2.58E-07	10	11	2.8327	1.99E-07
	$m_2$	2	3	0.4025	0.00E+00	6	8	2.4220	8.27E-08	8	9	0.5146	8.32E-07
	$m_3$	2	3	0.4997	0.00E+00	6	8	0.4001	1.27E-08	7	8	0.4550	2.49E-07
	$m_4$	2	3	0.2278	0.00E+00	9	11	1.2758	2.47E-08	10	11	0.6024	9.96E-08
	$m_5$	2	3	1.4800	0.00E+00	9	11	1.1169	2.47E-08	10	11	0.6103	9.96E-08
	$m_6$	2	3	0.1908	0.00E+00	6	8	1.1200	5.53E-08	8	9	0.3045	1.00E-07
	$m_7$	2	3	2.2168	0.00E+00	9	11	0.4203	2.47E-08	10	11	2.8192	9.96E-08
	$m_8$	2	3	0.5285	0.00E+00	9	11	1.6755	2.47E-08	10	11	0.5757	9.96E-08

| Show Table

DownLoad: CSV

Table 5. Numerical results of the three algorithms on Problem 4.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	2	3	0.0296	0.00E+00	1	2	0.0073	0.00E+00	1	2	0.0291	0
	$m_2$	1	2	0.0027	0	1	2	0.0035	0.00E+00	1	2	0.0031	0
	$m_3$	2	3	0.0042	4.97E-16	1	2	0.0034	0.00E+00	1	2	0.0026	0.00E+00
	$m_4$	3	4	0.0049	0.00E+00	5	7	0.0093	2.75E-07	7	8	0.0060	6.10E-07
	$m_5$	3	4	0.0055	0.00E+00	5	7	0.0105	2.75E-07	7	8	0.0078	6.10E-07
	$m_6$	2	3	0.0034	0.00E+00	4	6	0.0067	1.70E-08	5	6	0.0046	2.61E-07
	$m_7$	3	4	0.0049	0.00E+00	5	7	0.0072	2.75E-07	7	8	0.0064	6.10E-07
	$m_8$	3	4	0.0071	0.00E+00	5	7	0.0083	2.84E-07	7	8	0.0056	6.18E-07
5000	$m_1$	2	3	0.0332	0.00E+00	1	2	0.0646	0.00E+00	1	2	0.0367	0
	$m_2$	1	2	0.0099	0	1	2	0.0213	0.00E+00	1	2	0.0174	0
	$m_3$	2	3	1.0121	4.97E-16	1	2	0.5463	0.00E+00	1	2	0.3020	0.00E+00
	$m_4$	3	4	0.0686	0.00E+00	5	7	0.6076	6.23E-07	8	9	0.2798	1.25E-07
	$m_5$	3	4	0.0281	0.00E+00	5	7	0.0214	6.23E-07	8	9	0.3643	1.25E-07
	$m_6$	2	3	0.0487	0.00E+00	4	6	0.9525	1.57E-08	5	6	0.1231	2.41E-07
	$m_7$	3	4	0.0160	0.00E+00	5	7	0.1311	6.23E-07	8	9	0.0235	1.25E-07
	$m_8$	3	4	0.3845	0.00E+00	5	7	0.0238	6.27E-07	8	9	0.0280	1.25E-07
10000	$m_1$	2	3	0.0432	0.00E+00	1	2	0.0152	0.00E+00	1	2	0.1671	0
	$m_2$	1	2	0.0130	0	1	2	0.6404	0.00E+00	1	2	0.2411	0
	$m_3$	2	3	0.3455	4.97E-16	1	2	0.1039	0.00E+00	1	2	0.0879	0.00E+00
	$m_4$	3	4	0.0962	0.00E+00	5	7	0.1264	8.83E-07	8	9	0.1373	1.76E-07
	$m_5$	3	4	0.0206	0.00E+00	5	7	0.0717	8.83E-07	8	9	0.0563	1.76E-07
	$m_6$	2	3	0.0129	0.00E+00	4	6	0.0410	1.56E-08	5	6	0.0300	2.39E-07
	$m_7$	3	4	2.1280	0.00E+00	5	7	2.0752	8.83E-07	8	9	0.0345	1.76E-07
	$m_8$	3	4	0.0435	0.00E+00	5	7	0.3836	8.85E-07	8	9	0.2295	1.77E-07
50000	$m_1$	2	3	0.1215	0.00E+00	1	2	0.0786	0.00E+00	1	2	0.2465	0
	$m_2$	1	2	0.9649	0	1	2	0.0727	0.00E+00	1	2	0.1039	0
	$m_3$	2	3	1.1914	4.97E-16	1	2	0.1050	0.00E+00	1	2	0.0993	0.00E+00
	$m_4$	3	4	0.0787	0.00E+00	6	8	0.4597	1.96E-08	8	9	0.9459	3.95E-07
	$m_5$	3	4	0.5360	0.00E+00	6	8	1.9008	1.96E-08	8	9	0.1311	3.95E-07
	$m_6$	2	3	0.0827	0.00E+00	4	6	1.4476	1.54E-08	5	6	1.6395	2.37E-07
	$m_7$	3	4	0.5580	0.00E+00	6	8	0.8577	1.96E-08	8	9	1.1280	3.95E-07
	$m_8$	3	4	0.1724	0.00E+00	6	8	0.1708	1.96E-08	8	9	0.1320	3.95E-07
100000	$m_1$	2	3	0.1040	0.00E+00	1	2	0.2292	0.00E+00	1	2	0.0971	0
	$m_2$	1	2	0.5020	0	1	2	0.3929	0.00E+00	1	2	0.0608	0
	$m_3$	2	3	0.6183	4.97E-16	1	2	0.5716	0.00E+00	1	2	1.2024	0.00E+00
	$m_4$	3	4	0.6508	0.00E+00	6	8	0.8017	2.77E-08	8	9	1.1241	5.58E-07
	$m_5$	3	4	2.7106	0.00E+00	6	8	0.2833	2.77E-08	8	9	0.5422	5.58E-07
	$m_6$	2	3	0.2422	0.00E+00	4	6	0.4995	1.54E-08	5	6	0.1569	2.37E-07
	$m_7$	3	4	0.4045	0.00E+00	6	8	0.8315	2.77E-08	8	9	0.6378	5.58E-07
	$m_8$	3	4	1.2673	0.00E+00	6	8	0.2407	2.77E-08	8	9	2.2426	5.58E-07

| Show Table

DownLoad: CSV

Table 6. Numerical results of the three algorithms on Problem 5.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	24	26	0.0584	9.23E-07	4	6	0.0278	4.28E-07	6	7	0.0106	3.94E-07
	$m_2$	23	25	0.0523	9.30E-07	4	6	0.0085	2.03E-07	7	8	0.0083	3.89E-07
	$m_3$	12	14	0.0252	5.60E-07	6	8	0.0112	1.37E-08	7	8	0.0103	5.04E-07
	$m_4$	12	14	0.0248	8.98E-07	7	9	0.0125	4.92E-08	8	9	0.0077	1.11E-07
	$m_5$	12	14	0.0275	8.98E-07	7	9	0.0142	4.92E-08	8	9	0.0118	1.11E-07
	$m_6$	19	21	0.0250	8.22E-07	8	10	0.0347	8.90E-09	8	9	0.0077	2.35E-07
	$m_7$	12	14	0.0258	8.98E-07	7	9	0.0133	4.92E-08	8	9	0.0090	1.11E-07
	$m_8$	12	14	0.0186	8.96E-07	7	9	0.0120	5.06E-08	8	9	0.0098	1.12E-07
5000	$m_1$	28	30	0.1528	8.04E-07	4	6	0.0764	9.58E-07	6	7	0.1398	8.81E-07
	$m_2$	27	29	0.6379	8.10E-07	4	6	0.3117	4.53E-07	7	8	0.4172	8.70E-07
	$m_3$	15	17	0.0647	5.18E-07	6	8	0.2211	1.30E-08	8	9	0.0829	5.70E-08
	$m_4$	16	18	0.1284	7.87E-07	7	9	0.0740	1.11E-07	8	9	0.1978	2.50E-07
	$m_5$	16	18	1.3222	7.87E-07	7	9	0.2236	1.11E-07	8	9	0.5069	2.50E-07
	$m_6$	15	17	0.0576	6.47E-07	7	9	0.7438	5.65E-07	8	9	0.9363	8.56E-07
	$m_7$	16	18	0.5002	7.87E-07	7	9	0.5365	1.11E-07	8	9	0.7172	2.50E-07
	$m_8$	16	18	0.2607	7.86E-07	7	9	0.0450	1.12E-07	8	9	0.0370	2.51E-07
10000	$m_1$	29	31	2.2261	9.03E-07	5	7	0.5164	7.20E-09	7	8	0.0696	6.32E-08
	$m_2$	28	30	0.1845	9.10E-07	4	6	0.2108	6.40E-07	8	9	0.3887	6.24E-08
	$m_3$	15	17	0.1033	6.58E-07	6	8	0.0598	1.23E-08	8	9	1.9183	8.06E-08
	$m_4$	17	19	1.0910	8.89E-07	7	9	1.2817	1.58E-07	8	9	0.0453	3.54E-07
	$m_5$	17	19	0.1082	8.89E-07	7	9	0.0601	1.58E-07	8	9	0.2457	3.54E-07
	$m_6$	15	17	0.4385	7.90E-07	8	10	0.2200	1.32E-08	9	10	0.4761	5.62E-08
	$m_7$	17	19	0.5594	8.89E-07	7	9	0.1719	1.58E-07	8	9	0.0590	3.54E-07
	$m_8$	17	19	2.5253	8.89E-07	7	9	0.0791	1.58E-07	8	9	0.4212	3.54E-07
50000	$m_1$	32	34	2.0660	9.88E-07	5	7	0.2138	1.61E-08	7	8	0.2570	1.41E-07
	$m_2$	31	33	1.7560	9.95E-07	5	7	1.2728	7.61E-09	8	9	0.3223	1.39E-07
	$m_3$	18	20	1.1115	9.18E-07	6	8	0.6486	1.15E-08	8	9	4.0013	1.80E-07
	$m_4$	20	22	1.9429	9.68E-07	7	9	0.4158	3.53E-07	8	9	0.3523	7.92E-07
	$m_5$	20	22	0.5584	9.68E-07	7	9	0.9673	3.53E-07	8	9	0.2946	7.92E-07
	$m_6$	17	19	1.4478	9.79E-07	8	10	0.2156	4.69E-08	9	10	0.2300	5.16E-08
	$m_7$	20	22	4.0180	9.68E-07	7	9	1.8746	3.53E-07	8	9	0.6637	7.92E-07
	$m_8$	20	22	0.6509	9.68E-07	7	9	0.2024	3.54E-07	8	9	0.1919	7.92E-07
100000	$m_1$	34	36	3.6145	8.70E-07	5	7	0.9745	2.28E-08	7	8	2.1130	2.00E-07
	$m_2$	33	35	2.2467	8.77E-07	5	7	2.2109	1.08E-08	8	9	1.3419	1.97E-07
	$m_3$	16	18	1.2758	1.67E-07	6	8	0.4651	1.13E-08	8	9	0.2564	2.55E-07
	$m_4$	22	24	2.1591	8.54E-07	7	9	0.8911	5.00E-07	9	10	0.7061	5.68E-08
	$m_5$	22	24	2.1281	8.54E-07	7	9	0.3940	5.00E-07	9	10	0.7502	5.68E-08
	$m_6$	21	23	1.4187	9.59E-07	8	10	2.1780	5.35E-08	8	9	0.2659	9.73E-07
	$m_7$	22	24	2.3113	8.54E-07	7	9	0.8201	5.00E-07	9	10	2.4092	5.68E-08
	$m_8$	22	24	1.9841	8.54E-07	7	9	0.5536	5.00E-07	9	10	0.7111	5.68E-08

| Show Table

DownLoad: CSV

Table 7. Numerical results of the three algorithms on Problem 6.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	1	2	0.0438	0.00E+00	5	7	0.0154	3.13E-08	6	7	0.0362	5.08E-07
	$m_2$	2	3	0.0055	0.00E+00	4	6	0.0116	4.07E-09	6	7	0.0090	7.74E-08
	$m_3$	1	2	0.0059	3.14E-16	4	6	0.0074	1.95E-08	6	7	0.0126	1.35E-07
	$m_4$	2	3	0.0056	0.00E+00	10	12	0.0173	6.36E-08	9	10	0.0108	2.10E-08
	$m_5$	2	3	0.0070	0.00E+00	10	12	0.0204	6.36E-08	9	10	0.0153	2.10E-08
	$m_6$	2	3	0.0053	0.00E+00	6	8	0.0132	1.63E-08	8	9	0.0082	2.34E-07
	$m_7$	2	3	0.0056	0.00E+00	10	12	0.0232	6.36E-08	9	10	0.0091	2.10E-08
	$m_8$	2	3	0.0099	0.00E+00	10	12	0.0185	9.54E-08	9	10	0.0085	2.17E-08
5000	$m_1$	1	2	0.0374	0.00E+00	5	7	1.0327	7.00E-08	7	8	0.5082	2.23E-08
	$m_2$	2	3	0.0205	0.00E+00	4	6	0.0221	9.11E-09	6	7	1.3592	1.73E-07
	$m_3$	1	2	0.0747	3.14E-16	4	6	0.3235	1.95E-08	6	7	0.2540	1.35E-07
	$m_4$	2	3	0.5673	0.00E+00	11	13	0.3735	4.00E-09	9	10	0.0230	4.77E-08
	$m_5$	2	3	0.1877	0.00E+00	11	13	0.2003	4.00E-09	9	10	0.2913	4.77E-08
	$m_6$	2	3	0.0243	0.00E+00	6	8	0.5030	1.63E-08	8	9	0.0371	2.34E-07
	$m_7$	2	3	0.0166	0.00E+00	11	13	0.3037	4.00E-09	9	10	0.0259	4.77E-08
	$m_8$	2	3	0.1480	0.00E+00	11	13	0.7863	4.34E-09	9	10	1.1625	4.80E-08
10000	$m_1$	1	2	1.5394	0.00E+00	5	7	0.3123	9.90E-08	7	8	0.1095	3.15E-08
	$m_2$	2	3	0.3070	0.00E+00	4	6	0.7642	1.29E-08	6	7	0.0387	2.45E-07
	$m_3$	1	2	0.0315	3.14E-16	4	6	0.0600	1.95E-08	6	7	0.6297	1.35E-07
	$m_4$	2	3	0.2660	0.00E+00	11	13	0.4963	1.34E-08	9	10	2.2974	6.75E-08
	$m_5$	2	3	0.3858	0.00E+00	11	13	1.1199	1.34E-08	9	10	0.0373	6.75E-08
	$m_6$	2	3	0.1753	0.00E+00	6	8	0.2915	1.63E-08	8	9	2.9723	2.34E-07
	$m_7$	2	3	0.5146	0.00E+00	11	13	0.4325	1.34E-08	9	10	0.1870	6.75E-08
	$m_8$	2	3	0.2806	0.00E+00	11	13	0.1227	1.42E-08	9	10	0.0536	6.78E-08
50000	$m_1$	1	2	1.1916	0.00E+00	5	7	0.2469	2.21E-07	7	8	0.1870	7.04E-08
	$m_2$	2	3	0.9393	0.00E+00	4	6	1.1379	2.88E-08	6	7	1.6610	5.47E-07
	$m_3$	1	2	0.0926	3.14E-16	4	6	0.2889	1.95E-08	6	7	0.1471	1.35E-07
	$m_4$	2	3	1.2706	0.00E+00	11	13	1.1981	2.10E-07	9	10	1.4916	1.51E-07
	$m_5$	2	3	0.1308	0.00E+00	11	13	0.5316	2.10E-07	9	10	0.1756	1.51E-07
	$m_6$	2	3	0.8749	0.00E+00	6	8	0.7440	1.63E-08	8	9	2.3692	2.34E-07
	$m_7$	2	3	0.1153	0.00E+00	11	13	0.3875	2.10E-07	9	10	0.1597	1.51E-07
	$m_8$	2	3	1.6229	0.00E+00	11	13	1.9796	2.12E-07	9	10	0.2254	1.51E-07
100000	$m_1$	1	2	0.1545	0.00E+00	5	7	0.3623	3.13E-07	7	8	0.9581	9.96E-08
	$m_2$	2	3	0.5466	0.00E+00	4	6	0.3900	4.07E-08	6	7	0.2321	7.74E-07
	$m_3$	1	2	0.4549	3.14E-16	4	6	0.6795	1.95E-08	6	7	0.4348	1.35E-07
	$m_4$	2	3	1.1598	0.00E+00	11	13	0.5917	5.11E-07	9	10	0.9209	2.14E-07
	$m_5$	2	3	0.2296	0.00E+00	11	13	2.1472	5.11E-07	9	10	0.9297	2.14E-07
	$m_6$	2	3	0.5503	0.00E+00	6	8	0.3154	1.63E-08	8	9	1.7438	2.34E-07
	$m_7$	2	3	1.1678	0.00E+00	11	13	1.3542	5.11E-07	9	10	1.0237	2.14E-07
	$m_8$	2	3	0.2391	0.00E+00	11	13	0.6503	5.14E-07	9	10	0.3572	2.14E-07

| Show Table

DownLoad: CSV

Table 8. Numerical results of the three algorithms on Problem 7.

		iSDFM				DAIS1				MSGPALG
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$m_1$	19	21	0.0328	9.46E-07	67	69	0.1959	9.10E-07	67	68	0.0613	9.89E-07
	$m_2$	52	53	0.0584	9.20E-07	101	103	0.1120	9.84E-07	61	62	0.0533	9.84E-07
	$m_3$	57	59	0.0884	9.41E-07	76	78	0.1511	9.29E-07	87	88	0.0644	9.38E-07
	$m_4$	55	57	0.0840	6.99E-07	83	85	0.2007	9.36E-07	83	84	0.0643	8.91E-07
	$m_5$	56	58	0.0663	8.51E-07	81	83	0.1548	9.41E-07	83	84	0.0635	8.91E-07
	$m_6$	47	49	0.0655	9.14E-07	81	83	0.1694	9.87E-07	68	69	0.0403	8.99E-07
	$m_7$	55	57	0.0831	9.64E-07	85	87	0.1402	9.92E-07	83	84	0.0378	8.91E-07
	$m_8$	54	56	0.0986	8.26E-07	83	85	0.2083	8.98E-07	82	83	0.0417	9.64E-07
5000	$m_1$	44	46	0.4975	9.72E-07	101	103	4.4357	9.03E-07	47	48	3.9616	7.95E-07
	$m_2$	41	43	2.1529	9.63E-07	91	93	0.4886	9.98E-07	93	94	0.1564	9.74E-07
	$m_3$	50	52	0.1946	9.94E-07	94	96	1.1680	9.55E-07	80	81	0.4414	9.74E-07
	$m_4$	53	55	0.3218	8.21E-07	89	91	4.2258	8.31E-07	65	66	0.7041	9.19E-07
	$m_5$	51	53	0.3422	9.93E-07	90	92	0.5222	8.89E-07	65	66	0.5813	9.19E-07
	$m_6$	56	58	1.4298	9.63E-07	73	75	0.8748	9.42E-07	56	57	0.2107	9.25E-07
	$m_7$	54	56	0.2158	9.01E-07	96	98	1.8010	6.68E-07	65	66	4.0542	9.19E-07
	$m_8$	52	54	0.5704	8.94E-07	89	91	1.7997	9.43E-07	67	68	1.2837	9.44E-07
10000	$m_1$	49	51	3.5965	8.84E-07	106	108	2.0185	8.15E-07	52	53	0.2338	8.65E-07
	$m_2$	39	41	0.3701	8.64E-07	91	93	1.3319	9.24E-07	94	95	1.4506	9.72E-07
	$m_3$	54	56	1.3825	8.64E-07	95	97	1.5026	9.24E-07	70	71	0.6796	8.77E-07
	$m_4$	56	58	3.5432	6.98E-07	96	98	2.1861	9.18E-07	57	58	4.7632	8.81E-07
	$m_5$	52	54	0.9342	8.00E-07	97	99	1.6761	9.85E-07	57	58	0.4527	8.81E-07
	$m_6$	59	60	1.5782	8.65E-07	91	93	1.8383	9.48E-07	58	59	0.5386	7.32E-07
	$m_7$	57	59	0.4248	7.88E-07	103	105	1.7186	7.81E-07	57	58	0.4381	8.81E-07
	$m_8$	58	60	1.2930	9.43E-07	99	101	2.3954	9.07E-07	53	54	4.5234	6.99E-07
50000	$m_1$	46	48	5.1900	8.32E-07	74	76	5.7164	9.08E-07	36	37	0.6755	6.71E-07
	$m_2$	55	57	2.7338	9.19E-07	96	98	14.8517	9.62E-07	95	96	2.5677	9.62E-07
	$m_3$	62	64	2.8549	8.89E-07	101	103	7.6005	9.46E-07	88	89	2.2433	9.13E-07
	$m_4$	59	61	2.7944	7.58E-07	105	107	7.4685	9.10E-07	81	82	2.4065	9.25E-07
	$m_5$	61	63	2.9541	7.11E-07	103	105	5.6730	9.09E-07	81	82	2.6295	9.25E-07
	$m_6$	54	56	2.7075	9.94E-07	102	104	5.4700	9.54E-07	56	57	1.5636	8.15E-07
	$m_7$	63	65	3.1767	8.45E-07	101	103	5.2194	9.21E-07	81	82	2.1452	9.25E-07
	$m_8$	60	62	2.8825	9.99E-07	96	98	5.2916	9.82E-07	81	82	2.2370	9.30E-07
100000	$m_1$	35	36	3.2271	9.76E-07	110	112	11.6739	9.71E-07	50	51	2.3679	9.55E-07
	$m_2$	58	60	4.4230	9.75E-07	103	105	13.3138	8.57E-07	67	68	2.9833	7.51E-07
	$m_3$	57	59	5.5342	8.21E-07	80	82	10.3855	9.39E-07	72	73	3.3898	9.29E-07
	$m_4$	55	56	5.0986	9.16E-07	88	90	10.7673	7.47E-07	88	89	4.4502	9.38E-07
	$m_5$	55	57	5.3401	9.05E-07	99	101	9.7451	9.95E-07	88	89	4.9388	9.38E-07
	$m_6$	57	59	5.4860	9.28E-07	103	105	11.7762	7.17E-07	74	75	3.1878	9.23E-07
	$m_7$	59	60	5.4163	8.97E-07	88	90	9.7121	8.46E-07	88	89	3.9409	9.38E-07
	$m_8$	54	56	5.1927	9.03E-07	99	101	10.0416	9.68E-07	88	89	3.6687	9.40E-07

| Show Table

DownLoad: CSV

With the help of the Dolan and Moré performance profile ^[33], we present the information in Tables 2–8 graphically for better and easier visualization of each algorithm's performance. These graphs are plotted in Figures 1–3. It can be clearly observed from Figure 1 that the iSDFM algorithm solved about 72% of the problems with the least ITER, as compared to the DAIS1 and MSGPALG with around 30% and 12%, respectively. Figure 2 shows that the iSDFM outperformed the other two methods by solving around 70% of the problems with the least FVAL. In terms of TIME, iSDFM algorithm competes favorably with the MSGPALG algorithm. In general, our proposed iSDFM algorithm shows better efficiency as compared to the DAIS1 and MSGPALG algorithms. This might not be unconnected with taking the convex combination of the BB-like parameters incorporated with the inertial technique.

Figure 1. Performance profile on number of iterations.

DownLoad: Full-Size Img PowerPoint

Figure 2. Performance profile on function evaluations.

DownLoad: Full-Size Img PowerPoint

Figure 3. Performance profile on CPU 21:31:38.

DownLoad: Full-Size Img PowerPoint

4. Application in motion control

In robotics, manipulators and effectors are the aspects in which some parts of the robots interact with other objects by performing different tasks such as picking from one point and placing on another. For stability and accuracy in the robots' movement, the characteristics of motor dynamics, which are contemporarily used as actuators in $n-$ link and $1-$ link robot systems, need to be considered ^[34,35]. This is a tracking control problem of a nonlinear system, and the motor dynamics are required to satisfy the condition that the actual output of the system can track the desired trajectory with least possible error ^[36].

Some of the developed methods for tracking control problems of nonlinear systems include proportional-integral-derivative (PID) control ^[37,38], feedback linearization ^[39,40] and optimal output tracking control by using approximation approach ^[41].

We present an application of our proposed algorithm in motion control of two planar robotic manipulators. Consider the following model:

$\begin{equation} \min\limits_{m\in \mathbb{R}^n }f(m), \end{equation}$

(4.1)

where $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth and convex function. In the case of motion control, the function in (4.1) has the form $f(v_k): = \frac{1}{2}\|\upsilon_k-y_{uk}\|_{2}^2.$ Subsequently, we minimize

$\begin{equation} \min\limits_{\upsilon_k \in \mathbb{R}^2 }\frac{1}{2}\|\upsilon_k-y_{uk}\|_{2}^2 \end{equation}$

(4.2)

at each computational time interval $\tau_k\in [0, \tau_f].$

As described in ^[42], the discrete-time kinematics equation of a two-joint planar robot manipulator at the position level is given as

$\begin{equation} \psi (\theta_k) = \upsilon_k. \end{equation}$

(4.3)

The kinematics map $\psi(\cdot)$ is given as

$\begin{equation} \psi(\theta) = \begin{bmatrix} l_1\cos(\theta_1) + l_2\cos(\theta_1+\theta_2)\\ l_1\sin(\theta_1) + l_2\sin(\theta_1+\theta_2) \end{bmatrix}, \end{equation}$

(4.4)

where $l_1$ and $l_2$ are the lengths of the rod links, and $\theta_k \in \mathbb{R}^2$ is the joint angle vector.

From (4.2), the term $\upsilon_k$ is controlled to track a Lissajous curve:

$\begin{equation} y_{uk} = \begin{bmatrix} \frac{3}{2} + \frac{1}{5}\sin(3 \tau_k)\\ \frac{\sqrt{3}}{2} + \frac{1}{5}\sin (2 \tau_k) \end{bmatrix}. \end{equation}$

(4.5)

For our algorithm to fit (4.1), we present its slight modification as follows:

Algorithm 2: Modified iSDFM.
Input: Let $\alpha_k = 0$ for $k\geq 0$ in . Consider the same inputs as in , and let $\Gamma(m_k) = \nabla f(m_k)$ . Replace Step 2 and Step 3 of the Algorithm 1 with the followings:
Step 2: Compute $\Lambda_k=\kappa\varsigma^{j},$ where $j$ is the smallest non–negative integer such that
$\begin{equation} f(m_k + \kappa\varsigma^{j}t_k) - f(m_k) \leq \sigma \kappa\varsigma^{j} \Gamma(m_k)^T t_k. \; \; \; \; \; \; \; (4.6)\end{equation}$
Step 3: To update the next iterate, use
$\begin{equation} m_{k+1}:=m_k+\Lambda_kt_k. \; \; \; \; \; \; \; (4.7)\end{equation}$

| Show Table

DownLoad: CSV

Remark 4.1. Assume that the solution to problem (4.1) exists and that the level set $\{m\in \mathbb{R}^n: f(m)\leq f(m_0)\}$ is bounded. Using Assumption $A_{2}$ and Theorem 2 of ^[43], we conclude that converges, that is, $\liminf_{k\to \infty}\|\Gamma(m_k)\| = 0$ holds.

In this experiment, we chose initial joint angle vector $\theta_0 = [0, \frac{\pi}{3}]^T,$ $l_1 = l_2 = 1,$ $\varsigma = 0.2$ and $\sigma = 0.08.$ The task duration $[0, 20]$ is subdivided into 200 equal parts. The performance of Algorithm 2 in the motion control problem is shown in the Figure 4.

Figure 4. Performance generated by Algorithm 2: (a) robot trajectories; (b) end effector trajectory and desired path; (c) residual error on x-axis; (d) residual error on y-axis.

DownLoad: Full-Size Img PowerPoint

Figure 4(a) represents the robot trajectories synthesized by Algorithm 2. Figure 4(b) shows the end effector and the desired path. Figure 4(, ) shows the performance errors on x and y axes, respectively. These figures indicate that efficiently performed the assignment with an error as low as $10^{-5}$ on both the x-axis and y-axis.

5. Conclusions

In conclusion, we have proposed an inertial-based spectral method for solving a system of nonlinear equations. The inertial-step introduced is believed to have enhanced the performance of the new method. We have discussed the global convergence of the proposed algorithm under the monotonicity and Lipschitz continuity assumptions. We have also presented some numerical experiments which depicted the efficiency of the proposed algorithm. The proposed algorithm is reported to have better numerical performance than the methods in ^[21,22]. Subsequently, we have demonstrated the applicability of the new method in motion control problems arising from robotics. However, the numerical results showed that the proposed method won about 72% and 70% of the experiments in terms of ITER and FVAL. Therefore, we recommend further research in order to improve its performance to the optimum.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The sixth author was funded by Chiang Mai University and the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (grant number B05F640183).

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	X. Cui, Y. Luo, Y. Yang, Y. Guo, H. Wang, X. Liu, Energy saving mechanism and energy saving approach of asynchronous motor under periodic variable working conditions, Proc. Chin. J. Electr. Eng., 28 (2008), 90–97.
[2]	D. Yang, H. R. Karimi, L. Gelman, A fuzzy fusion rotating machinery fault diagnosis framework based on the enhancement deep convolutional neural networks, Sensors, 22 (2022), 671. https://doi.org/10.3390/s22020671 doi: 10.3390/s22020671
[3]	Z. Lv, Online monitoring of batch processes combining subspace design of latent variables with support vector data description, Complex Eng. Syst., 1 (2021). https://org.doi/10.20517/ces.2021.02 doi: 10.20517/ces.2021.02
[4]	J. Yu, X. Yan, Data-feature-driven nonlinear process monitoring based on joint deep learning models with dual-scale, Inf. Sci., 591 (2022), 381–399. https://doi.org/10.1016/j.ins.2021.12.106 doi: 10.1016/j.ins.2021.12.106
[5]	J. Yu, X. Yan, Active features extracted by deep belief network for process monitoring, ISA Trans., 84 (2018), 247–261. https://doi.org/10.1016/j.isatra.2018.10.011 doi: 10.1016/j.isatra.2018.10.011
[6]	J. Yu, X. Yan, Multiscale intelligent fault detection system based on agglomerative hierarchical clustering using stacked denoising autoencoder with temporal information, Appl. Soft Comput., 95 (2020), 106525. https://doi.org/10.1016/j.asoc.2020.106525 doi: 10.1016/j.asoc.2020.106525
[7]	Y. Zhao, X. Li, S. Yang, Minimum allowable values of energy efficiency and energy efficiency grades for small and medium three-phase asynchronous motors 18613, 2012.
[8]	C. Luo, W. B. Ma, J. Zhao, Evaluation and analysis of the influence factors on the energy consumption of the motor system, Motor Control Appl., 43 (2016), 98–103.
[9]	C. Li, Research on energy saving evaluation index system of motor system, Motor Control Appl., 43 (2016), 74–77.
[10]	L. X. Ma, M. Y. Lv, Research on intelligent quantification and classification method of power energy efficiency, Power Sci. Eng., 33 (2017), 46–49.
[11]	J. Cervantes, F. Garcia-Lamont, L. Rodríguez-Mazahua, A. Lopez, A comprehensive survey on support vector machine classification: Applications, challenges and trends, Neurocomputing, 408 (2020), 189–215. https://doi.org/10.1016/j.neucom.2019.10.118 doi: 10.1016/j.neucom.2019.10.118
[12]	G. G. Wang, S. Deb, Z. Cui, Monarch butterfly optimizations, Neural Comput. Appl., 31 (2019), 1995–2014. https://doi.org/10.1007/s00521-015-1923-y doi: 10.1007/s00521-015-1923-y
[13]	S. Mirjalili, A. Lewis, The whale optimization algorithm, Adv. Eng. Software, 95 (2016), 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008 doi: 10.1016/j.advengsoft.2016.01.008
[14]	S. H. S. Moosavi, V. K. Bardsiri, Satin bowerbird optimizer: A new optimization algorithm to optimize ANFIS for software development effort estimation, Eng. Appl. Artif. Intell., 60 (2017), 1–15. https://doi.org/10.1016/j.engappai.2017.01.006 doi: 10.1016/j.engappai.2017.01.006
[15]	G. G. Wang, Moth search algorithm: a bio-inspired metaheuristic algorithm for global optimization problems, Memetic Comput., 10 (2018), 151–164. https://doi.org/10.1007/s12293-016-0212-3 doi: 10.1007/s12293-016-0212-3
[16]	F. A. Hashim, E. H. Houssein, M. S. Mabrouk, W. Al‐Atabany, S. Mirjalili, Henry gas solubility optimization: A novel physics-based algorithm, Future Gener. Comput. Syst., 101 (2019), 646–667. https://doi.org/10.1016/j.future.2019.07.015 doi: 10.1016/j.future.2019.07.015
[17]	A. A. Heidari, S. Mirjalili, H. Faris, I. Aljarah, M. Mafarja, H. Chen, Harris hawks optimization: Algorithm and applications, Future Gener. Comput. Syst., 97 (2019), 849–872. https://doi.org/10.1016/j.future.2019.02.028 doi: 10.1016/j.future.2019.02.028
[18]	S. Li, H. Chen, M. Wang, A. A. Heidari, S. Mirjalili, Slime mould algorithm: A new method for stochastic optimization, Future Gener. Comput. Syst., 111 (2020), 300–323. https://doi.org/10.1016/j.future.2020.03.055 doi: 10.1016/j.future.2020.03.055
[19]	I. Ahmadianfar, A. A. Heidari, A. H. Gandomi, X. Chu, H. Chen, RUN beyond the metaphor: An efficient optimization algorithm based on Runge Kutta method, Expert Syst. Appl., 181 (2021), 115079. https://doi.org/10.1016/j.eswa.2021.115079 doi: 10.1016/j.eswa.2021.115079
[20]	J. Tu, H. Chen, M. Wang, A. H. Gandomi, The colony predation algorithmg, J. Bionic Eng., 18 (2021), 674–710. https://doi.org/10.1007/s42235-021-0050-y doi: 10.1007/s42235-021-0050-y
[21]	I. Ahmadianfar, A. AsgharHeidari, S. Noshadian, H. Chen, A. HGandomi, INFO: An efficient optimization algorithm based on weighted mean of vectors, Expert Syst. Appl., 195 (2022), 116516. https://doi.org/10.1016/j.eswa.2022.116516 doi: 10.1016/j.eswa.2022.116516
[22]	G. G. Wang, S. Deb, X. Z. Gao, L. Coelho, A new metaheuristic optimization algorithm motivated by elephant herding behavior, Bio-Inspir. Comput., 8 (2016), 394–409. https://doi.org/10.1504/IJBIC.2016.081335 doi: 10.1504/IJBIC.2016.081335
[23]	Z. Zhang, H. Wang, H. Zhou, S. You, Parameter estimation of chaotic systems based on multi-mechanism hybrid image group Algorithm, Microelectron. Comput., 6 (2020), 40–45.
[24]	M. S. Tavazoei, M. Haeri, Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms, Appl. Math. Comput., 187 (2007), 1076–1085. https://doi.org/10.1016/j.amc.2006.09.087 doi: 10.1016/j.amc.2006.09.087
[25]	F. Chakraborty, P. K. Roy, D. Nandi, Oppositional elephant herding optimization with dynamic Cauchy mutation for multilevel image thresholding, Evol. Intell., 12 (2019), 445–467. https://doi.org/10.1007/s12065-019-00238-1 doi: 10.1007/s12065-019-00238-1
[26]	W. Luo, H. Jin, H. Li, R. Zhou, Blind source separation of radar signals based on chaotic adaptive firework algorithm, Syst. Eng. Electr., 42 (2020), 2497–2505.
[27]	F. Marini, B. Walczak, Particle swarm optimization (PSO). A tutorial, Chemom. Intell. Lab. Syst., 149 (2015), 153–165. https://doi.org/10.1016/j.chemolab.2015.08.020 doi: 10.1016/j.chemolab.2015.08.020
[28]	J. Liang, B. Qu, P. Suganthan, Problem Definitions and Evaluation Criteria for the CEC 2014 Special Session and Competition on Single Objective RealParameter Numerical Optimization, in Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, (2014).
[29]	N. Lynn, P. N. Suganthan, Heterogeneous comprehensive learning particle swarm optimization with enhanced exploration and exploitation, Swarm Evol. Comput., 24 (2015), 11–24. https://doi.org/10.1016/j.swevo.2015.05.002 doi: 10.1016/j.swevo.2015.05.002
[30]	Y. Xue, J. Jiang, B. Zhao, T. Ma, A self-adaptive artificial bee colony algorithm based on global best for global optimization, Soft Comput., 22 (2018), 2935–2952. https://doi.org/10.1007/s00500-017-2547-1 doi: 10.1007/s00500-017-2547-1
[31]	W. F. Gao, L. L. Huang, S. Y. Liu, C. Dai, Artificial bee colony algorithm based on information learning, IEEE Trans. Cybern, 45 (2015), 2827–2839. https://doi.org/10.1109/TCYB.2014.2387067 doi: 10.1109/TCYB.2014.2387067
[32]	Y. luo, H. Jin, H. Li, H. Rong, Blind source separation of radar signals based on chaotic adaptive fireworks algorithm, Syst. Eng. Electron. Technol., 42 (2020), 95–103.
[33]	J. Peng, Y. Zhou, C. L. P. Chen, Region-kernel-based support vector machines for hyperspectral image classification, IEEE Trans. Geosci. Remote Sensing, 53 (2015), 4810–4824. https://doi.org/10.1109/TGRS.2015.2410991 doi: 10.1109/TGRS.2015.2410991
[34]	Z. Lv, X. Yan, Q. Jiang, Batch process monitoring based on self-adaptive subspace support vector data description, Chemom. Intell. Lab. Syst., 170 (2017), 25–31. https://doi.org/10.1016/j.chemolab.2017.09.009 doi: 10.1016/j.chemolab.2017.09.009
[35]	J. Yu, X. Yan, Whole process monitoring based on unstable neuron output information in hidden layers of deep belief network, IEEE Trans. Cybern., 50 (2020), 3998–4007. https://doi.org/10.1109/TCYB.2019.2948202 doi: 10.1109/TCYB.2019.2948202
[36]	Y. Shao, C. Zhang, X. Wang, N. Deng, Improvements on twin support vector machines, IEEE Trans. Neural Networks, 22 (2020), 962–968. https://doi.org/10.1109/TNN.2011.2130540 doi: 10.1109/TNN.2011.2130540
[37]	S. Zhang, X. Li, M. Zong, X. Zhu, R. Wang, Efficient kNN classification with different numbers of nearest neighbors, IEEE Trans. Neural Networks Learn. Syst., 29 (2018), 1774–1785. https://doi.org/10.1109/TNNLS.2017.2673241 doi: 10.1109/TNNLS.2017.2673241

This article has been cited by:

Xiaowei Fang, A derivative-free RMIL conjugate gradient method for constrained nonlinear systems of monotone equations, 2025, 10, 2473-6988, 11656, 10.3934/math.2025528

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)