A Two-Step Iteration Method for Vertical Linear Complementarity Problems

Abstract

In this paper, a two-step iteration method was established which can be viewed as a generalisation of the existing modulus-based methods for vertical linear complementarity problems. The convergence analysis of the proposed method is presented, which can enlarge the convergence domain of the parameter matrix compared to the recent results. Numerical examples show that the proposed method is efficient with the two-step technique and confirm the improvement of the theoretical results.

1. Introduction

As a generalisation of a linear complementarity problem (LCP) [1], the vertical linear complementarity problem (VLCP) has wide applications in many fields of science and technology, such as control theory, nonlinear networks and economics; see [2,3,4,5,6,7] for details. Let $A_1,\dots ,A_{\ell }\in {\mathbb{R}}^{n\times n}$ and $q_1,\dots ,q_{\ell }\in {\mathbb{R}}^n$, where ℓ is a positive integer. The VLCP${}_{\ell }$ seeks to find $z,w_1,\dots ,w_{\ell }\in {\mathbb{R}}^n$ to satisfy

$$w_i=A_{i}z+q_i,i=1,\dots ,\ell ,\mathrm{with}min\{z,w_1,\dots ,w_{\ell }\}=0,$$

(1)

where the minimum operation is taken component-wise, which implies that all the involved vectors are non-negative and at least one entry of the ith ($i=1,2,\dots ,n$) component is zero. Note that the VLCP${}_1$ is exactly the LCP.

Recently, for solving the VLCP, a modulus-based formulation was introduced by Mezzadi in [8], which can result in a class of modulus-based matrix splitting (MMS) iteration methods, shown to be more efficient than smooth Newton method [9]. The kind of modulus-based methods can be viewed as the generalisation of the one for solving LCP [10]. For other existing methods of solving the VLCP, one can refer to the recent studies [8,11,12] and the references therein. The MMS methods have been successfully used to solve many kinds of complementarity problems due to the high efficiency in solving linear modulus equations in each iteration. There were also many accelerated techniques, such as double splitting [13], precondition [14], two-step splitting [15,16,17,18], and relaxation [19] to improve the convergence rate of MMS methods in recent works.

In this work, we focus on applying the two-step splitting technique to the equivalent modulus equation of the VLCP. The advantage of the two-step splitting is to make full use of the information of the system matrix in each iteration. Such a technique was successfully used in the LCP [15,16], nonlinear complementarity problem [17], and horizontal LCP [18]. Numerical results showed that the computation time could be saved significantly by the two-step technique comparing to the original iteration method. Hence, we aim to construct the two-step modulus-based matrix splitting (TMMS) iteration for the VLCP, which will be done in Section 2. The convergence theorems of the proposed method will be given in Section 3, and shown to improve the existing ones of the MMS method. By numerical tests in the Section 4, the efficiency of the proposed method is presented. Concluding remarks are given in Section 5.

Some necessary notations, definitions and known results are given first. Let $e={(1,1,\dots ,1)}^T\in {\mathbb{R}}^n$. Let $A=(a_{ij})\in {\mathbb{R}}^{n\times n}$ and $A=D_A-C_A=D_A-L_A-U_A$, where $D_A,-C_A,-L_A$ and $-U_A$ denote the diagonal, nondiagonal, strictly lower triangle and strictly upper triangle parts of A, respectively. By $\rho (A)$, we denote the spectral radius of A. For $A=(a_{ij})\in {\mathbb{R}}^{n\times n}$, $A>(\ge )0$ means that $a_{ij}>(\ge )0$ for all $i,j$. For two matrices $A=(a_{ij}),B=(b_{ij})\in {\mathbb{R}}^{m\times n}$ the order $A\ge (>)B$ means $a_{ij}\ge (>)b_{ij}$ for any i and $j.$ By $|A|$, we denote $|A|=(|a_{ij}|)$ and the comparison matrix of A is $⟨A⟩=(⟨a_{ij}⟩)$, where $⟨a_{ij}⟩=|a_{ij}|$ if $i=j$ and $⟨a_{ij}⟩=-|a_{ij}|$ if $i\ne j$. A is called a Z-matrix if $a_{ij}\le 0$ for any $i\ne j$, a nonsingular M-matrix if it is a nonsingular Z-matrix with $A^{-1}\ge 0$, an H-matrix if $⟨A⟩$ is a nonsingular M-matrix, a strictly diagonal dominant (s.d.d.) matrix if $|a_{ii}|>{\displaystyle \sum _{j\ne i}}|a_{ij}|$ for all $1\le i\le n$ (e.g., see [20]), and an $H_{+}$-matrix if A is an H-matrix with $a_{ii}>0$ for every i (e.g., see [21]). $A=M-N$ is called an H-splitting if $⟨M⟩-|N|$ is a nonsingular M-matrix (e.g., see [22]). It is known that the VLCP has a unique solution if the row-representative matrices $\{A_1,A_2,\dots ,A_{\ell }\}$ satisfy row $\mathcal{W}$-property; see [7]. In the following discussion, we always assume that both the system matrices of the VLCP and their row-representative matrices are $H_{+}$-matrices, which is a sufficient condition of the row $\mathcal{W}$-property, including many typical situations where the solution is unique; see [7,8].

2. New Method

First, we introduce the MMS method for solving VLCP${}_{\ell }$.

Let $A_i=F_i-G_i$$(i=1,2,\dots ,\ell )$ be ℓ splittings of $A_i$, $\mathrm{\Omega }$ be a diagonal matrix with positive diagonal entries and $\gamma$ be a positive constant. Then, with

$$\left\{\begin{array}{lll}z& =& \frac{1}{\gamma }(|x_1|+x_1),\\ w_j& =& {\displaystyle \sum _{i=1}^j}\frac{\mathrm{\Omega }}{\gamma }(|x_i|-x_i)+\frac{\mathrm{\Omega }}{\gamma }(|x_{j+1}|+x_{j+1}),j=1,2,\dots ,\ell -1,\\ w_{\ell }& =& {\displaystyle \sum _{i=1}^{\ell }}\frac{\mathrm{\Omega }}{\gamma }(|x_i|-x_i),\end{array}\right.$$

(2)

VLCP${}_{\ell }$ can be equivalently transformed into a system of fixed-point equations

$$\left\{\begin{array}{ccc}(2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i+F_{\ell })x_1& =\hfill & ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i+G_{\ell })x_1+(2^{\ell -1}\mathrm{\Omega }-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i\hfill \\ & \hfill & -A_{\ell })|x_1|+\mathrm{\Omega }{\displaystyle \sum _{i=2}^{\ell }}{2}^{\ell -i+1}|x_i|-\gamma ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{q}_i+q_{\ell }),\hfill \\ x_j& =\hfill & \frac{1}{2}{\mathrm{\Omega }}^{-1}[(A_{j-1}-A_j)(|x_1|+x_1)+\gamma (|x_{j+1}|+x_{j+1})\hfill \\ & \hfill & +\gamma q_{j-1}-\gamma q_j],j=2,3,\dots ,\ell -1,\hfill \\ x_{\ell }& =\hfill & \frac{1}{2}{\mathrm{\Omega }}^{-1}[(A_{\ell -1}-A_{\ell })(|x_1|+x_1)+\gamma q_{\ell -1}-\gamma q_{\ell }],\hfill \end{array}\right.$$

(3)

see [8] for more details. Based on (2) and (3), the MMS method is presented as follows:

Method 1

([8]). MMS method for VLCP${}_{\ell }$

Let$A_i=F_i-G_i(i=1,2,\dots ,\ell )$be ℓ splittings of$A_i\in {\mathbb{R}}^{n\times n}$,$\mathrm{\Omega }\in {\mathbb{R}}^{n\times n}$be a diagonal matrix with positive diagonal entries and γ be a positive constant. Given$x_1^{(0)}\in {\mathbb{R}}^n$, for$k=0,1,2,\dots$, compute$x_2^{(k)},\dots ,x_{\ell }^{(k)}$by

$$\left\{\begin{array}{ccc}{x}_{\ell }^{(k)}& =\hfill & \frac{1}{2}{\mathrm{\Omega }}^{-1}[(A_{\ell -1}-A_{\ell })(|x_1^{(k)}|+x_1^{(k)})+\gamma q_{\ell -1}-\gamma q_{\ell }],\hfill \\ x_j^{(k)}& =\hfill & \frac{1}{2}{\mathrm{\Omega }}^{-1}[(A_{j-1}-A_j)(|x_1^{(k)}|+x_1^{(k)})+\gamma (|x_{j+1}^{(k)}|+x_{j+1}^{(k)})+\gamma q_{j-1}-\gamma q_j],\hfill \\ & \hfill & j=\ell -1,\ell -2,\dots ,2.\hfill \end{array}\right.$$

(4)

and compute$x_1^{(k+1)}\in {\mathbb{R}}^n$by

$$\begin{array}{lll}& & (2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i+F_{\ell })x_1^{(k+1)}\\ & =& ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i+G_{\ell })x_1^{(k)}+(2^{\ell -1}\mathrm{\Omega }-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i-A_{\ell })|x_1^{(k)}|\\ & & +\mathrm{\Omega }{\displaystyle \sum _{i=2}^{\ell }}{2}^{\ell -i+1}|x_i^{(k)}|-\gamma ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{q}_i+q_{\ell }).\end{array}$$

Then, set

$$\left\{\begin{array}{lll}{z}^{(k+1)}& =& \frac{1}{\gamma }(|x_1^{(k+1)}|+x_1^{(k+1)}),\\ w_j^{(k+1)}& =& {\displaystyle \sum _{i=1}^j}\frac{\mathrm{\Omega }}{\gamma }(|x_i^{(k+1)}|-x_i^{(k+1)})+\frac{\mathrm{\Omega }}{\gamma }(|x_{j+1}^{(k+1)}|+x_{j+1}^{(k+1)}),j=1,2,\dots ,\ell -1,\\ w_{\ell }^{(k+1)}& =& {\displaystyle \sum _{i=1}^{\ell }}\frac{\mathrm{\Omega }}{\gamma }(|x_i^{(k+1)}|-x_i^{(k+1)}),\end{array}\right.$$

(5)

until the iteration is convergent.

In order to make full use of the information in the system matrices, by the two-step matrix splitting technique, we construct the two-step modulus-based matrix splitting (TMMS) iteration method as follows:

Method 2.

TMMS method for VLCP${}_{\ell }$

Let$\mathrm{\Omega }\in {\mathbb{R}}^{n\times n}$be a diagonal matrix with positive diagonal entries, γ be a positive constant, and$A_i=F_i^{(t)}-G_i^{(t)}(t=1,2)$be two splittings of$A_i(i=1,2,\dots ,\ell )$. Given an initial vector$x_1^{(0)}\in {\mathbb{R}}^n$, for$k=0,1,2,\dots$, compute$x_1^{(k+1)}\in {\mathbb{R}}^n$by

$$\left\{\begin{array}{lll}(2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i^{(1)}+F_{\ell }^{(1)})x_1^{(k+\frac{1}{2})}& =& ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i^{(1)}+G_{\ell }^{(1)})x_1^{(k)}+(2^{\ell -1}\mathrm{\Omega }\\ & & -{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i-A_{\ell })|x_1^{(k)}|+\mathrm{\Omega }{\displaystyle \sum _{i=2}^{\ell }}{2}^{\ell -i+1}|x_i^{(k)}|\\ & & -\gamma ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{q}_i+q_{\ell }),\\ (2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i^{(2)}+F_{\ell }^{(2)})x_1^{(k+1)}& =& ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i^{(2)}+G_{\ell }^{(2)})x_1^{(k+\frac{1}{2})}+(2^{\ell -1}\mathrm{\Omega }\\ & & -{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i-A_{\ell })|x_1^{(k+\frac{1}{2})}|+\mathrm{\Omega }{\displaystyle \sum _{i=2}^{\ell }}{2}^{\ell -i+1}|x_i^{(k+\frac{1}{2})}|\\ & & -\gamma ({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{q}_i+q_{\ell }),\end{array}\right.$$

(6)

where$x_2^{(k)},\dots ,x_{\ell }^{(k)}$are computed by (4). Then, set the same sequences as (5) until the iteration is convergent.

Clearly, if we take $F_i^{(1)}-G_i^{(1)}=F_i^{(2)}-G_i^{(2)}$, Method 2 reduces to Method 1. For the simplest case, when $\ell =2$, by (4), the main iteration (6) reduces to

$$\begin{array}{l}\left\{\begin{array}{lll}(2\mathrm{\Omega }+F_1^{(1)}+F_2^{(1)})x_1^{(k+\frac{1}{2})}& =& (G_1^{(1)}+G_2^{(1)})x_1^{(k)}+(2\mathrm{\Omega }-A_1-A_2)|x_1^{(k)}|\\ & & +|(A_1-A_2)(|x_1^{(k)}|+x_1^{(k)})+\gamma q_1-\gamma q_2|\\ & & -\gamma (q_1+q_2),\\ (2\mathrm{\Omega }+F_1^{(2)}+F_2^{(2)})x_1^{(k+1)}& =& (G_1^{(2)}+G_2^{(2)})x_1^{(k+\frac{1}{2})}+(2\mathrm{\Omega }-A_1-A_2)|x_1^{(k+\frac{1}{2})}|\\ & & +|(A_1-A_2)(|x_1^{(k+\frac{1}{2})}|+x_1^{(k+\frac{1}{2})})+\gamma q_1-\gamma q_2|\\ & & -\gamma (q_1+q_2),\end{array}\right.\end{array}$$

(7)

Moreover, by specifically choosing the matrix splittings of the system matrices, one can obtain some TMMS relaxation methods. For $i=1,2,\dots ,\ell$, taking

$$\left\{\begin{array}{l}{F}_i^{(1)}=\frac{1}{\alpha }(D_{A_i}^{(1)}-\beta L_{A_i}^{(1)}),G_i^{(1)}=F_i^{(1)}-A_i,\\ F_i^{(2)}=\frac{1}{\alpha }(D_{A_i}^{(2)}-\beta U_{A_i}^{(2)}),G_i^{(2)}=F_i^{(2)}-A_i,\end{array}\right.$$

(8)

we can obtain the two-step modulus-based accelerated overrelaxation (TMAOR) iteration method. Taking $(\alpha ,\beta )=(\alpha ,\alpha )$, $(\alpha ,\beta )=(1,1)$ and $(\alpha ,\beta )=(1,0)$, the TMAOR reduces to the two-step modulus-based successive overrelaxation (TMSOR), Gauss–Seidel (TMMGS) and Jacobi (TMJ) iteration methods, respectively.

3. Convergence Analysis

Lemma 3

([20]). Assume that A is a Z-matrix. Then, the following three statements are equivalent:

(1) 

A is a nonsingular M-matrix;

(2) 

There exists a diagonal matrix D with positive diagonal entries, such that$AD$is an s.d.d. matrix with positive diagonal entries.

(3) 

If$A=F-G$satisfy$F^{-1}\ge 0$and$G\ge 0$, then$\rho (F^{-1}G)<1$.

Lemma 4

([23]). Let A be an H-matrix. Then $|A^{-1}{|\le ⟨A⟩}^{-1}$.

Lemma 5

([24]). Let $B\in {\mathbb{R}}^{n\times n}$ be an s.d.d. matrix. Then, $\forall C\in {\mathbb{R}}^{n\times n}$,

$$||B^{-1}C{||}_{\infty }\le \underset{1\le i\le n}{max}\frac{{(|C|e)}_i}{{(⟨B⟩e)}_i}.$$

We first give the convergence result for $\ell =2$.

Theorem 6.

Let$A_1,A_2$and all their row-representative matrices be$H_{+}$-matrices. Let$A_i=F_i^{(t)}-G_i^{(t)}(t=1,2)$be two splittings of$A_i(i=1,2)$. Assume that:

(I) 

For$t=1,2$,$A_1=F_1^{(t)}-G_1^{(t)}$is a splitting of$A_1$satisfying$D_{F_1^{(t)}}>0$, and$A_2=F_2^{(t)}-G_2^{(t)}$is an H-splitting of$A_2$;

(II) 

For$t=1,2$,$⟨F_1^{(t)}⟩\ge ⟨F_2^{(t)}⟩$and$|G_2^{(t)}|\ge |G_1^{(t)}|$;

(III) 

There exists a positive diagonal matrix T with positive diagonal entries such that both$(⟨F_2^{(1)}⟩-|G_2^{(1)}|)T$and$(⟨F_2^{(2)}⟩-|G_2^{(2)}|)T$are s.d.d. matrices;

(IV) 

$$\mathrm{\Omega }Te>\underset{t=1,2}{max}\{[\frac{1}{2}(D_{F_1^{(t)}}+D_{F_2^{(t)}})-(⟨F_2^{(t)}⟩-|G_2^{(t)}|)]Te\}.$$

(9)

Then, Method 2 converges to the solution of the VLCP${}_2$.

Proof. 

For $t=1,2$, by Assumptions (II) and (III), we have

$$\begin{array}{lll}& & ⟨2\mathrm{\Omega }+F_1^{(t)}+F_2^{(t)}⟩Te\\ & >& ⟨F_1^{(t)}+F_2^{(t)}⟩Te\\ & \ge & (⟨F_1^{(t)}⟩+⟨F_2^{(t)}⟩)Te\\ & \ge & 2⟨F_2^{(t)}⟩Te\\ & \ge & (2⟨F_2^{(t)}⟩-|G_2^{(t)}|)Te\\ & >& 0.\end{array}$$

Therefore, $⟨2\mathrm{\Omega }+F_1^{(t)}+F_2^{(t)}⟩T$ is an s.d.d matrix, which implies that $2\mathrm{\Omega }+F_1^{(t)}+F_2^{(t)}$ is an H-matrix. Then, by Lemma 4, we have the error at iteration $k+1$:

Let $x_1^{*}$ be the solution of (3). By the first equation of (7), we can obtain the error at the iteration $(k+1)$:

$$\left\{\begin{array}{lll}(2\mathrm{\Omega }+F_1^{(1)}+F_2^{(1)})(x_1^{(k+\frac{1}{2})}-x_1^{*})& =& (G_1^{(1)}+G_2^{(1)})(x_1^{(k)}-x_1^{*})+(2\mathrm{\Omega }-A_1-A_2)\\ & & \times (|x_1^{(k)}|-|x_1^{*}|)+|(A_1-A_2)(|x_1^{(k)}|+x_1^{(k)})\\ & & +\gamma q_1-\gamma q_2|-|(A_1-A_2)(|x_1^{*}|+x_1^{*})\\ & & +\gamma q_1-\gamma q_2|,\\ (2\mathrm{\Omega }+F_1^{(2)}+F_2^{(2)})(x_1^{(k+1)}-x_1^{*})& =& (G_1^{(2)}+G_2^{(2)})(x_1^{(k+\frac{1}{2})}-x_1^{*})+(2\mathrm{\Omega }-A_1-A_2)\\ & & \times (|x_1^{(k+\frac{1}{2})}|-|x_1^{*}|)+|(A_1-A_2)(|x_1^{(k+\frac{1}{2})}|+x_1^{(k+\frac{1}{2})})\\ & & +\gamma q_1-\gamma q_2|-|(A_1-A_2)(|x_1^{*}|+x_1^{*})\\ & & +\gamma q_1-\gamma q_2|,\end{array}\right.$$

Then, by Lemma 4, we obtain

$$|x_1^{(k+1)}-x_1^{*}|\le {\mathcal{P}}^{(2)}{\mathcal{P}}^{(1)}|x_1^{(k)}-x_1^{*}|,$$

(10)

where

$$\left\{\begin{array}{ll}{\mathcal{P}}^{(t)}& ={{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)},\\ {\mathcal{F}}^{(t)}& =⟨2\mathrm{\Omega }+F_1^{(t)}+F_2^{(t)}⟩,\\ {\mathcal{G}}^{(t)}& =|G_1^{(t)}+G_2^{(t)}|+|2\mathrm{\Omega }-A_1-A_2|+2|A_1-A_2|.\end{array}\right.$$

By Lemma 5, we have

$$||T^{-1}{\mathcal{P}}^{(t)}{T||}_{\infty }=||T^{-1}{{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)}{T||}_{\infty }=||{({\mathcal{F}}^{(t)}T)}^{-1}({\mathcal{G}}^{(t)}T){||}_{\infty }\le \underset{1\le i\le n}{max}\frac{{({\mathcal{G}}^{(t)}Te)}_i}{{({\mathcal{F}}^{(t)}Te)}_i}.$$

(11)

Still considering assumption (II), we can obtain

$$\left\{\begin{array}{l}|G_1^{(t)}+G_2^{(t)}|+|G_1^{(t)}-G_2^{(t)}|=2|G_2^{(t)}|,\\ |C_{F_1^{(t)}}+C_{F_2^{(t)}}|+|C_{F_1^{(t)}}-C_{F_2^{(t)}}|=2|C_{F_2^{(t)}}|,\\ |D_{F_1^{(t)}}-D_{F_2^{(t)}}|=D_{F_1^{(t)}}-D_{F_2^{(t)}}.\end{array}\right.$$

(12)

Then, we have

$$\begin{array}{lll}& & {\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\\ \hfill & =& (⟨2\mathrm{\Omega }+F_1^{(t)}+F_2^{(t)}⟩-|G_1^{(t)}+G_2^{(t)}|-|2\mathrm{\Omega }-A_1-A_2|-2|A_1-A_2|)Te\\ \hfill & =& [2\mathrm{\Omega }+D_{F_1^{(t)}}+D_{F_2^{(t)}}-|C_{F_1^{(t)}}+C_{F_2^{(t)}}|-|G_1^{(t)}+G_2^{(t)}|\\ \hfill & & -|2\mathrm{\Omega }-D_{F_1^{(t)}}-D_{F_2^{(t)}}-C_{F_1^{(t)}}-C_{F_2^{(t)}}+G_1^{(t)}+G_2^{(t)}|\\ \hfill & & -2|D_{F_1^{(t)}}+C_{F_1^{(t)}}-G_1^{(t)}-(D_{F_2^{(t)}}+C_{F_2^{(t)}}-G_2^{(t)})|]Te\\ \hfill & \ge & (2\mathrm{\Omega }+D_{F_1^{(t)}}+D_{F_2^{(t)}}-|C_{F_1^{(t)}}+C_{F_2^{(t)}}|-|G_1^{(t)}+G_2^{(t)}|\\ \hfill & & -|2\mathrm{\Omega }-D_{F_1^{(t)}}-D_{F_2^{(t)}}|-|C_{F_1^{(t)}}+C_{F_2^{(t)}}|+2|G_1^{(t)}+G_2^{(t)}|\\ \hfill & & -2|D_{F_1^{(t)}}-D_{F_2^{(t)}}|-2|C_{F_1^{(t)}}-C_{F_2^{(t)}}|-2|G_1^{(t)}-G_2^{(t)}|)Te\\ \hfill & =& (2\mathrm{\Omega }+3D_{F_2^{(t)}}-D_{F_1^{(t)}}-|2\mathrm{\Omega }-D_{F_1^{(t)}}-D_{F_2^{(t)}}|\\ \hfill & & -2|G_1^{(t)}+G_2^{(t)}|-2|G_1^{(t)}-G_2^{(t)}|-2|C_{F_1^{(t)}}+C_{F_2^{(t)}}|-2|C_{F_1^{(t)}}-C_{F_2^{(t)}}|)Te\\ \hfill & =& (2\mathrm{\Omega }+3D_{F_2^{(t)}}-D_{F_1^{(t)}}-|2\mathrm{\Omega }-D_{F_1^{(t)}}-D_{F_2^{(t)}}|-4|G_2^{(t)}|-4|C_{F_2^{(t)}}|)Te,\end{array}$$

(13)

where the last two equalities hold by (12).

When

$$\mathrm{\Omega }\ge \frac{1}{2}(D_{F_1^{(t)}}+D_{F_2^{(t)}}),$$

by (13), we have

$$\begin{array}{lll}& & {\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\\ \hfill & \ge & [2\mathrm{\Omega }+3D_{F_2^{(t)}}-D_{F_1^{(t)}}-(2\mathrm{\Omega }-D_{F_1^{(t)}}-D_{F_2^{(t)}})-4|G_2^{(t)}|-4|C_{F_2^{(t)}}|]Te\\ \hfill & =& (4D_{F_2^{(t)}}-4|G_2^{(t)}|-4|C_{F_2^{(t)}}|)Te\\ \hfill & =& 4(⟨F_2^{(t)}⟩-|G_2^{(t)}|)Te>0.\end{array}$$

(14)

When

$$[\frac{1}{2}(D_{F_1^{(t)}}+D_{F_2^{(t)}})-(⟨F_2^{(t)}⟩-|G_2^{(t)}|)]Te<\mathrm{\Omega }Te<\frac{1}{2}(D_{F_1^{(t)}}+D_{F_2^{(t)}})Te,$$

by (13), we have

$$\begin{array}{lll}& & {\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\\ \hfill & \ge & [2\mathrm{\Omega }+3D_{F_2^{(t)}}-D_{F_1^{(t)}}-(D_{F_1^{(t)}}+D_{F_2^{(t)}}-2\mathrm{\Omega })-4|G_2^{(t)}|-4|C_{F_2^{(t)}}|]Te\\ \hfill & =& (4\mathrm{\Omega }+2D_{F_2^{(t)}}-2D_{F_1^{(t)}}-4|G_2^{(t)}|-4|C_{F_2^{(t)}|})Te\\ \hfill & =& 4[\mathrm{\Omega }-\frac{1}{2}(D_{F_1^{(t)}}+D_{F_2^{(t)}})+⟨F_2^{(t)}⟩-|G_2^{(t)}|]Te>0.\end{array}$$

(15)

Combining (14) and (15), we have ${\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te>0$ provided that the assumption (IV) holds. Then, by (11), we have $||T^{-1}{{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)}T{||}_{\infty }<1$. Then, we have the next inequality:

$$||\rho ({\mathcal{P}}^{(2)}{\mathcal{P}}^{(1)})||\le ||T^{-1}{\mathcal{P}}^{(2)}{\mathcal{P}}^{(1)}{T||}_{\infty }\le ||T^{-1}{\mathcal{P}}^{(2)}{T||}_{\infty }||T^{-1}{\mathcal{P}}^{(1)}T{||}_{\infty }<1,$$

which implies that $\underset{k\to +\infty }{lim}{x}_1^{(k)}=x_1^{*}$ by (10), ending the proof. □

Remark 7.

By the proof of Theorem 6, (II) is relevant in the derivation of the first chain of inequalities. A simple example can be given to make readers understand it easily. Let

$$A_1=\left(\begin{array}{cccc}4& -1& & \\ -1& 4& & \\ & & 4& -1\\ & & -1& 4\end{array}\right)\mathit{and}{A}_2=\left(\begin{array}{cccc}4& -1& -1& \\ -1& 4& & -1\\ -1& & 4& -1\\ & -1& -1& 4\end{array}\right).$$

Consider the two-step SOR splitting, where

$$F_1^{(1)}=\left(\begin{array}{cccc}\frac{4}{\alpha }& & & \\ -1& \frac{4}{\alpha }& & \\ & & \frac{4}{\alpha }& \\ & & -1& \frac{4}{\alpha }\end{array}\right),G_1^{(1)}=\left(\begin{array}{cccc}\frac{4}{\alpha }-4& 1& & \\ & \frac{4}{\alpha }-4& & \\ & & \frac{4}{\alpha }-4& 1\\ & & & \frac{4}{\alpha }-4\end{array}\right),$$

$$F_1^{(2)}=\left(\begin{array}{cccc}\frac{4}{\alpha }& -1& & \\ & \frac{4}{\alpha }& & \\ & & \frac{4}{\alpha }& -1\\ & & & \frac{4}{\alpha }\end{array}\right),G_1^{(2)}=\left(\begin{array}{cccc}\frac{4}{\alpha }-4& & & \\ 1& \frac{4}{\alpha }-4& & \\ & & \frac{4}{\alpha }-4& \\ & & 1& \frac{4}{\alpha }-4\end{array}\right),$$

$$F_2^{(1)}=\left(\begin{array}{cccc}\frac{4}{\alpha }& & & \\ -1& \frac{4}{\alpha }& & \\ -1& & \frac{4}{\alpha }& \\ & -1& -1& \frac{4}{\alpha }\end{array}\right),G_2^{(1)}=\left(\begin{array}{cccc}\frac{4}{\alpha }-4& 1& 1& \\ & \frac{4}{\alpha }-4& & 1\\ & & \frac{4}{\alpha }-4& 1\\ & & & \frac{4}{\alpha }-4\end{array}\right),$$

$$F_2^{(2)}=\left(\begin{array}{cccc}\frac{4}{\alpha }& -1& -1& \\ & \frac{4}{\alpha }& & -1\\ & & \frac{4}{\alpha }& -1\\ & & & \frac{4}{\alpha }\end{array}\right),G_2^{(2)}=\left(\begin{array}{cccc}\frac{4}{\alpha }-4& & & \\ 1& \frac{4}{\alpha }-4& & \\ 1& & \frac{4}{\alpha }-4& \\ & 1& 1& \frac{4}{\alpha }-4\end{array}\right).$$

Clearly, for the matrices presented above, (II) is satisfied. Moreover, by simple computation, one can easily determine that$⟨A_1⟩\ge ⟨A_2⟩$with two-step AOR splittings is a sufficient condition of (II).

Remark 8.

If we take$F_i^{(1)}-G_i^{(1)}=F_i^{(2)}-G_i^{(2)}$, all the assumptions in Theorem 6 can reduce to those in Theorem 4.1 of [8]. Clearly, (IV) is weaker than the corresponding one in Theorem 4.1 of [8], where Ω was assumed to satisfy$\mathrm{\Omega }\ge \frac{1}{2}(D_{F_1}+D_{F_2})$. On the other hand,$2\mathrm{\Omega }+F_1^{(t)}+F_2^{(t)}$is proved to be an H-matrix in Theorem 6, not set to be an assumption as that in [8].

Remark 9.

In view of the assumptions in Theorem 6, Assumption (III) seems to be a special one. In fact, for some special cases, the matrix T given in the Assumption (III) can be computed. Taking the TMAOR method where the matrix splittings are given by (8), for example, we have

$$⟨F_2^{(1)}⟩-|G_2^{(1)}|=⟨F_2^{(2)}⟩-|G_2^{(2)}|=\frac{1-|1-\alpha |}{\alpha }{D}_{A_2}-|C_{A_2}|.$$

Since$A_2$is an$H_{+}$-matrix, by Lemma 3, we have$\rho (D_{A_2}^{-1}|C_{A_2}|)<1$. By simple computation, if$0<\beta \le \alpha <\frac{2}{1+\rho (D_{A_2}^{-1}|C_{A_2}|)}$, we can obtain$\frac{1-|1-\alpha |}{\alpha }{D}_{A_2}-|C_{A_2}|$is an M-matrix. Then, letting

$$T=\mathrm{diag}[(\frac{1-|1-\alpha |}{\alpha }{D}_{A_2}-|C_{A_2}|{)}^{-1}e],$$

(16)

Assumption (III) of Theorem 6 can be satisfied.

By the similar proof technique, we can obtain the convergence theorem for VLCP${}_{\ell }(\ell \ge 3)$. We then first show the idea of the proof when $\ell =3$.

First, by (6) and the first equation of (3), we can determine the error at the iteration $(k+1)$:

$$\left\{\begin{array}{lll}|x_1^{(k+\frac{1}{2})}-x_1^{*}|& \le & |4\mathrm{\Omega }+2F_1^{(1)}+F_2^{(1)}+F_3^{(1)}{|}^{-1}(|2G_1^{(1)}+G_2^{(1)}+G_3^{(1)}|\\ & & +|4\mathrm{\Omega }-2A_1-A_2-A_3|+2|2A_1-A_2-A_3|+4|A_2-A_3|)\\ & & \times |x_1^{(k)}-x_1^{*}|,\\ |x_1^{(k+1)}-x_1^{*}|& \le & |4\mathrm{\Omega }+2F_1^{(2)}+F_2^{(2)}+F_3^{(2)}{|}^{-1}(|2G_1^{(2)}+G_2^{(2)}+G_3^{(2)}|\\ & & +|4\mathrm{\Omega }-2A_1-A_2-A_3|+2|2A_1-A_2-A_3|+4|A_2-A_3|)\\ & & \times |x_1^{(k+\frac{1}{2})}-x_1^{*}|,\end{array}\right.$$

If there exists a diagonal matrix T with positive diagonal entries such that $(⟨F_3^{(t)}⟩-|G_3^{(t)}|)T,t=1,2,$ are s.d.d. matrices, we obtain

$$|x_1^{(k+1)}-x_1^{*}|\le {\mathcal{P}}^{(2)}{\mathcal{P}}^{(1)}|x_1^{(k)}-x_1^{*}|,$$

where

$$\left\{\begin{array}{ll}{\mathcal{P}}^{(t)}& ={{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)},\\ {\mathcal{F}}^{(t)}& =⟨4\mathrm{\Omega }+2F_1^{(t)}+F_2^{(t)}+F_3^{(t)}⟩,\\ {\mathcal{G}}^{(t)}& =|2G_1^{(t)}+G_2^{(t)}+G_3^{(t)}|+|4\mathrm{\Omega }-2A_1-A_2-A_3|+4|A_2-A_3|.\end{array}\right.$$

If $2⟨F_1^{(t)}⟩\ge ⟨F_2^{(t)}+F_3^{(t)}⟩,⟨F_2^{(t)}⟩\ge ⟨F_3^{(t)}⟩,2|G_1^{(t)}|\le |G_2^{(t)}+G_3^{(t)}|$, and $|G_2^{(t)}|\le |G_3^{(t)}|$ hold, we can also have that $⟨4\mathrm{\Omega }+2F_1^{(t)}+F_2^{(t)}+F_3^{(t)}⟩T$ is an s.d.d. matrix and

$$||T^{-1}{{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)}T{||}_{\infty }\le \underset{1\le i\le n}{max}\frac{{({\mathcal{G}}^{(t)}Te)}_i}{{({\mathcal{F}}^{(t)}Te)}_i}.$$

Similarly to (13), we can obtain

$$\begin{array}{lll}& & {\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\\ & \ge & (4\mathrm{\Omega }-2D_{F_1^{(t)}}-D_{F_2^{(t)}}+7D_{F_3^{(t)}}-|4\mathrm{\Omega }-2D_{F_1^{(t)}}-D_{F_2^{(t)}}-D_{F_3^{(t)}}|-8|G_3^{(t)}|-8|C_{F_3^{(t)}}|)Te.\end{array}$$

Then, we can also distinguish two cases with respect to $\mathrm{\Omega }$, where

$$\mathrm{\Omega }\ge \frac{1}{4}(2D_{F_1^{(t)}}+D_{F_2^{(t)}}+D_{F_3^{(t)}})$$

and

$$[\frac{1}{4}(2D_{F_1^{(t)}}+D_{F_2^{(t)}}+D_{F_3^{(t)}})-(⟨F_3^{(t)}⟩-|G_3^{(t)}|)]Te<\mathrm{\Omega }Te<[\frac{1}{4}(2D_{F_1^{(t)}}+D_{F_2^{(t)}}+D_{F_3^{(t)}})]Te,$$

and obtain

$${\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\ge 8(⟨F_3^{(t)}⟩-|G_3^{(t)}|)Te>0$$

and

$${\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te=8[\mathrm{\Omega }-\frac{1}{4}(2D_{F_1^{(t)}}+D_{F_2^{(t)}}+D_{F_3^{(t)}})+⟨F_3^{(t)}⟩-|G_3^{(t)}|]Te>0,$$

respectively.

In summary, we have the next result.

Theorem 10.

Let$A_1,A_2,A_3$and all their row-representative matrices be$H_{+}$-matrices. Let$A_i=F_i^{(t)}-G_i^{(t)}(t=1,2)$be two splittings of$A_i(i=1,2,3)$. Assume that:

(I) 

$D_{F_1^{(t)}}>0$,$D_{F_2^{(t)}}>0$, and$A_3=F_3^{(t)}-G_3^{(t)}$are an H-splitting of$A_3$;

(II) 

$2⟨F_1^{(t)}⟩\ge ⟨F_2^{(t)}+F_3^{(t)}⟩,⟨F_2^{(t)}⟩\ge ⟨F_3^{(t)}⟩,2|G_1^{(t)}|\le |G_2^{(t)}+G_3^{(t)}|$, and$|G_2^{(t)}|\le |G_3^{(t)}|$;

(III) 

There exists a diagonal matrix T with positive diagonal entries such that$(⟨F_3^{(t)}⟩-|G_3^{(t)}|)T,t=1,2,$are s.d.d. matrices;

(IV) 

$\mathrm{\Omega }Te\ge [\frac{1}{4}(2D_{F_1^{(t)}}+D_{F_2^{(t)}}+D_{F_3^{(t)}})-(⟨F_3^{(t)}⟩-|G_3^{(t)}|)]Te$.

Then, Method 2 converges to the solution of the VLCP${}_3$.

Furthermore, by deduction, for a general ℓ, we can also show the main steps of the proof.

In fact, the errors at the iteration $(k+1)$ are

$$\left\{\begin{array}{lll}|x_1^{(k+\frac{1}{2})}-x_1^{*}|& \le & |2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i^{(1)}+F_{\ell }^{(1)}{|}^{-1}(|{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i^{(1)}+G_{\ell }^{(1)}|\\ & & +|2^{\ell -1}\mathrm{\Omega }-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i-A_{\ell }|\\ & & +2[|A_{\ell -1}-A_{\ell }|+{\displaystyle \sum _{j=2}^{\ell -2}}{2}^{\ell -j-1}|2^{j-1}{A}_{\ell -j}-{\displaystyle \sum _{s=\ell -j+1}^{\ell -1}}{2}^{\ell -s-1}{A}_s-A_{\ell }|\\ & & +|2^{\ell -2}{A}_1-{\displaystyle \sum _{s=2}^{\ell -1}}{2}^{\ell -s-1}{A}_s-A_{\ell }|])|x_1^{(k)}-x_1^{*}|,\\ |x_1^{(k+1)}-x_1^{*}|& \le & |2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i^{(2)}+F_{\ell }^{(2)}{|}^{-1}(|{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i^{(2)}+G_{\ell }^{(2)}|\\ & & +|2^{\ell -1}\mathrm{\Omega }-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i-A_{\ell }|\\ & & +2[|A_{\ell -1}-A_{\ell }|+{\displaystyle \sum _{j=2}^{\ell -2}}{2}^{\ell -j-1}|2^{j-1}{A}_{\ell -j}-{\displaystyle \sum _{s=\ell -j+1}^{\ell -1}}{2}^{\ell -s-1}{A}_s-A_{\ell }|\\ & & +|2^{\ell -2}{A}_1-{\displaystyle \sum _{s=2}^{\ell -1}}{2}^{\ell -s-1}{A}_s-A_{\ell }|])|x_1^{(k+\frac{1}{2})}-x_1^{*}|.\end{array}\right.$$

If there exists a diagonal matrix T with positive diagonal entries such that $(⟨F_{\ell }^{(t)}⟩-|G_{\ell }^{(t)}|)T,t=1,2,$ are s.d.d. matrices, we obtain

$$|x_1^{(k+1)}-x_1^{*}|\le {\mathcal{P}}^{(2)}{\mathcal{P}}^{(1)}|x_1^{(k)}-x_1^{*}|,$$

where

$$\left\{\begin{array}{lll}{\mathcal{P}}^{(t)}& =& {{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)},\\ {\mathcal{F}}^{(t)}& =& ⟨2^{\ell -1}\mathrm{\Omega }+{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{F}_i^{(t)}+F_{\ell }^{(t)}⟩,\\ {\mathcal{G}}^{(t)}& =& |{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{G}_i^{(1)}+G_{\ell }^{(1)}|+|2^{\ell -1}\mathrm{\Omega }-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{A}_i-A_{\ell }|\\ & & +2[|A_{\ell -1}-A_{\ell }|+{\displaystyle \sum _{j=2}^{\ell -2}}{2}^{\ell -j-1}|2^{j-1}{A}_{\ell -j}-{\displaystyle \sum _{s=\ell -j+1}^{\ell -1}}{2}^{\ell -s-1}{A}_s-A_{\ell }|\\ & & +|2^{\ell -2}{A}_1-{\displaystyle \sum _{s=2}^{\ell -1}}{2}^{\ell -s-1}{A}_s-A_{\ell }|].\end{array}\right.$$

If

$$\left\{\begin{array}{l}{2}^{\ell -j}⟨F_{j-1}^{(t)}⟩\ge ⟨{\displaystyle \sum _{i=j}^{\ell -1}}{2}^{\ell -i-1}{F}_i^{(t)}+F_{\ell }^{(t)}⟩,(j=2,3,\dots ,\ell -1)\\ ⟨F_{\ell -1}^{(t)}⟩\ge ⟨F_{\ell }^{(t)}⟩,\end{array}\right.$$

(17)

and

$$\left\{\begin{array}{l}{2}^{\ell -j}|G_j^{(t)}|\le |{\displaystyle \sum _{i=j}^{\ell -1}}{2}^{\ell -i-1}{G}_i^{(t)}+G_{\ell }^{(t)}|,(j=2,3,\dots ,\ell -1)\\ |G_{\ell -1}^{(t)}|\le |G_{\ell }^{(t)}|;\end{array}\right.$$

(18)

hold, we can also have ${\mathcal{F}}^{(t)}T$ as an s.d.d. matrix and

$$||T^{-1}{{\mathcal{F}}^{(t)}}^{-1}{\mathcal{G}}^{(t)}T{||}_{\infty }\le \underset{1\le i\le n}{max}\frac{{({\mathcal{G}}^{(t)}Te)}_i}{{({\mathcal{F}}^{(t)}Te)}_i}.$$

Similarly to (13), we can obtain

$$\begin{array}{lll}& & {\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\\ & \ge & [2^{\ell -1}\mathrm{\Omega }+(2^{\ell }-1)D_{F_{\ell }^{(t)}}-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}-2^{\ell }|C_{F_{\ell }^{(t)}}|-2^{\ell }|G_{\ell }^{(t)}|\\ & & -|2^{\ell -1}\mathrm{\Omega }-{\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}-D_{F_{\ell }^{(t)}}|]Te.\end{array}$$

Then, we can also distinguish two cases with respect to $\mathrm{\Omega }$, where

$$\mathrm{\Omega }\ge 2^{1-\ell }({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}+D_{F_{\ell }^{(t)}})$$

and

$$\begin{array}{lll}& & [2^{1-\ell }({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}+D_{F_{\ell }^{(t)}})-(⟨F_{\ell }^{(t)}⟩-|G_{\ell }^{(t)}|)]Te\\ & <& \mathrm{\Omega }Te<2^{1-\ell }({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}+D_{F_{\ell }^{(t)}})Te,\end{array}$$

and obtain

$${\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\ge 2^{\ell }(⟨F_{\ell }^{(t)}⟩-|G_{\ell }^{(t)}|)Te>0$$

and

$${\mathcal{F}}^{(t)}Te-{\mathcal{G}}^{(t)}Te\ge [2^{\ell }\mathrm{\Omega }-2({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}+D_{F_{\ell }^{(t)}})+2^{\ell }(⟨F_{\ell }^{(t)}⟩-|G_{\ell }^{(t)}|)]Te>0,$$

respectively. Finally, we have the next theorem.

Theorem 11.

Let$A_1,A_2,\dots ,A_{\ell }$and all their row-representative matrices be$H_{+}$-matrices. Let$A_i=F_i^{(t)}-G_i^{(t)}(t=1,2)$be two splittings of$A_i(i=1,2,\dots ,\ell )$. Assume that:

(I) 

$D_{F_i^{(t)}}>0$,$i=1,2,\dots ,\ell -1$, and$A_{\ell }=F_{\ell }^{(t)}-G_{\ell }^{(t)}$are an H-splitting of$A_{\ell }$;

(II) 

(17) and (18) are satisfied;

(III) 

There exists a diagonal matrix T with positive diagonal entries such that$(⟨F_{\ell }^{(t)}⟩-|G_{\ell }^{(t)}|)T,t=1,2,$are s.d.d. matrices;

(IV) 

$\mathrm{\Omega }Te\ge [2^{1-\ell }({\displaystyle \sum _{i=1}^{\ell -1}}{2}^{\ell -i-1}{D}_{F_i^{(t)}}+D_{F_{\ell }^{(t)}})-(⟨F_{\ell }^{(t)}⟩-|G_{\ell }^{(t)}|)]Te$.

Then, Method 2 converges to the solution of the VLCP${}_{\ell }$.

Same comments as in Remarks 8 and 9 can be given for Theorems 10 and 11.

4. Numerical Examples

In this section, numerical examples are given to show the efficiency of the proposed method.

Consider the two following examples similar to [8], where Examples 12 and 13 are of the symmetry and asymmetry cases, respectively.

Example 12.

Let$n=m^2$. Consider the VLCP whose system matrices are given by

$$A_1=\left(\begin{array}{cccc}S& & & \\ & S& & \\ & & \ddots & \\ & & & S\end{array}\right)+I_n\in {\mathbb{R}}^{n\times n},A_2=\left(\begin{array}{cccc}S& -I_m& & \\ -I_m& S& \ddots & \\ & \ddots & \ddots & -I_m\\ & & -I_m& S\end{array}\right)\in {\mathbb{R}}^{n\times n},$$

where

$$S=\left(\begin{array}{cccc}4& -1& & \\ -1& 4& \ddots & \\ & \ddots & \ddots & -1\\ & & -1& 4\end{array}\right)\in {\mathbb{R}}^{m\times m}.$$

Example 13.

Let$n=m^2$. Consider the VLCP whose system matrices are given by

$$A_1=\left(\begin{array}{cccc}S& & & \\ & S& & \\ & & \ddots & \\ & & & S\end{array}\right)+I_n\in {\mathbb{R}}^{n\times n},A_2=\left(\begin{array}{cccc}S& -0.5I_m& & \\ -1.5I_m& S& \ddots & \\ & \ddots & \ddots & -0.5I_m\\ & & -1.5I_m& S\end{array}\right)\in {\mathbb{R}}^{n\times n},$$

where

$$S=\left(\begin{array}{cccc}4& -0.5& & \\ -1.5& 4& \ddots & \\ & \ddots & \ddots & -0.5\\ & & -1.5& 4\end{array}\right)\in {\mathbb{R}}^{m\times m}.$$

The numerical tests are performed on a computer, which has Intel(R) Core(TM) i7-9700 CPU 3.00 GHz with 8 GB memory. Denote the total computation time (in seconds) and the iteration steps by T and $IT$, respectively. Let $\gamma =1$, $x_1^{(0)}=e$ and the tolerance be ${10}^{-6}$. By “$SAVE$”, we denote the per centum of total computation time saved by the TMSOR method from the MSOR method, where

$$SAVE=\frac{T_{MSOR}-T_{TMSOR}}{T_{MSOR}}\times 100\%.$$

The numerical results are presented in Table 1, Table 2 and Table 3, where the notations “MSOR${}_{\alpha }$” and “TMSOR${}_{\alpha }$” denote the MSOR and TMSOR methods with relaxation parameter $\alpha$, respectively, and the parameter matrix $\mathrm{\Omega }$ is chosen as

$$\mathrm{\Omega }=\frac{\tau }{2}(D_{F_1}+D_{F_2}),$$

$\tau =0.8,0.9,1.0$.

Table 1. Numerical results when $\tau =1$ (the “optimal” computation times of the MSOR and TMSOR are set to bold for each dimension and each example).

Example	Method	$\mathit{m}=128$			$\mathit{m}=256$			$\mathit{m}=512$
		$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$	$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$	$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$
Example 12	MSOR${}_{0.9}$	48	0.1999		49	0.9145		51	5.1585
	TMSOR${}_{0.9}$	24	0.1335	33%	25	0.6761	26%	26	4.0739	21%
	MSOR${}_{1.0}$	41	0.1818		42	0.7773		44	4.3167
	TMSOR${}_{1.0}$	21	0.1149	36%	21	0.5360	31%	22	3.0526	29%
	MSOR${}_{1.1}$	35	0.1569		36	0.6566		38	3.9291
	TMSOR${}_{1.1}$	18	0.1107	29%	20	0.5843	11%	19	2.7612	30%
	MSOR${}_{1.2}$	52	0.2079		52	0.9520		54	5.5109
	TMSOR${}_{1.2}$	26	0.1374	34%	29	0.7632	20%	28	4.0793	26%
Example 13	MSOR${}_{0.9}$	42	0.1792		36	0.6613		45	4.6083
	TMSOR${}_{0.9}$	24	0.1321	26%	20	0.5607	15%	25	3.6112	22%
	MSOR${}_{1.0}$	35	0.1539		30	0.5444		38	3.8549
	TMSOR${}_{1.0}$	20	0.1317	14%	17	0.4273	22%	22	3.0377	21%
	MSOR${}_{1.1}$	30	0.1417		25	0.4919		32	3.5243
	TMSOR${}_{1.1}$	18	0.1070	24%	15	0.4131	16%	19	2.7911	21%
	MSOR${}_{1.2}$	42	0.1899		29	0.5140		45	4.5890
	TMSOR${}_{1.2}$	28	0.1828	4%	16	0.4884	5%	30	4.3746	5%

Table 2. Numerical results when $\tau =0.9$ (the “optimal” computation times of the MSOR and TMSOR are set to bold for each dimension and each example).

Example	Method	$\mathit{m}=128$			$\mathit{m}=256$			$\mathit{m}=512$
		$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$	$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$	$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$
Example 12	MSOR${}_{0.9}$	44	0.1426		46	0.8142		47	4.6364
	TMSOR${}_{0.9}$	22	0.1030	27%	23	0.6159	24%	24	3.2768	29%
	MSOR${}_{1.0}$	38	0.1378		39	0.6936		40	3.9927
	TMSOR${}_{1.0}$	19	0.0881	36%	20	0.5315	23%	20	2.8169	29%
	MSOR${}_{1.1}$	42	0.1375		41	0.7498		44	4.4324
	TMSOR${}_{1.1}$	22	0.0944	31%	24	0.6519	13%	23	3.3152	25%
	MSOR${}_{1.2}$	78	0.2284		81	1.4617		79	8.0223
	TMSOR${}_{1.2}$	37	0.1443	36%	41	1.1061	24%	40	5.7297	28%
Example 13	MSOR${}_{0.9}$	39	0.1391		33	0.5767		41	4.1800
	TMSOR${}_{0.9}$	22	0.0981	29%	19	0.4896	15%	24	3.4328	17%
	MSOR${}_{1.0}$	32	0.1116		27	0.4928		34	3.4535
	TMSOR${}_{1.0}$	19	0.0897	19%	16	0.4133	16%	20	2.8645	17%
	MSOR${}_{1.1}$	36	0.1236		25	0.4295		39	4.0004
	TMSOR${}_{1.1}$	23	0.1098	11%	14	0.3647	15%	24	3.5276	11%
	MSOR${}_{1.2}$	55	0.1740		35	0.6576		59	5.8532
	TMSOR${}_{1.2}$	41	0.1664	4%	20	0.5234	20%	44	6.2346	-6%

Table 3. Numerical results when $\tau =0.8$ (the “optimal” computation times of the MSOR and TMSOR are set to bold for each dimension and each example).

Example	Method	$\mathit{m}=128$			$\mathit{m}=256$			$\mathit{m}=512$
		$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$	$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$	$\mathit{IT}$	$\mathit{T}$	$\mathit{SAVE}$
Example 12	MSOR${}_{0.9}$	41	0.1400		42	0.7212		44	4.645
	TMSOR${}_{0.9}$	21	0.0977	30%	21	0.5731	21%	22	3.3496	28%
	MSOR${}_{1.0}$	35	0.1218		36	0.6060		38	3.9238
	TMSOR${}_{1.0}$	18	0.0853	29%	20	0.5141	15%	20	2.9174	26%
	MSOR${}_{1.1}$	59	0.1803		59	0.9876		61	6.3516
	TMSOR${}_{1.1}$	29	0.1177	35%	32	0.8357	15%	31	4.6185	27%
	MSOR${}_{1.2}$	184	0.5213		192	3.3582		184	18.8629
	TMSOR${}_{1.2}$	80	0.3141	40%	83	2.3453	30%	80	11.6375	38%
Example 13	MSOR${}_{0.9}$	35	0.1238		30	0.5136		38	3.9445
	TMSOR${}_{0.9}$	20	0.0963	22%	17	0.4464	13%	22	3.2718	17%
	MSOR${}_{1.0}$	31	0.1100		24	0.4548		33	3.3400
	TMSOR${}_{1.0}$	19	0.0911	17%	15	0.3597	21%	20	2.9164	13%
	MSOR${}_{1.1}$	46	0.1566		31	0.5378		49	4.9155
	TMSOR${}_{1.1}$	32	0.1368	13%	17	0.4551	15%	34	5.0096	-2%
	MSOR${}_{1.2}$	79	0.2616		46	0.8248		85	8.5703
	TMSOR${}_{1.2}$	93	0.3573	-37%	28	0.7918	4%	93	13.7466	-60%

One can see that all methods are convergent for different dimensions. Since there are two linear systems solved in each iteration of the TMMS method, most of the number of iteration steps of the MMS method is nearly twice or a little less than twice as long as that of the TMMS method in each comparison. Meanwhile, the TMMS method converges faster than the MMS method except for a few cases. Specially, we can see that the CPU time is saved larger than 20% for most cases. Therefore, the two-step technique works for the improvement of the MMS method. On the other hand, one can see that the relaxation parameter may affect the computation efficiencies of the MMS and TMMS.

Although there are some cases of Example 13 where the values of “SAVE” are small or negative, the values of “SAVE” can be larger than 15% for the “optimal” relaxation parameters of both two examples, set to bold. Nevertheless, by Table 2 and Table 3, both the MMS and TMMS methods are convergent for all cases when $\tau <1$, which confirms the improvement of the proposed convergence theorem as Remark 8 commented. However, the theoretical analysis of the relaxation parameter is still difficult even for the LCP. It may be an interesting work in the future.

5. Concluding Remarks

The two-step splittings are successfully applied to the MMS iteration method for solving the VLCP. The convergence analysis is given where the convergence domain of the parameter matrix is larger than the existing one. Numerical results show that the proposed method can improve the convergence rate of the MMS iteration method. In two recent works [25,26], the modulus-based transformation was also used for tensor complementarity problems (TCP). One can thus expect that some accelerated technique such as two-step splittings can be also used for the TCP.