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Self adaptive spectral conjugate gradient method for solving nonlinear monotone equations

  • M. Koorapetse
  • P. KaeloEmail author
Open Access
Original Research
  • 59 Downloads

Abstract

In this paper, we propose a self adaptive spectral conjugate gradient-based projection method for systems of nonlinear monotone equations. Based on its modest memory requirement and its efficiency, the method is suitable for solving large-scale equations. We show that the method satisfies the descent condition \(F_{k}^{T}d_{k}\leq -c\|F_{k}\|^{2}, c>0\), and also prove its global convergence. The method is compared to other existing methods on a set of benchmark test problems and results show that the method is very efficient and therefore promising.

Keywords

Self adaptive Spectral conjugate gradient method Nonlinear monotone equations 

Abbreviations

CPU

CPU time is seconds

DIM

Dimension

ITDM

Improved three-term derivative-free method

MLS

Modified Liu-Storey method

NFE

Number of function evaluations

NI

Number of iterations

SASCGM

Self adaptive spectral conjugate gradient-based projection method

SDYP

Spectral DY-type projection method

SP

Starting (initial) point

AMS Subject Classification

90C06 90C30 90C56 65K05 65K10 

Introduction

In this paper, we focus on solving large-scale nonlinear system of equations
$$\begin{array}{*{20}l} F(x) = 0, \end{array} $$
(1)
where \(F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is continuous and monotone. A function F is monotone if it satisfies the monotonicity condition
$$\begin{array}{*{20}l} (F(x)-F(y))^{T}(x-y) \geq 0, \quad \forall x,y\in\mathbb{R}^{n}. \end{array} $$
(2)

Nonlinear monotone equations arise in many practical applications, for example, chemical equilibrium systems [1], economic equilibrium problems [2], and some monotone variational inequality problems [3]. A number of computational methods have been proposed to solve nonlinear equations. Among them, Newton’s method, quasi-Newton method, Gauss-Newton method, and their variants are very popular due to their local superlinear convergence property (see, for example, [4, 5, 6, 7, 8, 9]). However, they are not suitable for large-scale nonlinear monotone equations as they need to solve a linear system of equations using the second derivative information (Jacobian matrix or an approximation of it).

Due to their modest memory requirements, conjugate gradient-based projection methods are suitable for solving large-scale nonlinear monotone equations (1). Conjugate gradient-based projection methods generate a sequence {xk} by exploring the monotonicity of the function F. Let zk=xk+αkdk, where αk>0 is the step length that is determined by some line search and
$$d_{k}=\left\{\begin{array}{ll}-F_{k},& k=0,\\ -F_{k} + \beta_{k}d_{k-1},& k\geq 1,\end{array}\right.$$
Fk=F(xk) and βk is a parameter, is the search direction. Then by monotonicity of F, the hyperplane
$$\begin{array}{*{20}l} H_{k}=\{x\in\mathbb{R}^{n}|F(z_{k})^{T}(x-z_{k})=0\} \end{array} $$
strictly separates the current iterate xk from the solution set of (1). Projecting xk on this hyperplane generates the next iterate xk+1 as
$$\begin{array}{*{20}l} x_{k+1}=x_{k}-\frac{F(z_{k})^{T}(x_{k}-z_{k})}{\|F(z_{k})\|^{2}}F(z_{k}). \end{array} $$
(3)

This projection concept on the hyperplane Hk was first presented by Solodov and Svaiter [10].

Following Solodov and Svaiter [10], a lot of work has been done, and continues to be done, to come up with a number of conjugate gradient-based projection methods for nonlinear monotone equations. For example, Hu and Wei [11] proposed a conjugate gradient-based projection method for nonlinear monotone equations (1) where the search direction dk is given as
$$\begin{array}{*{20}l} d_{k}=\left\{\begin{array}{ll} -F_{k}, & k=0, \\ -F_{k}+\frac{F_{k}^{T}y_{k-1}d_{k-1}-d_{k-1}^{T}F_{k}y_{k-1}} {\max\left(\gamma\|d_{k-1}\|\|y_{k-1}\|,d_{k-1}^{T}y_{k-1},-d_{k-1}^{T}F_{k-1}\right)}, & k\geq 1, \end{array}\right. \end{array} $$
yk−1=FkFk−1 and γ>0. This method was shown to perform well numerically and its global convergence was established using the line search
$$\begin{array}{*{20}l} -F(z_{k})^{T}d_{k}\geq\sigma\alpha_{k}\|F(z_{k})\|\| d_{k}\|^{2}, \end{array} $$
(4)

with σ>0 being a constant.

Recently, three term conjugate gradient-based projection methods have also been presented. One such method is that by Feng et al. [12] who presented their direction as
$$d_{k}=\left\{\begin{array}{ll}-F_{k},&k=0,\\ -\left(1+\beta_{k}\frac{F_{k}^{T}d_{k-1}}{\|F_{k}\|^{2}}\right)F_{k}+\beta_{k}d_{k-1},&k\geq 1,\end{array}\right.$$
where \(|\beta _{k}|\leq t\frac {\|F_{k}\|}{\|d_{k-1}\|},\,\, \forall k\geq 1\), and t>0 is a constant. The global convergence of this method was also established using the line search (4). For other conjugate gradient-based projection methods, the reader is referred to [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

In this paper, following the work of Abubakar and Kumam [21], Hu and Wei [11] and that of Liu and Li [22], we propose a self adaptive spectral conjugate gradient-based projection method for solving systems of nonlinear monotone Eq. (1). This method is presented in the next section and the rest of the paper is organized as follows. In “Convergence analysis” section, we show that the proposed method satisfies the descent property \(F_{k}^{T}d_{k}\leq -c\|F_{k}\|^{2}, c>0\), and also establish its global convergence. In “Numerical experiments” section, we present the numerical results and lastly, conclusion is presented in “Conclusion” section.

Algorithm

In this section, we give the details of the proposed method. We start by briefly reviewing the work of Abubakar and Kumam [21] and that of Liu and Li [22].

Most recently, Abubakar and Kumam [21] proposed the direction
$$\begin{array}{*{20}l} d_{k}=\left\{\begin{array}{ll} -F_{k}, & k=0, \\ -F_{k}+\frac{F_{k}^{T}w_{k-1}d_{k-1}-F_{k}^{T}d_{k-1}w_{k-1}}{\max\left(\mu\|d_{k-1}\|\|w_{k-1}\|,w_{k}^{T}d_{k-1}\right)}, & k\geq 1, \end{array}\right. \end{array} $$
where μ is a positive constant and
$$w_{k-1}=y_{k-1}+t\|F_{k}\|s_{k-1},\,\,\, t=1+\|F_{k}\|^{-1}\max\left(0,-\frac{y_{k-1}^{T}s_{k-1}}{\|s_{k-1}\|^{2}}\right)$$
and sk−1=zk−1xk−1=αk−1dk−1. This method was shown to perform well numerically and its global convergence was established using line search (4). In 2015, Liu and Li [22] proposed a spectral DY-type projection method for nonlinear monotone system of Eq. (1) with the search direction dk as
$$\begin{array}{*{20}l} d_{k}=\left\{\begin{array}{ll} -F_{k}, & k=0,\\ -\lambda_{k} F_{k}+\beta_{k}^{DY}d_{k-1}, & k\geq 1, \end{array}\right. \end{array} $$
where\(\beta _{k}^{DY}=\frac {\|F_{k}\|^{2}}{d_{k-1}^{T}u_{k-1}}\), uk−1=yk−1+tdk−1, \(t=1+\max \left \{0,-\frac {d_{k-1}^{T}y_{k-1}}{d_{k-1}^{T}d_{k-1}}\right \}\), yk−1=FkFk−1+rsk−1 with sk−1=xkxk−1, r>0 being a constant and \(\lambda _{k}=\frac {s_{k-1}^{T}s_{k-1}}{s_{k-1}^{T}y_{k-1}}\). The global convergence of this method was established using the line search
$$\begin{array}{*{20}l} -F(z_{k})^{T}d_{k}\geq\sigma\alpha_{k}\| d_{k}\|^{2}. \end{array} $$
(5)
Motivated by the work of Abubakar and Kumam [21], Hu and Wei [11] and that of Liu and Li [22], in this paper we present our direction as
$$\begin{array}{*{20}l} d_{k}=\left\{\begin{array}{ll} -F_{k}, & k=0,\\ -\lambda_{k}^{*} F_{k}+\beta_{k}^{MP}d_{k-1}-\delta_{k}^{MP}y_{k-1}, & k\geq 1, \end{array}\right. \end{array} $$
(6)
where
$$\begin{array}{*{20}l} \beta_{k}^{MP}=\frac{F_{k}^{T}y_{k-1}}{\max\{\mu_{k}d_{k-1}^{T}y_{k-1},-\eta F_{k-1}^{T}d_{k-1}+\mu_{k}\|d_{k-1}\|\|y_{k-1}\|\}} \end{array} $$
(7)
and
$$\begin{array}{*{20}l} \delta_{k}^{MP}=\frac{F_{k}^{T}d_{k-1}}{\max\{\mu_{k}d_{k-1}^{T}y_{k-1},-\eta F_{k-1}^{T}d_{k-1}+\mu_{k}\|d_{k-1}\|\|y_{k-1}\|\}} \end{array} $$
(8)

with η>0 being a constant and the parameters \(\lambda _{k}^{*}=\frac {s_{k-1}^{T}y_{k-1}}{s_{k-1}^{T}s_{k-1}}\) and \(\mu _{k}>\frac {1}{\lambda _{k}^{*}}\) where sk−1=xkxk−1 and yk−1=FkFk−1+rsk−1, r∈(0,1). With dk defined by (6), (7), and (8), we now present our algorithm.

Throughout this paper, we assume that the following assumption holds.

Assumption 1

(i) The function F(·) is monotone on \(\mathbb {R}^{n}\), i.e. \((F(x)-F(y))^{T}(x-y) \geq 0, \forall x,y\in \mathbb {R}^{n}\). (ii) The solution set of (1) is nonempty. (iii) The function F(·) is Lipschitz continuous on \(\mathbb {R}^{n}\), i.e. there exists a positive constant L such that
$$\begin{array}{*{20}l} \parallel F(x)-F(y)\parallel \, \leq L\parallel x-y\parallel, \quad \forall\, x,y\in\mathbb{R}^{n}. \end{array} $$
(9)

Convergence analysis

In this section we present the descent property and global convergence of the proposed method.

Lemma 1

For all k≥0, we have
$$\begin{array}{*{20}l} r\leq\,\lambda_{k}^{*}\leq\,L+r. \end{array} $$
(10)

Proof

From the definition of yk−1, we get that
$$\begin{array}{*{20}l} s_{k-1}^{T}y_{k-1}=(F_{k}-F_{k-1})^{T}(x_{k}-x_{k-1})+r\|s_{k-1}\|^{2}, \end{array} $$
which using the monotonicity of F it follows that
$$\begin{array}{*{20}l} s_{k-1}^{T}y_{k-1}\geq\,r\|s_{k-1}\|^{2}. \end{array} $$
(11)
Also, from the Lipschitz continuity we obtain that
$$\begin{array}{*{20}l} s_{k-1}^{T}y_{k-1}\,\leq\,(L+r)\|s_{k-1}\|^{2}. \end{array} $$
(12)

Combining (11) and (12) we get the inequality (10). This, therefore, means that \(\lambda _{k}^{*}\) is well defined.

Lemma 2

Suppose that Assumption 1 holds. Let the sequence {xk} be generated by Algorithm 1. Then the search direction dk satisfies the descent condition
$$\begin{array}{*{20}l} F_{k}^{T}d_{k}\,\leq\,-r\|F_{k}\|^{2}, \quad \forall\,k\geq0. \end{array} $$
(13)

Proof

Since d0=−F0, we have \(F_{0}^{T}d_{0}=-\|F_{0}\|^{2}\), which satisfies (13). For k≥ 1, we have from (6) that
$$\begin{array}{*{20}l} F_{k}^{T}d_{k}=-\lambda_{k}^{*}\|F_{k}\|^{2}+\beta_{k}^{MP}F_{k}^{T}d_{k-1}-\delta_{k}^{MP}F_{k}^{T}y_{k-1}. \end{array} $$
(14)
Using (7) and (8) we obtain
$$\begin{array}{*{20}l} F_{k}^{T}d_{k}&=-\lambda_{k}^{*}\|F_{k}\|^{2}+\frac{\left(F_{k}^{T}y_{k-1}\right)\left(F_{k}^{T}d_{k-1}\right)}{\max\left\{\mu_{k}d_{k-1}^{T}y_{k-1},-\eta F_{k-1}^{T}d_{k-1}+\mu_{k}\|d_{k-1}\|\|y_{k-1}\|\right\}}\\ &\quad\, -\frac{\left(F_{k}^{T}d_{k-1}\right)\left(F_{k}^{T}y_{k-1}\right)}{\max\left\{\mu_{k}d_{k-1}^{T}y_{k-1},-\eta F_{k-1}^{T}d_{k-1}+\mu_{k}\|d_{k-1}\|\|y_{k-1}\|\right\}}\\ &=-\lambda_{k}^{*}\|F_{k}\|^{2}\\ &\leq -r\|F_{k}\|^{2}. \end{array} $$

Lemma 3

For all k≥0, we have
$$\begin{array}{*{20}l} r\|F_{k}\|\leq\|d_{k}\|\leq\,3(L+r)\|F_{k}\|. \end{array} $$
(15)

Proof

From (13) and Cauchy-Schwarz inequality, we have
$$\begin{array}{*{20}l} \|d_{k}\|\geq\,r\|F_{k}\|. \end{array} $$
(16)
Also, we have that
$$\begin{array}{*{20}l} {}\max\left\{\mu_{k}d_{k-1}^{T}y_{k-1},-\eta F_{k-1}^{T}d_{k-1}+\mu_{k}\|d_{k-1}\|\|y_{k-1}\|\right\}\!\geq\!-\eta F_{k-1}^{T}d_{k-1}\,+\,\mu_{k}\|d_{k-1}\|\|y_{k-1}\|. \end{array} $$
It then follows from (6), (7), and (8) that
$$\begin{array}{*{20}l} \|d_{k}\|&\leq\,\lambda_{k}^{*}\|F_{k}\|+|\beta_{k}|\|d_{k-1}\|+|\delta_{k}|\|y_{k-1}\|\\ &\leq\,\lambda_{k}^{*}\|F_{k}\|+\frac{\|F_{k}\|\|y_{k-1}\|}{\mu_{k}\|d_{k-1}\|\|y_{k-1}\|}\|d_{k-1}\|+\frac{\|F_{k}\|\|d_{k-1}\|}{\mu_{k}\|d_{k-1}\|\|y_{k-1}\|}\|y_{k-1}\|\\ &=\lambda_{k}^{*}\|F_{k}\|+\frac{2}{\mu_{k}}\|F_{k}\|\\ &\leq\,3\lambda_{k}^{*}\|F_{k}\|\\ &\leq\,3(L+r)\|F_{k}\|. \end{array} $$

Lemma 4

Suppose Assumption 1 holds and let {xk} be generated by Algorithm 1. Then the steplength αk is well defined and satisfies the inequality
$$\begin{array}{*{20}l} \alpha_{k}\,\geq\,\min\left\{\kappa,\frac{\rho r}{9(L+\sigma)(L+r)^{2}}\right\}. \end{array} $$
(17)

Proof

Suppose that, at kth iteration, xk is not a solution, that is, Fk≠0, and for all i=0,1,2,..., inequality (5) fails to hold, that is
$$\begin{array}{*{20}l} -F(x_{k}+\kappa\rho^{i} d_{k})^{T}d_{k}<\sigma\kappa\rho^{i}\parallel d_{k}\parallel^{2}. \end{array} $$
(18)
Since F is continuous, taking limits as i on both sides of (18) yields
$$\begin{array}{*{20}l} -F(x_{k})^{T}d_{k}\leq\,0, \end{array} $$
(19)
which contradicts Lemma 2. So, the steplength αk is well defined and can be determined within a finite number of trials. Now, we prove inequality (17). If αkκ, then \(\alpha '_{k}=\frac {\alpha _{k}}{\rho }\) does not satisfy (5), that is
$$\begin{array}{*{20}l} -F(x_{k}+\alpha'_{k}d_{k})^{T}d_{k}<\sigma\alpha'_{k}\parallel d_{k}\parallel^{2}. \end{array} $$
Using (9), (13) and (15) we have that
$$\begin{array}{*{20}l} r\|F_{k}\|^{2}&\,\leq\,-F_{k}^{T}d_{k}\\ &=\left(F\left(x_{k}+\alpha'_{k}d_{k}\right)-F_{k}\right)^{T}d_{k}-F\left(x_{k}+\alpha'_{k}d_{k}\right)^{T}d_{k}\\ &\leq\, L\alpha'_{k}\|d_{k}\|^{2}+\sigma\alpha'_{k}\|d_{k}\|^{2}\\ &=(L+\sigma)\alpha_{k}\rho^{-1}\|d_{k}\|^{2}\\ &\leq (L+\sigma)\alpha_{k}\rho^{-1}\left(3(L+r)\|F_{k}\|\right)^{2}. \end{array} $$
Thus
$$\begin{array}{*{20}l} \alpha_{k}\,\geq\,\min\left\{\kappa,\frac{\rho r}{9(L+\sigma)(L+r)^{2}}\right\}. \end{array} $$

The following lemma shows that if the sequence {xk} is generated by Algorithm 1, and x is a solution of (1), i.e. F(x)=0, then the sequence {∥xkx∥} is decreasing and convergent. Thus, the sequence {xk} is bounded.

Lemma 5

Suppose Assumption 1 holds and the sequence {xk} is generated by Algorithm 1. For any x such that F(x)=0, we have that
$$\begin{array}{*{20}l} \|x_{k+1}-x^{*}\|^{2}\leq\|x_{k}-x^{*}\|^{2}-\|x_{k+1}-x_{k}\|^{2} \end{array} $$
(20)
and the sequence {xk} is bounded. Furthermore, either {xk} is finite and the last iterate is a solution of (1), or {xk} is infinite and
$$\begin{array}{*{20}l} \sum_{k=0}^{\infty}\|x_{k+1}-x_{k}\|^{2}<\infty, \end{array} $$
which means
$$\begin{array}{*{20}l} {\lim}_{k\rightarrow\infty}\|x_{k+1}-x_{k}\| = 0. \end{array} $$
(21)

Proof

The conclusion follows from Theorem 2.1 in [10].

Theorem 1

Let {xk} be the sequence generated by Algorithm 1. Then
$$\begin{array}{*{20}l} {\lim}_{k\rightarrow\infty}\inf\| F_{k}\| = 0. \end{array} $$
(22)

Proof

Suppose that the inequality (22) is not true. Then there exists a constant ε1>0 such that
$$\begin{array}{*{20}l} \|F_{k}\|\,\geq\,\epsilon_{1}, \quad\,\forall\,k\,\geq\,0. \end{array} $$
This together with (13) implies that
$$\begin{array}{*{20}l} \|d_{k}\|\,\geq\,r\|F_{k}\|\,\geq\,r\epsilon_{1} >0, \quad \quad \forall\,k\,\geq\,0. \end{array} $$
(23)
This and (21) gives that
$$\begin{array}{*{20}l} {\lim}_{k\rightarrow\infty}\alpha_{k} = 0. \end{array} $$
(24)
On the other hand, Lemma 5 implies that
$$\begin{array}{*{20}l} \alpha_{k}\,\geq\,\min\left\{\kappa,\frac{\rho r}{9(L+\sigma)(L+r)^{2}}\right\}>0, \end{array} $$

which contradicts (24). Therefore (22) is true.

Numerical experiments

In this section, results of our proposed method SASCGM are presented together with those of improved three-term derivative-free method (ITDM) [21], the modified Liu-Storey conjugate gradient projection (MLS) method [11], and the spectral DY-type projection method (SDYP) [22]. All algorithms are coded in MATLAB R2016a. In our experiments, we set ε=10−4, i.e., the algorithms are stopped whenever the inequality ∥Fk∥≤10−4 is satisfied, or the total number of iterations exceeds 1000. The method SASCGM is implemented with the parameters σ=10−4, ρ=0.5, r=10−3, \(\mu _{k}=\frac {1}{\lambda _{k}^{*}}+0.1\) and κ=1, while parameters for algorithms ITDM, MLS, and SDYP are set as in respective papers.

The methods are compared using number of iterations, number of function evaluations and CPU time taken for each method to reach the optimal value or termination. We test the algorithms on ten (10) test problems with their dimensions varied from 5000 to 20000, and with four (4) different starting points \(x_{0}=\left (\frac {1}{n},\frac {1}{n},\ldots,\frac {1}{n}\right)^{T}\), x1=(−1,−1,…,−1)T, x2=(0.5,0.5,…,0.5)T and x3=(−0.5,−0.5,…,−0.5)T. The test functions are listed as follows:

Problem 1. Sun and Liu [19] The mapping F is given by
$$\begin{array}{*{20}l} F(x) = Ax+g(x), \end{array} $$
where
$$\begin{array}{*{20}l} A=\left(\begin{array}{ccccc} 2 & -1 & & & \\ -1 & 2 & -1 & & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & -1 \\ & & & -1 & 2 \end{array} \right) \end{array} $$

and \(\phantom {\dot {i}\!}g(x)=(2e^{x_{1}}-1, 3e^{x_{2}}-1,\ldots,3e^{x_{n-1}}-1, 2e^{x_{n}}-1)^{T}\).

Problem 2. Liu and Li [22] Let F be defined by
$$\begin{array}{*{20}l} F(x) = Ax+g(x), \end{array} $$
where \(\phantom {\dot {i}\!}g(x)=(e^{x_{1}}-1, e^{x_{2}}-1,\ldots,e^{x_{n}}-1)^{T}\) and
$$\begin{array}{*{20}l} A=\left(\begin{array}{ccccc} 2 & -1 & & & \\ -1 & 2 & -1 & & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & -1 \\ & & & -1 & 2 \end{array} \right). \end{array} $$
Problem 3. Liu and Feng [18] The mapping F is given by
$$\begin{array}{*{20}l} F_{1}(x) = & 3x_{1}^{3}+2x_{2}-5+\sin(x_{1}-x_{2})\sin(x_{1}+x_{2}),\\ F_{i}(x) = & -x_{i-1}e^{(x_{i-1}-x_{i})}+x_{i}(4+3x_{i}^{2})+2x_{i+1}\\ &+\sin(x_{i}-x_{i+1})\sin(x_{i}+x_{i+1})-8,\,\,\,\, i = 2,3,\ldots,n-1,\\ F_{n}(x) = & -x_{n-1}e^{(x_{n-1}-x_{n})}+4x_{n}-3. \end{array} $$
Problem 4. Liu and Li [20] The mapping F is given by
$$\begin{array}{*{20}l} F_{1}(x) &= 2x_{1}-x_{2}+e^{x_{1}}-1,\\ F_{i}(x) &= -x_{i-1}+2x_{i}-x_{i+1}+e^{x_{i}}-1, \,\,\,\, i=2, 3,\ldots, n-1,\\ F_{n}(x) &= -x_{n-1}+2x_{n}+e^{x_{n}}-1. \end{array} $$
Problem 5. Abubakar and Kumam [21] The mapping F is given by
$$\begin{array}{*{20}l} F_{i}(x) = e^{x_{i}} - 1, \,\,\,\, i=1,2,3,\ldots,n. \end{array} $$
Problem 6. Hu and Wei [11] The mapping F is given by
$$\begin{array}{*{20}l} F_{1}(x) &= 2.5x_{1}+x_{2}-1,\\ F_{i}(x) &= x_{i-1}+2.5x_{i}+x_{i+1}-1, \,\,\,\, i=2, 3,\ldots, n-1,\\ F_{n}(x) &= x_{n-1}+2.5x_{n}-1. \end{array} $$
Problem 7. Hu and Wei [11] The mapping F is given by
$$\begin{array}{*{20}l} F_{1}(x) &= 2x_{1}+0.5h^{2}(x_{1}+h)^{3}-x_{2},\\ F_{i}(x) &= 2x_{i}+0.5h^{2}(x_{i}+hi)^{3}-x_{i-1}+x_{i+1},\,\,\,\,i=2,3,...,n-1,\\ F_{n}(x) &= 2x_{n}+0.5h^{2}(x_{n}+hn)^{3}-x_{n-1}, \end{array} $$

where \(h=\frac {1}{n+1}\).

Problem 8. Wang and Guan [25] The mapping F is given by
$$\begin{array}{*{20}l} F_{i}(x) &= 2x_{i}-\sin|x_{i}-1|, \,\,\,\, i=1,2,3,...,n. \end{array} $$
Problem 9. Wang and Guan [25] The mapping F is given by
$$\begin{array}{*{20}l} F_{i}(x) = e^{x_{i}} - 2, \,\,\,\, i=1,2,3,...,n. \end{array} $$
Problem 10. Gao and He [24] The mapping F is given by
$$\begin{array}{*{20}l} F_{i}(x) &= x_{i}-\sin(|x_{i}|-1), \,\,\,\, i=1,2,3,...,n. \end{array} $$
The numerical results are reported in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, where “SP” represents the starting point (initial point), “DIM” denotes the dimension of the problem, “NI” refers to the number of iterations, “NFE” stands for the number of function evaluations, and “CPU” is the CPU time in seconds. In Table 3, “*” indicates that the algorithm did not converge within the maximum number of iterations. From the tables, we observe that the proposed method performs better than the other methods in Problems 2, 3, 4, 6, 9, and 10. The proposed method performs slightly lower in Problems 1, 5, 7, and 8. However, overall, the proposed method shows that it is very competitive with the other methods and can be a good addition to the existing methods in the literature.
Table 1

Numerical results of Problem 1

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

19

21

20

18

60

69

62

39

3.9873

4.5649

4.2163

2.5931

 

10000

17

20

22

15

50

65

69

30

13.6563

17.7531

18.4265

8.2387

 

20000

18

27

21

17

54

98

66

34

61.1241

111.0820

72.5027

38.6235

x1

5000

18

20

18

16

55

67

58

34

3.8213

4.6693

4.1661

2.3705

 

10000

18

19

18

21

52

63

58

56

14.7603

17.7104

16.7535

15.7107

 

20000

17

22

21

18

49

74

68

41

56.5528

85.1238

78.3744

47.1014

x2

5000

20

20

17

16

62

68

51

34

4.6887

4.7276

3.7336

2.3715

 

10000

18

20

17

21

58

65

52

49

16.3762

18.3691

14.6588

13.7502

 

20000

20

19

21

18

66

59

68

38

75.9882

67.8322

78.6551

44.4579

x3

5000

18

22

20

19

58

73

63

43

4.0671

5.0897

4.9479

3.0129

 

10000

20

23

21

18

61

79

67

40

17.1879

22.1984

18.8722

11.2565

 

20000

17

23

23

18

52

77

74

38

60.0939

88.9299

85.4446

43.6871

Table 2

Numerical results of Problem 2

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

8

7

6

6

17

19

14

13

1.1809

1.3228

1.1286

0.9069

 

10000

7

7

5

6

15

19

11

13

4.2963

5.3774

3.1084

3.6528

 

20000

7

6

5

6

15

15

11

13

17.1297

17.3641

12.7289

14.7579

x1

5000

19

24

20

22

57

78

60

54

3.9909

5.4833

4.3748

3.7601

 

10000

19

29

22

26

52

119

70

68

14.6032

33.9007

19.6976

19.0850

 

20000

21

25

21

25

57

83

63

64

65.7284

95.5594

72.3957

73.8244

x2

5000

18

22

21

18

52

75

66

43

3.6378

5.2738

5.0433

3.0073

 

10000

18

22

17

21

58

72

50

55

16.3369

20.2522

14.0884

15.5107

 

20000

18

25

17

17

57

104

51

35

65.4977

119.4127

58.7641

40.4790

x3

5000

19

22

21

18

56

72

66

38

4.1432

5.0871

4.9204

2.7117

 

10000

19

24

19

20

56

80

57

44

15.7323

22.3916

15.9898

12.3388

 

20000

19

25

21

17

55

83

65

35

63.0373

94.0844

74.6803

39.6888

Table 3

Numerical results of Problem 3

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

5

21

20

*

17

129

122

*

0.0951

0.0555

0.1255

*

 

10000

5

19

19

*

17

117

116

*

0.0219

0.1717

0.1598

*

 

20000

5

19

19

*

17

117

116

*

0.0391

0.3119

0.1863

*

x1

5000

5

22

18

*

16

136

109

*

0.0068

0.0586

0.0464

*

 

10000

5

22

22

*

16

138

136

*

0.0137

0.1423

0.1151

*

 

20000

5

23

21

*

16

147

131

*

0.0260

0.2613

0.2114

*

x2

5000

14

19

18

*

109

119

110

*

0.0592

0.0635

0.0469

*

 

10000

14

17

18

*

105

106

111

*

0.0844

0.0922

0.0938

*

 

20000

14

19

17

*

106

118

104

*

0.2302

0.2238

0.2718

*

x3

5000

4

20

18

*

9

124

109

*

0.0040

0.0530

0.0465

*

 

10000

4

20

18

*

9

122

109

*

0.0080

0.1049

0.0926

*

 

20000

4

22

20

*

9

138

123

*

0.0150

0.2312

0.2057

*

Table 4

Numerical results of Problem 4

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

8

7

6

6

17

19

14

13

0.0047

0.0054

0.0098

0.0031

 

10000

7

7

5

6

15

19

11

13

0.0082

0.0104

0.0059

0.0061

 

20000

7

6

5

6

15

15

11

13

0.0150

0.0155

0.0109

0.0111

x1

5000

19

24

20

22

57

78

60

54

0.0144

0.0215

0.0160

0.0124

 

10000

19

29

22

26

52

119

70

68

0.0269

0.0617

0.0356

0.0311

 

20000

21

25

21

25

57

83

63

64

0.0553

0.0836

0.0611

0.0549

x2

5000

18

22

21

18

52

75

66

43

0.0133

0.0204

0.0173

0.0099

 

10000

18

22

17

21

58

72

50

55

0.0289

0.0389

0.0261

0.0251

 

20000

18

25

17

17

57

104

51

35

0.0530

0.0998

0.0495

0.0309

x3

5000

19

22

21

18

56

72

66

38

0.0207

0.0278

0.0234

0.0134

 

10000

19

24

19

20

56

80

57

44

0.0331

0.0430

0.0406

0.0206

 

20000

19

25

21

17

55

83

65

35

0.0618

0.0814

0.0627

0.0309

Table 5

Numerical results of Problem 5

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

8

6

6

6

17

13

13

13

0.0027

0.0022

0.0317

0.0016

 

10000

7

6

6

6

15

13

13

13

0.0039

0.0039

0.0033

0.0024

 

20000

7

5

5

5

15

11

11

11

0.0069

0.0055

0.0048

0.0034

x1

5000

7

5

5

13

9

6

6

26

0.0019

0.0014

0.0013

0.0032

 

10000

8

5

5

13

11

6

6

26

0.0040

0.0026

0.0021

0.0051

 

20000

8

5

5

13

11

6

6

26

0.0070

0.0045

0.0039

0.0085

x2

5000

17

14

14

10

35

29

29

19

0.0056

0.0053

0.0046

0.0024

 

10000

17

14

14

10

35

29

29

19

0.0100

0.0094

0.0079

0.0040

 

20000

18

14

14

10

37

29

29

19

0.0179

0.0161

0.0140

0.0063

x3

5000

5

4

4

13

6

5

5

26

0.0014

0.0012

0.0011

0.0034

 

10000

5

4

4

14

6

5

5

28

0.0024

0.0020

0.0018

0.0056

 

20000

5

4

4

14

6

5

5

28

0.0042

0.0036

0.0031

0.0097

Table 6

Numerical results of Problem 6

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

20

30

19

20

75

184

57

46

0.0177

0.0433

0.0207

0.0068

 

10000

19

22

20

20

72

77

64

46

0.0228

0.0373

0.0203

0.0196

 

20000

21

21

21

21

81

70

67

49

0.0572

0.0469

0.0594

0.0259

x1

5000

23

26

21

21

85

90

66

46

0.0133

0.0169

0.0118

0.0069

 

10000

23

24

27

24

86

81

82

58

0.0263

0.0295

0.0292

0.0228

 

20000

22

28

23

23

87

143

78

52

0.0713

0.0951

0.0684

0.0413

x2

5000

20

25

19

20

78

99

59

46

0.0120

0.0178

0.0106

0.0068

 

10000

20

22

18

22

80

97

53

54

0.0302

0.0450

0.0187

0.0225

 

20000

22

24

18

18

84

93

56

40

0.0694

0.0593

0.0429

0.0213

x3

5000

21

29

22

23

77

150

68

53

0.0120

0.0272

0.0122

0.0122

 

10000

21

24

21

21

80

80

65

47

0.0328

0.0275

0.0304

0.0130

 

20000

22

22

24

23

90

70

72

53

0.0476

0.0455

0.0441

0.0265

Table 7

Numerical results of Problem 7

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

3

5

5

5

11

16

16

12

0.0096

0.0166

0.0201

0.0147

 

10000

2

5

5

5

7

16

16

12

0.0162

0.0284

0.0380

0.0205

 

20000

7

4

4

4

27

13

13

10

0.1062

0.0535

0.0440

0.0331

x1

5000

41

31

30

57

160

117

111

190

0.1409

0.1063

0.0994

0.1685

 

10000

41

31

32

58

160

114

117

193

0.3351

0.2038

0.2106

0.3586

 

20000

39

30

32

57

152

109

118

187

0.5987

0.3983

0.4295

0.7145

x2

5000

38

30

28

55

148

112

103

183

0.1305

0.1064

0.0927

0.2220

 

10000

38

27

30

55

148

99

109

182

0.2898

0.1833

0.2068

0.3617

 

20000

36

28

30

54

140

103

108

176

0.5344

0.3519

0.4036

0.6686

x3

5000

38

30

28

55

148

112

103

183

0.1394

0.1235

0.1278

0.1581

 

10000

38

27

30

55

148

99

109

182

0.2931

0.1757

0.2005

0.3389

 

20000

36

28

30

54

140

103

108

176

0.4699

0.3714

0.4239

0.5990

Table 8

Numerical results of Problem 8

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

6

3

3

9

24

10

10

21

0.0026

0.0013

0.0189

0.0023

 

10000

6

3

3

9

24

10

10

21

0.0042

0.0022

0.0021

0.0036

 

20000

6

3

3

10

24

10

10

23

0.0071

0.0039

0.0035

0.0078

x1

5000

7

4

4

10

28

13

13

26

0.0031

0.0018

0.0018

0.0027

 

10000

7

5

5

11

28

17

17

28

0.0050

0.0040

0.0036

0.0047

 

20000

7

5

5

11

28

17

17

28

0.0085

0.0070

0.0062

0.0077

x2

5000

4

6

6

10

16

23

23

23

0.0018

0.0031

0.0029

0.0026

 

10000

5

6

6

10

20

23

23

23

0.0036

0.0050

0.0046

0.0039

 

20000

5

6

6

10

20

23

23

23

0.0059

0.0089

0.0080

0.0065

x3

5000

6

4

4

10

24

13

13

24

0.0027

0.0018

0.0017

0.0027

 

10000

7

4

4

10

28

13

13

24

0.0050

0.0030

0.0027

0.0040

 

20000

7

4

4

10

28

14

14

24

0.0084

0.0055

0.0049

0.0067

Table 9

Numerical results of Problem 9

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

8

11

11

10

27

33

33

21

0.0039

0.0056

0.0087

0.0029

 

10000

9

11

11

10

31

33

33

21

0.0071

0.0094

0.0085

0.0045

 

20000

9

11

11

11

31

33

33

23

0.0120

0.0166

0.0150

0.0085

x1

5000

4

12

12

11

10

35

35

23

0.0016

0.0060

0.0054

0.0031

 

10000

5

12

12

11

14

35

35

23

0.0034

0.0099

0.0089

0.0048

 

20000

5

13

13

11

14

39

39

23

0.0058

0.0194

0.0172

0.0084

x2

5000

4

11

11

9

12

34

34

19

0.0017

0.0055

0.0051

0.0025

 

10000

4

11

11

9

12

34

34

19

0.0027

0.0089

0.0087

0.0038

 

20000

4

11

11

9

12

34

34

19

0.0046

0.0166

0.0147

0.0068

x3

5000

6

12

12

12

17

36

36

26

0.0022

0.0057

0.0052

0.0030

 

10000

6

13

13

12

17

39

39

26

0.0041

0.0109

0.0094

0.0050

 

20000

7

12

12

13

21

36

36

28

0.0085

0.0177

0.0157

0.0097

Table 10

Numerical results of Problem 10

SP

DIM

NI

   

NFE

   

CPU

   
  

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

SASCGM

DLPM

MLS

SDYP

x0

5000

7

13

13

11

21

40

40

23

0.0027

0.0060

0.0092

0.0027

 

10000

7

13

13

11

21

40

40

23

0.0043

0.0098

0.0087

0.0041

 

20000

8

13

13

11

24

40

40

23

0.0082

0.0168

0.0150

0.0065

x1

5000

6

12

12

6

18

37

37

12

0.0021

0.0052

0.0046

0.0013

 

10000

6

13

13

6

18

40

40

12

0.0036

0.0098

0.0087

0.0022

 

20000

6

13

13

7

18

40

40

14

0.0060

0.0170

0.0149

0.0042

x2

5000

6

10

10

7

16

29

29

14

0.0020

0.0044

0.0037

0.0017

 

10000

6

10

10

7

16

29

29

14

0.0034

0.0071

0.0064

0.0024

 

20000

6

14

14

7

16

42

42

14

0.0057

0.0179

0.0155

0.0040

x3

5000

5

9

9

7

15

28

28

15

0.0019

0.0040

0.0036

0.0015

 

10000

5

9

9

7

15

28

28

15

0.0029

0.0065

0.0057

0.0026

 

20000

5

10

10

7

15

31

31

15

0.0047

0.0129

0.0112

0.0043

The performance of the three methods is further presented graphically in Figs. 1, 2, and 3 based on the number of iterations (NI), number of function evaluations (NFE), and the CPU time, respectively, using the performance profile of Dolan and Mor\(\acute {e}\) [28]. That is, we plot the probability ρS(τ) of the test problems for which each of the three methods was within a factor τ. Figures 1, 2, and 3 clearly show the efficiency of the proposed SASCGM method as compared to the other three methods.
Fig. 1

Iterations performance profile

Fig. 2

Function evaluations performance profile

Fig. 3

Cpu time performance profile

Conclusion

In this paper, we proposed a self adaptive spectral conjugate gradient-based projection (SASCGM) method for solving systems of large-scale nonlinear monotone equations. The proposed method is free from derivative evaluations and also satisfies the descent condition \(F_{k}^{T}d_{k}\leq -c\|F_{k}\|^{2}, c>0\), independent of any line search. The global convergence of the proposed method was also established. The proposed algorithm was tested on some benchmark problems with different initial points and different dimensions and the numerical results show that the method is competitive.

Notes

Acknowledgements

The authors appreciate the work of the referees, for their valuable comments and suggestions that led to the improvement of this paper. The authors would also like to thank Prof J. Liu who provided the MATLAB code for SDYP.

Authors’ contributions

The authors jointly worked on the results, read, and approved the final manuscript.

Funding

None.

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    Meintjes, K., Morgan, A. P.: A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22, 333–361 (1987).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dirkse, S. P., Ferris, M. C.: A collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5, 319–345 (1995).CrossRefGoogle Scholar
  3. 3.
    Zhao, Y. B., Li, D.: Monotonicity of fixed point and normal mapping associated with variational inequality and its applications. SIAM J. Optim. 11, 962–973 (2001).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dehghan, M., Hajarian, M.: New iterative method for solving non-linear equations with fourth-order convergence. Int. J. Comput. Math. 87, 834–839 (2010).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Li, D., Fukushima, M.: A global and superlinear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152–172 (1999).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dehghan, M., Hajarian, M.: On some cubic convergence iterative formulae without derivatives for solving nonlinear equations. Int. J. Numer. Methods Biomed. Eng. 27, 722–731 (2011).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dehghan, M., Hajarian, M.: Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations. Comput. Appl. Math. 29, 19–30 (2010).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhou, G., Toh, K. C.: Superlinear convergence of a Newton-type algorithm for monotone equations. J. Optim. Theory Appl. 125, 205–221 (2005).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhou, W. J., Liu, D. H.: A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77, 2231–2240 (2008).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Solodov, M. V., Svaiter, B. F.: A globally convergent inexact newton method for systems of monotone equations, Reformulation: Nonsmooth, Piecewise Smooth, Semismoothing methods. Springer US (1998).  https://doi.org/10.1007/978-1-4757-6388-1_18.CrossRefGoogle Scholar
  11. 11.
    Hu, Y., Wei, Z.: A modified Liu-Storey conjugate gradient projection algorithm for nonlinear monotone equations. Int. Math. Forum. 9, 1767–1777 (2014).CrossRefGoogle Scholar
  12. 12.
    Feng, D., Sun, M., Wang, X.: A family of conjugate gradient methods for large scale nonlinear equations. J. Inqual. Appl. 2017, 236 (2017).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ding, Y., Xiao, Y., Li, J.: A class of conjugate gradient methods for convex constrained monotone equations. Optim. 66(12), 2309–2328 (2017).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Koorapetse, M., Kaelo, P.: Globally convergent three-term conjugate gradient projection methods for solving nonlinear monotone equations. Arab. J. Math. 7, 289–301 (2018).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, J., Li, S.: Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. J. Ind. Manag. Optim. 13, 283–295 (2017).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wang, X. Y., Li, S. J., Kou, X. P.: A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints. Calcolo. 53, 133–145 (2016).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Xiao, Y., Zhu, H.: A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405, 310–319 (2013).MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, J., Feng, Y.: A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algor. 82, 245–262 (2019).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sun, M., Liu, J.: Three derivative-free projection methods for nonlinear equations with convex constraints. J. Appl. Math. Comput. 47, 265–276 (2015).MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu, J., Li, S.: A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 53, 427–450 (2016).MathSciNetCrossRefGoogle Scholar
  21. 21.
    Abubakar, A. B., Kumam, P.: An improved three-term derivative-free method for solving nonlinear equations. Comput. Appl. Math. 37(5), 6760–6773 (2018).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liu, J., Li, S.: Spectral DY-type projection method for nonlinear monotone systems of equations. J. Comput. Math. 33, 341–354 (2015).MathSciNetCrossRefGoogle Scholar
  23. 23.
    Abubakar, A. B., Kumam, P.: A descent Dai-Liao conjugate gradient method for nonlinear equations. Numer. Algor. 81, 197–210 (2019).MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gao, P., He, C.: An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo. 55, 53 (2018).  https://doi.org/10.1007/s10092-018-0291-2.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, S, Guan, H: A scaled conjugate gradient method for solving monotone nonlinear equations with convex constraints. J. Appl. Math. 2013, 286486 (2013).MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ou, Y., Li, J.: A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints. J. Appl. Math. Comput. 56, 195–216 (2018).MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yuan, G., Zhang, M.: A three-terms Polak-Ribiere-Polyak conjugate gradient algorithm for large-scale nonlinear equations. J. Comput. Appl. Math. 286, 186–195 (2015).MathSciNetCrossRefGoogle Scholar
  28. 28.
    Dolan, E. D., More, J. J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002).MathSciNetCrossRefGoogle Scholar

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© The Author(s) 2020

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BotswanaGaboroneBotswana

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