Accelerated modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

Abstract

In this paper, we reformulate a class of nonlinear complementarity problems as the implicit fixed-point equations. We demonstrate accelerated modulus-based matrix splitting iteration method. We show their convergence by assuming that the system matrix is positive definite or the splitting of the system matrix are $H_+$-compatible splitting and discuss the choice of the optimal parameter. Furthermore, we give two-step accelerated modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. Numerical experiments are presented to show the effectiveness of the method.

1 Introduction

In this paper, we consider the following nonlinear complementarity problems:

$$\begin{aligned} z\ge 0,\quad w=Az+\Psi (z)\ge 0,\quad z^Tw=0, \end{aligned}$$

(1.1)

where $z=(z_1,z_2,\ldots ,z_n)^T\in R^n$ is an unknown vector, $A=(a_{ij})\in R^{n\times n}$ is a given large and sparse matrix, $\Psi (z)$ is a nonlinear function and the notation “$\ge$” denotes the componentwise defined partial ordering between two vectors and the superscript “T” denotes the transpose of a vector.

When $\Psi (z)=q$ and $q\in R^n$, the nonlinear complementarity problems of the form (1.1) reduce to the linear complementarity problems. Many problems in scientific computing and engineering applications demand to compute solutions of complementarity problem, for example, the free boundary problem of fluid dynamics, contact problem in elasticity, economic transportation; see Ferri and Pang (1997), Cottle et al. (1992). Constructing efficient and feasible iteration methods for solving linear complementarity problem has been received widely attention. For example, the projected iterations (Hadjidimos and Tzoumas 2016; Bai 1996), the matrix muti-splitting iterations (Bai 1999; Bai and Evans 2001, 2002; Bai and Zhang 2013) and the general fixed point iterations (Dong and Jiang 2009; Mangasarian 1997; Noor 1988).

Recently, modulus-based iteration methods are very popular; see, e.g., Bai (2010), Zhang (2011), Zhang and Ren (2013), Xia and Li (2015), Huang and Ma (2016), Xie and Xu (2016), Ma and Huang (2016), Hong and Li (2016), Xu (2015) and Zheng and Yin (2013, 2014). Because these methods avoid the projections of the iterative used in the projected relaxation iterations and the general fixed-point iterations. Frommer and Mayer (1989) researched a modulus-based nonsmooth Newton’s method to the equivalent reformulation of the linear complementarity problems and established its locally quadratical convergence theory. Ma and Huang (2016) proposed modified modulus-based matrix splitting iteration method for a class of weakly nondifferentiable nonlinear complementarity problems and studies the convergence property when the system matrices are $H_+$-matrices. Zheng and Yin (2013, 2014) developed the accelerated modulus-based matrix splitting iteration method for the solution of the large sparse linear complementarity problem and derived the convergence results. Xie and Xu (2016) proved the convergence of two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems.

Inspired by the previous work, in this paper, by reformulating the nonlinear complementarity problem (1.1) as an implicit fixed point equation, we give accelerated modulus-based matrix splitting iteration methods for solving (1.1). The convergence conditions when the system is either a positive definite matrix or an $H_+$-matrix are presented. Moreover, we discuss the choice of the optimal parameter.

This paper is organized as follows: We first present modulus-based matrix splitting iteration methods for solving a class of nonlinear complementarity problems (1.1) in Sect. 2. The convergence conditions and optimal parameter when the system matrix is a positive definite matrix is presented in Sect. 3. In Sect. 4, we derive the convergence theory when the system is an $H_+$-matrix. And the optimal parameter of AMAOR method is established in this section. The numerical experiments of the proposed methods are shown and analyzed in Sect. 5 and some conclusions are given in Sect. 6.

2 Accelerated modulus-based matrix splitting iteration method

In this section, by reformulating the problem (1.1) as an implicit fixed point equation based on the splitting of A, we present accelerated modulus-based matrix splitting iteration method.

We first give some definitions, notations and lemmas used in the sequel convergence analysis. For the matrices $A=(a_{ij})$, $B=(b_{ij})\in R^{n\times n}$, we call $A\ge B~(A>B)$, if $a_{ij}\ge b_{ij}~(a_{ij}>b_{ij})$ holds for all $1\le i\le n$, $1\le j\le n$. |A| denotes a nonnegative matrix with entries $|a_{ij}|$. If O is a null matrix and $A\ge O~(A>O)$, A is called a nonnegative matrix. We write $\Vert A\Vert$ and $A^{-1}$ to denote norm and the inverse of matrix A, respectively. $|x|=(|x_1|,|x_2|,\ldots , |x_n|)^T\in R^n$ denotes the absolute value of the vector x . I denotes the identity matrix of the proper size implied by context.

Let $Z^{n\times n}$ denote the set of all real $n\times n$ matrices having all nonpositive off-diagonal entries. A nonsingular matrix $A\in R^{n\times n}$ is called an M-matrix Berman and Plemmons (1979) if $A\in Z^{n\times n}$ and $A^{-1}\ge 0$. Matrix $A\in R^{n\times n}$ is called an H-matrix if its comparison matrix $\langle A\rangle =(\widetilde{a}_{ij})\in R^{n\times n}$ is an M-matrix, where

$$\begin{aligned} \widetilde{a}_{ij}=\left\{ \begin{array}{ll} |a_{ii}|, &{}\quad \mathrm {for}~ i=j\\ -|a_{ij}|, &{}\quad \mathrm {for}~ i\ne j \end{array}\quad i,j=1,2,\ldots ,n. \right. \end{aligned}$$

In particular, an H-matrix is called an $H_{+}$-matrix if the diagonal entries are all positive.

Let $\sigma (A)$ and $\rho (A)$ be the spectrum and the spectral radius of the matrix A, respectively. For a given matrix $A\in R^{n\times n}$, $A=M-N$ is called a splitting of the matrix A if M is nonsingular; a convergence splitting if $\rho (M^{-1}N)<1$; an M-splitting if M is a nonsingular M-matrix and $N\ge O$; and an H-compatible splitting if $\langle A\rangle =\langle M\rangle -|N|$. Clearly , if $A=M-N$ is an M-splitting and A is a nonsingular M-matrix, then $\rho (M^{-1}N)<1$, see (Bai 1999).

For the convergence proof, we need the following lemmas:

Lemma 2.1

(Frommer and Mayer 1989) Let $A, B\in R^{n\times n}$. If A is an M-matrix, $B\in Z^{n\times n}$ and $A\le B$, then B is an M-matrix.

Lemma 2.2

(Frommer and Szyld 1992) Let $A\in R^{n\times n}$ be an H-matrix and $A=D-B$, where D is the diagonal part of the matrix A. Then the following statements hold true:

• 1. A is nonsingular and $|A^{-1}|\le \langle A\rangle ^{-1}$;

• 2. |D| is nonsingular and $\rho (|D|^{-1}|B|)<1$.

Lemma 2.3

(Huang and Ma 2016) Let $A=M-N$ be a splitting of the matrix $A\in R^{n\times n}$, and $\Omega$, $\Gamma$ be $n\times n$ positive diagonal matrices. Then the following statements hold true:

• 1. If (w, z) is a solution of the complementarity problem (1.1), then $x=\dfrac{1}{2}(\Gamma ^{-1}z-\Omega ^{-1}w)$ satisfies the implicit fixed point equation

  $$\begin{aligned} (M\Gamma +\Omega )x=N\Gamma x+(\Omega -A \Gamma )|x|-\Psi (\Gamma (|x|+x)). \end{aligned}$$

  (2.1)

• 2. If x satisfies the implicit fixed point Eq. (2.1), then

  $$\begin{aligned} z=\Gamma (|x|+x),\quad w=\Omega (|x|-x), \end{aligned}$$

  is a solution of the complementarity problem (1.1).

According to Lemma 2.3, if $A=M_1-N_1=M_2-N_2$ be the splitting of A, we can reformulate the problem (1.1) as the following implicit fixed point equation

$$\begin{aligned} (M_1+\Omega )x=N_1 x+(\Omega -M_2 )|x|+N_2 |x|-\gamma \Psi (z), \end{aligned}$$

(2.2)

where $z=\frac{|x|+x}{\gamma }$, and $\omega =\frac{1}{\gamma }\Omega (|x|-x)$. By using the fixed point equation, we shall establish the following accelerated modulus-based matrix splitting iteration method for solving the problem (1.1).

Algorithm 2.1

Let $A=M_1-N_1=M_2-N_2$ be two splittings of the matrix $A\in R^{n\times n}$, let $\Omega$ be an $n\times n$ positive diagonal matrix and $\gamma$ be a positive constant. Given an initial vector $x^0\in R^n$, compute $z^0=(|x^0|+x^0)/\gamma$. For $k=0,1,2,\ldots$, until the iteration sequence $\{z^k\}_{k=0}^{+\infty }$ is convergence, compute $x^{k+1}\in R^n$ by solving the linear system

$$\begin{aligned} (M_1+\Omega )x^{k+1}=N_1x^k+(\Omega -M_2)|x^k|+N_2|x^{k+1}|-\gamma \Psi (z^k) \end{aligned}$$

(2.3)

and set

$$\begin{aligned} z^{k+1}=\dfrac{1}{\gamma }\left( |x^{k+1}|+x^{k+1}\right) . \end{aligned}$$

Algorithm 2.1 includes the modulus-based matrix splitting iteration method (Xia and Li 2015) with $M_2=A$ and $N_2=O$ as its special case. Moreover, with specific choices of the matrix splitting and iteration parameters, Algorithm 2.1 can yield a series of accelerated modulus-based matrix splitting methods. For example, let $A=D-L-U$ with D, $-L$ and $-U$ being the diagonal, the strictly lower-triangular and the strictly upper-triangular matrices of A, then

$$\begin{aligned} M_1=\dfrac{1}{\alpha }(D-\beta L), \quad N_1=\dfrac{1}{\alpha }[(1-\alpha )D+(\alpha -\beta )L+\alpha U],\quad M_2=D-U~\mathrm {and}~N_2=L \end{aligned}$$

Algorithm 2.1 reduces to the accelerated modulus-based accelerated overrelaxation (AMAOR) iteration method

$$\begin{aligned} (D+\alpha \Omega -\beta L)x^{k+1}= & {} [(1-\alpha )D+(\alpha -\beta )L+\alpha U]x^k+\alpha (\Omega -D+U)|x^k|\\&+\,\alpha L|x^{k+1}|-\alpha \gamma \Psi (z^k). \end{aligned}$$

It also gives the accelerated modulus-based successive overrelaxation (AMSOR) iteration method, the accelerated modulus-based Gauss–Seidel (AMGS) iteration method and the accelerated modulus-based Jacobi (AMJ) iteration method when $\alpha =\beta$, $\alpha =\beta =1$ and $\alpha =1, \beta =0$, respectively.

3 Convergence analysis for the case of positive-definite matrix

In this section, we consider A is positive definite matrix. To this end, we introduce the following functions:

$$\begin{aligned} \xi _1(\Omega )= & {} \Vert (\Omega +M_1)^{-1}N_1\Vert ,\quad \xi _2(\Omega )=\Vert (\Omega +M_1)^{-1}N_2\Vert , \\ \xi _3(\Omega )= & {} \Vert (\Omega +M_1)^{-1}(\Omega -M_1)\Vert ,\quad \xi _4(\Omega )=L\Vert (\Omega +M_1)^{-1}\Vert . \end{aligned}$$

Theorem 3.1

Let $A\in R^{n\times n}$ be a positive definite matrix, and $A=M_1-N_1=M_2-N_2$ be two splittings of the matrix A with $M_1\in R^{n\times n}$ being positive definite matrix. Assume that $\Omega \in R^{n\times n}$ is a positive diagonal matrix, $\gamma$ is a positive constant and $\Psi (z):R^n\rightarrow R^n$ is a Lipschitz continuous function with the Lipschitz constant L, that is, for any $z_1,z_2\in R^n$,

$$\begin{aligned} \Vert \Psi (z_1)-\Psi (z_2)\Vert \le L\Vert z_1-z_2\Vert \end{aligned}$$

holds. Let $\varrho (\Omega )=2[\xi _1(\Omega )+\xi _2(\Omega )+\xi _4(\Omega )]+\xi _3(\Omega )$. If the parameter matrix $\Omega$ satisfies $\varrho (\Omega )<1$, then the iteration sequence $\{z^k\}_{k=0}^{+\infty }\subseteq R^n_+$ generated by Algorithm 2.1 converges to the solution $z^* \in R^n_+$ of the problem (1.1) for any initial vector $x^0\in R^n$.

Proof

Assume that $(z^*,w^*)\in R^n\times R^n$ is a solution of the problem (1.1). By relationship (2.2), we have $z^*=\dfrac{\gamma }{2}(z^*-\Omega ^{-1}w^*)$ satisfies the fix point equation

$$\begin{aligned} (M_1+\Omega )x^{*}=N_1x^*+(\Omega -M_2)|x^*|+N_2|x^{*}|-\gamma \Psi (z^*), \end{aligned}$$

(3.1)

where $z^*=\frac{|x^{*}|+x^{*}}{\gamma }$. Together with Algorithm 2.1, subtracting (3.1) from (2.3), we obtain

$$\begin{aligned} (M_1+\Omega )(x^{k+1}-x^{*})= & {} N_1(x^k-x^{*})+(\Omega -M_2)(|x^k|-|x^{*}|)\nonumber \\&+\,N_2(|x^{k+1}|-|x^{*}|)-\gamma [\Psi (z^k)-\Psi (z^{*})]. \end{aligned}$$

(3.2)

Noticing that $M_1$ is a positive definite matrix and $\Omega$ is a positive diagonal matrix, which follows that $M_1+\Omega$ is also positive definite matrix; hence $M_1+\Omega$ is nonsingular. Then, together (3.2) with $A=M_1-N_1=M_2-N_2$ yields

$$\begin{aligned} x^{k+1}-x^{*}= & {} (M_1+\Omega )^{-1}\left[ N_1(x^k-x^{*})+(\Omega -M_2)(|x^k|-|x^{*}|)+N_2(|x^{k+1}|-|x^{*}|)\right] \nonumber \\&-\gamma (M_1+\Omega )^{-1}\left[ \Psi (z^k)-\Psi (z^{*})\right] \nonumber \\= & {} (M_1\!+\!\Omega )^{-1}\left[ N_1(x^k\!-\!x^{*})+(\Omega \!-M_2+N_2\!-\!N_2)(|x^k|\!-\!|x^{*}|)\right. \nonumber \\&\left. +N_2(|x^{k+1}|-|x^{*}|)\right] -\gamma (M_1+\Omega )^{-1}[\Psi (z^k)-\Psi (z^{*})]\nonumber \\= & {} (M_1\!+\!\Omega )^{-1}\left[ N_1(x^k\!-\!x^{*})+(\Omega \!-M_1\!+N_1\!-\!N_2)(|x^k|\!-\!|x^{*}|)\right. \nonumber \\&\left. +N_2(|x^{k+1}|\!-\!|x^{*}|)\right] -\gamma (M_1\!+\!\Omega )^{-1}[\Psi (z^k)\!-\!\Psi (z^{*})]\nonumber \\= & {} (M_1\!+\!\Omega )^{-1}\left[ N_1(x^k\!-\!x^{*})\!+\!(\Omega \!-\!M_1)(|x^k|\!-\!|x^{*}|)\right. \nonumber \\&\left. +N_1(|x^k|\!-\!|x^{*}|)\!+\!N_2(|x^{k\!+\!1}|\!-\!|x^{*}|)\right] -\gamma (M_1+\Omega )^{-1}[\Psi (z^k)-\Psi (z^{*})].\nonumber \\ \end{aligned}$$

(3.3)

Taking an arbitrary matrix norm on both sides of (3.3), we have

$$\begin{aligned} \Vert x^{k+1}-x^{*}\Vert\le & {} \!2\Vert (M_1+\Omega )^{-1}N_1\Vert \Vert x^k-x^{*}\Vert \!\nonumber \\&+ \Vert (M_1+\Omega )^{-1}N_2\Vert \Vert x^k-x^{*}\Vert + \Vert (M_1+\Omega )^{-1}N_2\Vert \Vert x^{k+1}-x^{*}\Vert \!\nonumber \\&+\Vert (M_1+\Omega )^{-1}) (\Omega -M_1)\Vert \Vert x^k-x^{*}\Vert \!+\!\gamma \Vert (M_1\!+\!\Omega )^{-1}\Vert \Vert \Psi (z^k)\!-\!\Psi (z^*)\Vert \nonumber \\\le & {} \!2\Vert (M_1+\Omega )^{-1}N_1\Vert \Vert x^k-x^{*}\Vert \!+\!\Vert (M_1+\Omega )^{-1}N_2\Vert \Vert x^k-x^{*}\Vert \!\nonumber \\&+ \Vert (M_1+\Omega )^{-1}N_2\Vert \Vert x^{k+1}-x^{*}\Vert +\Vert (M_1+\Omega )^{-1}) (\Omega -M_1)\Vert \Vert x^k-x^{*}\Vert \nonumber \\&+\gamma L\Vert (M_1+\Omega )^{-1}\Vert \Vert z^k-z^*\Vert , \end{aligned}$$

(3.4)

where the last inequality uses the fact that $\Psi (z)$ is a Lipschitz continuous function with the Lipschitz constant L. The inequality (3.4) can be rewritten as the following form:

$$\begin{aligned}{}[1-\Vert (M_1+\Omega )^{-1}N_2\Vert ] \Vert x^{k+1}-x^{*}\Vert\le & {} [\Vert 2(M_1+\Omega )^{-1}N_1\Vert +\Vert (M_1+\Omega )^{-1}N_2\Vert \nonumber \\&+\Vert (M_1+\Omega )^{-1}) (\Omega -M_1)\Vert ] \Vert x^{k}-x^{*}\Vert \\&+\gamma L\Vert (M_1+\Omega )^{-1}\Vert \Vert z^k-z^*\Vert . \end{aligned}$$

As $\gamma >0$, we have

$$\begin{aligned} \Vert z^k-z^*\Vert= & {} \Bigg \Vert \dfrac{|x^{k}|+x^{k}}{\gamma }-\dfrac{|x^{*}|+x^{*}}{\gamma }\Bigg \Vert \\= & {} \dfrac{1}{\gamma }\big \Vert |x^{k}|+x^{k}+|x^{*}|+x^{*}\big \Vert \\\le & {} \dfrac{1}{\gamma }\big [\Vert |x^{k}|-|x^{*}|\Vert +\Vert x^{k}-x^{*}\Vert \big ]\\\le & {} \dfrac{2}{\gamma }\Vert x^{k}-x^{*}\Vert . \end{aligned}$$

Hence

$$\begin{aligned}{}[1-\xi _2(\Omega )] \Vert x^{k+1}-x^{*}\Vert\le & {} [\Vert 2(M_1+\Omega )^{-1}N_1\Vert +\Vert (M_1+\Omega )^{-1}N_2\Vert \nonumber \\&+\Vert (M_1+\Omega )^{-1}) (\Omega -M_1)\Vert ] \Vert x^{k}-x^{*}\Vert \\&+2 L\Vert (M_1+\Omega )^{-1}\Vert \Vert x^k-x^*\Vert \\= & {} [2\xi _1(\Omega )+\xi _2(\Omega )+\xi _3(\Omega )+2\xi _4(\Omega )]\Vert x^k-x^*\Vert . \end{aligned}$$

Thereby, we can obtain

$$\begin{aligned} \Vert x^{k+1}-x^{*}\Vert\le & {} \dfrac{2\xi _1(\Omega )+\xi _2(\Omega )+\xi _3(\Omega )+2\xi _4(\Omega )}{1-\xi _2(\Omega )}\Vert x^k-x^*\Vert , \end{aligned}$$

with $\xi _2(\Omega )<1$. By the fact that

$$\begin{aligned} \dfrac{2\xi _1(\Omega )+\xi _2(\Omega )+\xi _3(\Omega )+2\xi _4(\Omega )}{1-\xi _2(\Omega )}<1 \end{aligned}$$

is equivalent to $\varrho (\Omega )=2[\xi _1(\Omega )+\xi _2(\Omega )+\xi _4(\Omega )]+\xi _3(\Omega )<1$, which shows that $\lim \nolimits _{k\rightarrow +\infty }x^k=x^*$. The proof is completed. $\square$

Remark 3.1

If $\Psi (z)=q$, where $q\in R^n$ is a constant vector, then the problem (1.1) reduce to the linear complementarity problem studied in Xu (2015). Because $L=0$, $\xi _1(\Omega )=\xi (\Omega )$, $\xi _2(\Omega )=\eta (\Omega )$, $\xi _3(\Omega )=\mu (\Omega )$, $\varrho (\Omega )=\delta (\Omega )$, where $\xi (\Omega )$, $\eta (\Omega )$, $\mu (\Omega )$, $\delta (\Omega )$ are defined in Xu (2015). At this case, Theorem 3.1 reduces to Theorem 4.1 in Xu (2015).

Remark 3.2

If we use the norm $\Vert \cdot \Vert _{\Omega ^{1/2},2}$ is defined by $\Vert x\Vert _{\Omega ^{1/2},2}=\Vert \Omega ^{1/2}x\Vert _2$ for a vector $x\in R^n$ and $\Vert X\Vert _{\Omega ^{1/2},2}=\Vert \Omega ^{1/2}X\Omega ^{-1/2}\Vert _2$ for a matrix $X\in R^{n\times n}$. Let $M_2=A$, $N_2=0$; then $\xi _1(\Omega )=\dfrac{\sigma _1}{2}$, $\xi _3(\Omega )=\sigma _1$, $\xi _4(\Omega )=\dfrac{\sigma _3}{2}$, where $\sigma _1$, $\sigma _2$, $\sigma _3$ are defined in Xie and Xu (2016). At this case, Theorem 3.1 reduce to Theorem 2.2 in Xie and Xu (2016).

In particular, if $M_1\in R^{n\times n}$ be a symmetric positive definite matrix and $\Omega =\omega I\in R^{n\times n}$ is a scalar matrix, Theorem 3.1 immediately gives another convergence result. For convenience, we introduce some parameters

$$\begin{aligned} \tau _1:=\Vert M_1^{-1}N_1\Vert _2,~\tau _2:=\Vert M_1^{-1}N_{2}\Vert _2,~\kappa :=\dfrac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}},~L_1:=\dfrac{L}{\lambda _{\mathrm{min}}},~\omega _1=\dfrac{\omega }{\lambda _{\mathrm{min}}}, \end{aligned}$$

(3.5)

and

$$\begin{aligned} \nu (\tau _1,\tau _2):=\dfrac{[(\tau _1+\tau _2)\kappa +L_1-1]+\sqrt{[(\tau _1+\tau _2)\kappa +L_1-1]^2+4\kappa [(\tau _1+\tau _2)+L_1]}}{2}, \end{aligned}$$

(3.6)

where $\lambda _{\mathrm{max}}$ and $\lambda _{\mathrm{min}}$ are the maximum and minimum eigenvalues of the matrix $M_1$, respectively. L is the Lipschitz constant of $\Psi (z)$. Combining the parameters in (3.5), (3.6) and Theorem 3.1, the convergence results can be described as follows:

Theorem 3.2

Let $A\in R^{n\times n}$ be a positive definite matrix and $A=M_1-N_1=M_2-N_2$ be two splittings of the matrix A with $M_1\in R^{n\times n}$ being symmetric positive definite matrix. Assume that $\Omega =\omega I\in R^{n\times n}$ is a positive scalar matrix, $\gamma$ is a positive constant and $\Psi (z):R^n\rightarrow R^n$ is a Lipschitz continuous function with the Lipschitz constant L. If $L_1<1$, then the iteration sequence $\{z^k\}_{k=0}^{+\infty }\subseteq R^n_+$ generated by Algorithm 2.1 converges to the solution $z^* \in R^n_+$ of the problem (1.1) for any initial vector $x^0\in R^n$, provided that the iterative parameter $\omega$ satisfies either of the following conditions:

• 1. when $0<\tau _1+\tau _2<\frac{1-L_1}{\sqrt{\kappa }}$,

  $$\begin{aligned} \nu (\tau _1,\tau _2)<\omega _1\le \sqrt{\kappa }; \end{aligned}$$

• 2. when $\dfrac{1-L_1}{\kappa }<\tau _1+\tau _2<\frac{1-L_1}{\sqrt{\kappa }}$,

  $$\begin{aligned} \sqrt{ \kappa }\le \omega _1<\dfrac{[1-L_1-(\tau _1+\tau _2)]\kappa }{(\tau _1+\tau _2)\kappa +L_1-1}; \end{aligned}$$

• 3. when $\tau _1+\tau _2\le \frac{1-L_1}{\kappa }$,

  $$\begin{aligned} \omega _1\ge \sqrt{\kappa }. \end{aligned}$$

Proof

From Theorem 3.1, we need to derive the condition $\varrho (\Omega )<1$ with 2-norm. From the properties of spectral norm, the fact that $M_1$ be a symmetric positive definite matrix and $\tau _1\ge 0$, $\tau _2\ge 0$, by directly calculation, we have

$$\begin{aligned} \xi _1(\Omega )= & {} \Vert (\omega I+M_1)^{-1}N_1\Vert _2=\Vert (\omega I+M_1)^{-1}M_1M_1^{-1}N_1\Vert _2\nonumber \\\le & {} \Vert (\omega I+M_1)^{-1}M_1\Vert _2\Vert M_1^{-1}N_1\Vert _2\nonumber \\= & {} \max \limits _{\lambda \in \sigma (M_1)}\dfrac{\tau _1\lambda }{\omega +\lambda }=\dfrac{\tau _1\lambda _{\mathrm{max}}}{\omega +\lambda _{\mathrm{max}}} \end{aligned}$$

(3.7)

and

$$\begin{aligned} \xi _2(\Omega )= & {} \Vert (\omega I+M_1)^{-1}N_2\Vert _2= \Vert (\omega I+M_1)^{-1}M_1M_1^{-1}N_2\Vert _2\nonumber \\\le & {} \Vert (\omega I+M_1)^{-1}M_1\Vert _2\Vert M_1^{-1}N_2\Vert _2\nonumber \\= & {} \max \limits _{\lambda \in \sigma (M_1)}\dfrac{\tau _2\lambda }{\omega +\lambda }=\dfrac{\tau _2\lambda _{\mathrm{max}}}{\omega +\lambda _{\mathrm{max}}} \end{aligned}$$

(3.8)

and by simple calculations

$$\begin{aligned} \xi _4(\Omega )=L\Vert (\omega I+M_1)^{-1}\Vert _2=L\max \limits _{\lambda \in \sigma (M_1)}\dfrac{1}{\omega +\lambda }=\dfrac{L}{\omega +\lambda _{\mathrm{min}}} \end{aligned}$$

(3.9)

and

$$\begin{aligned} \xi _3(\Omega )= & {} \Vert (\Omega +M_1)^{-1}(\Omega -M_1)\Vert _2 =\max \limits _{\lambda \in \sigma (M_1)} \dfrac{|\omega -\lambda |}{\omega +\lambda } \nonumber \\= & {} \max \Bigg \{\dfrac{|\omega -\lambda _{\mathrm{min}}|}{\omega +\lambda _{\mathrm{min}}},\dfrac{|\omega -\lambda _{\mathrm{max}}|}{\omega +\lambda _{\mathrm{max}}}\Bigg \} =\left\{ \begin{array}{ll} \dfrac{\lambda _{\mathrm{max}}-\omega }{\lambda _{\mathrm{max}}+\omega }, &{}\quad \mathrm {for} ~\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}, \\ \dfrac{\omega -\lambda _{\mathrm{min}}}{\omega +\lambda _{\mathrm{min}}}, &{}\quad \mathrm {for} ~\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}. \end{array} \right. \nonumber \\ \end{aligned}$$

(3.10)

Together (3.7), (3.8) with (3.9), (3.10) yields

$$\begin{aligned} \varrho (\Omega )= & {} 2[\xi _1(\Omega )+\xi _2(\Omega )+\xi _4(\Omega )]+\xi _3(\Omega ) \nonumber \\= & {} 2\left[ \dfrac{\tau _1\lambda _{\mathrm{max}}}{\omega +\lambda _{\mathrm{max}}}+\dfrac{\tau _2\lambda _{\mathrm{max}}}{\omega +\lambda _{\mathrm{max}}}+\dfrac{L}{\omega +\lambda _{\mathrm{min}}}\right] + \left\{ \begin{array}{ll} \dfrac{\lambda _{\mathrm{max}}-\omega }{\lambda _{\mathrm{max}}+\omega }, &{}\quad \mathrm {for} ~\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}, \\ \dfrac{\omega -\lambda _{\mathrm{min}}}{\omega +\lambda _{\mathrm{min}}}, &{}\quad \mathrm {for} ~\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}. \end{array} \right. \nonumber \\= & {} \left\{ \begin{array}{ll} \dfrac{2L}{\omega +\lambda _{\mathrm{min}}}+\dfrac{\lambda _{\mathrm{max}}(2\tau _1+2\tau _2+1)-\omega }{\lambda _{\mathrm{max}}+\omega }, &{}\quad \mathrm {for} ~\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}, \\ \dfrac{2L+\omega -\lambda _{\mathrm{min}}}{\omega +\lambda _{\mathrm{min}}}+\dfrac{2\lambda _{\mathrm{max}(\tau _1+\tau _2)}}{\omega +\lambda _{\mathrm{max}}}, &{}\quad \mathrm {for} ~\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}. \end{array} \right. \end{aligned}$$

(3.11)

From (3.11), we have

$$\begin{aligned} \varrho (\Omega )=\dfrac{\left[ 2L\!-\!\lambda _{\mathrm{min}}\!+\!\lambda _{\mathrm{max}}\!+\!2\tau _1\lambda _{\mathrm{max}}\!+\!2\tau _2\lambda _{\mathrm{max}}\right] \omega \!+\! \lambda _{\mathrm{max}}\left[ 2L\!+\!\lambda _{\mathrm{min}}\!+\!2\tau _1\lambda _{\mathrm{min}}\!+\!2\tau _2\lambda _{\mathrm{min}}\right] \!-\!\omega ^2}{(\omega +\lambda _{\mathrm{min}})(\omega +\lambda _{\mathrm{max}})}\nonumber \\ \end{aligned}$$

(3.12)

when $\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ and

$$\begin{aligned} \varrho (\Omega )=\dfrac{\omega ^2\!+\![2L\!-\!\lambda _{\mathrm{min}}\!+\!\lambda _{\mathrm{max}}\!+\!2\tau _1\lambda _{\mathrm{max}}\!+\!2\tau _2\lambda _{\mathrm{max}}]\omega \!+\! \lambda _{\mathrm{max}}[2L\!-\!\lambda _{\mathrm{min}}\!+\!2\tau _1\lambda _{\mathrm{min}}\!+\!2\tau _2\lambda _{\mathrm{min}}]}{(\omega \!+\!\lambda _{\mathrm{min}})(\omega +\lambda _{\mathrm{max}})}\nonumber \\ \end{aligned}$$

(3.13)

when $\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$.

We first consider the case that $\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$. Obviously, if $\varrho (\Omega )<1$, relationship (3.12) implies that

$$\begin{aligned} \omega ^2-[(\tau _1+\tau _2)\lambda _{\mathrm{max}}-\lambda _{\mathrm{min}}+L]w -\lambda _{\mathrm{max}}[(\tau _1+\tau _2)\lambda _{\mathrm{min}}+L]>0. \end{aligned}$$

The above inequality together with (3.5) yields

$$\begin{aligned} \omega _1^2-[(\tau _1+\tau _2)\kappa -1+L_1]w_1-\kappa [(\tau _1+\tau _2)+L_1]>0. \end{aligned}$$

(3.14)

This combined with $\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ gives that

$$\begin{aligned} \nu (\tau _1,\tau _2)<\omega _1\le \sqrt{\kappa }. \end{aligned}$$

With consideration that the obtained upper bound with respect to $\omega$ must be not less than the corresponding lower bound, it obtains that $\nu (\tau _1,\tau _2)<\sqrt{\kappa }$ when $0<\tau _1+\tau _2<\frac{1-L_1}{\sqrt{\kappa }}$.

Next, we consider the case that $\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$. Clearly, (3.13) implies that

$$\begin{aligned} {[}(\tau _1+\tau _2)\lambda _{\mathrm{max}}-\lambda _{\mathrm{min}}+L]\omega +[(\tau _1+\tau _2)-1] \lambda _{\mathrm{min}}\lambda _{\mathrm{max}}+L\lambda _{\mathrm{max}}<0, \end{aligned}$$

since $\varrho (\Omega )<1$. Using the notation (3.5), the above inequality can be rewritten as

$$\begin{aligned}{}[(\tau _1+\tau _2)\kappa -1+L_1]\omega _1+[(\tau _1+\tau _2)-1]\kappa +L_1\kappa <0. \end{aligned}$$

(3.15)

If $(\tau _1+\tau _2)\kappa -1+L_1>0$, i.e., $\dfrac{1-L_1}{\kappa }<\tau _1+\tau _2$, then we have

$$\begin{aligned} \omega _1<\dfrac{[1-L_1-(\tau _1+\tau _2)]\kappa }{(\tau _1+\tau _2)\kappa +L_1-1}. \end{aligned}$$

This together with $\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ gives that

$$\begin{aligned} \sqrt{ \kappa }\le \omega _1<\dfrac{[1-L_1-(\tau _1+\tau _2)]\kappa }{(\tau _1+\tau _2)\kappa +L_1-1}. \end{aligned}$$

Solving the following inequality:

$$\begin{aligned} \sqrt{ \kappa }<\dfrac{[1-L_1-(\tau _1+\tau _2)]\kappa }{(\tau _1+\tau _2)\kappa +L_1-1}, \end{aligned}$$

we have $\tau _1+\tau _2<\frac{1-L_1}{\sqrt{\kappa }}$. This together with $\frac{1-L_1}{\kappa }<\tau _1+\tau _2$ yields the second condition.

If $(\tau _1+\tau _2)\kappa -1+L_1\le 0$, i.e., $\frac{1-L_1}{\kappa }\ge \tau _1+\tau _2>0$. Obviously, (3.15) holds for any $\omega _1>0$. This together with the fact that $\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ yields the third condition. $\square$

Remark 3.3

If $\Psi (z)=q$, where $q\in R^n$ is a constant vector, then the problem (1.1) reduces to the linear complementarity problem studied in Xu (2015). Because $L=0$, then Theorem 3.1 reduces to Theorem 4.2 in Xu (2015).

Let $\mathcal {D}$ be the interval of convergence which has been obtained in Theorem 3.2. Then the optimal $\omega ^*$ which minimizes the value of $\varrho (\Omega )$ defined as (3.11) in the proof of Theorem 3.2 can be established. Since $\varrho (\Omega )$ is determined by parameter $\omega$, for convenience, denote $f(\omega )=\varrho (\Omega )$. That is

$$\begin{aligned} f(\omega )=\left\{ \begin{array}{ll} \dfrac{2L}{\omega +\lambda _{\mathrm{min}}}+\dfrac{\lambda _{\mathrm{max}}(2\tau _1+2\tau _2+1)-\omega }{\lambda _{\mathrm{max}}+\omega }, &{}\quad \mathrm {for} ~\omega \le \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}, \\ \dfrac{2L+\omega -\lambda _{\mathrm{min}}}{\omega +\lambda _{\mathrm{min}}}+\dfrac{2\lambda _{\mathrm{max}}(\tau _1+\tau _2)}{\omega +\lambda _{\mathrm{max}}}, &{}\quad \mathrm {for }~\omega \ge \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}. \end{array} \right. \end{aligned}$$

(3.16)

Based on the relation (3.16) and Theorem 3.2, we have the following theorem:

Theorem 3.3

With the same notations and interval of convergence for parameter $\omega$ in Theorem 3.2, we have

$$\begin{aligned} \omega ^*=\mathrm {arg}\min _{\omega \in \mathcal {D}}f(\omega )=\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}},~~f(\omega ^*)= \dfrac{[1+2(\tau _1+\tau _2)]\sqrt{\kappa }+2L_1-1}{\sqrt{\kappa }+1}, \end{aligned}$$

where $f(\omega )$ is defined as (3.16).

Proof

First, we claim that $f(\omega )$ is continuous at $\omega =\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$. From the relation (3.16), it is easy to see that $f(\omega )$ is continuous from the left and the right at $\omega =\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$, and

$$\begin{aligned} f(\omega _+)-f(\omega _-)= & {} \left[ \dfrac{2L+\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}-\lambda _{\mathrm{min}}}{\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}+\lambda _{\mathrm{min}}}+\dfrac{2\lambda _{\mathrm{max}}(\tau _1+\tau _2)}{\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}+\lambda _{\mathrm{max}}} \right] \\&- \left[ \dfrac{2L}{\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}+\lambda _{\mathrm{min}}}+\dfrac{\lambda _{\mathrm{max}}(2\tau _1+2\tau _2+1)-\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}}{\lambda _{\mathrm{max}}+\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}}\right] \\= & {} \dfrac{\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}-\lambda _{\mathrm{min}}}{\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}+\lambda _{\mathrm{min}}} -\dfrac{\lambda _{\mathrm{max}}-\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}}{\lambda _{\mathrm{max}}+\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}}\\= & {} \dfrac{(\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}-\lambda _{\mathrm{min}})^2}{\lambda _{\mathrm{min}}(\lambda _{\mathrm{max}}-\lambda _{\mathrm{min}})} -\dfrac{(\lambda _{\mathrm{max}}-\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}})^2}{\lambda _{\mathrm{max}}(\lambda _{\mathrm{max}}-\lambda _{\mathrm{min}})}\\= & {} 0. \end{aligned}$$

Hence, $f(\omega )$ is continuous at $\omega =\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$.

Next, we will find the minimum value point of $f(\omega )$ in the domain $\mathcal {D}$.

• 1.  If $\omega <\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$, that is, $0<\tau _1+\tau _2<\frac{1-L_1}{\sqrt{\kappa }}$, we have

  $$\begin{aligned} f'(\omega )= & {} -\dfrac{2L}{(\omega +\lambda _{\mathrm{min}})^2}+\dfrac{-(\omega +\lambda _{\mathrm{max}})-\lambda _{\mathrm{max}}(2\tau _1+2\tau _2+1)+\omega }{(\omega +\lambda _{\mathrm{max}})^2}\\\le & {} -\dfrac{2\lambda _{\mathrm{max}}(\tau _1+\tau _2+1)+2L}{(\omega +\lambda _{\mathrm{max}})^2}<0. \end{aligned}$$

  With consideration that obtained convergence domain for $\omega$ at this case, we know that $f(\omega )$ is a strictly monotonic decreasing function in the interval $(\lambda _{\mathrm{min}}\nu (\tau _1,\tau _2),\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}]$. Hence $f(\omega )$ attain the minimum value at $\omega =\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ when $0<\tau _1+\tau _2<\dfrac{1-L_1}{\sqrt{\kappa }}$.

• 2.  If $\omega >\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$, it then follows from (3.16) that

  $$\begin{aligned} f'(\omega )= & {} \dfrac{(\omega +\lambda _{\mathrm{min}})-(2L+\omega -\lambda _{\mathrm{min}})}{(\omega +\lambda _{\mathrm{min}})^2} -\dfrac{2\lambda _{\mathrm{max}}(\tau _1+\tau _2)}{(\omega +\lambda _{\mathrm{max}})^2} \\= & {} 2\dfrac{(\lambda _{\mathrm{min}}-L)(\omega +\lambda _{\mathrm{max}})^2-\lambda _{\mathrm{max}}(\tau _1+\tau _2)(\omega +\lambda _{\mathrm{min}})^2}{(\omega +\lambda _{\mathrm{max}})^2(\omega +\lambda _{\mathrm{min}})^2}. \end{aligned}$$

  Let

  $$\begin{aligned} g(\omega )=(\lambda _{\mathrm{min}}-L)(\omega +\lambda _{\mathrm{max}})^2-\lambda _{\mathrm{max}}(\tau _1+\tau _2)(\omega +\lambda _{\mathrm{min}})^2. \end{aligned}$$

  (3.17)

  Then

  $$\begin{aligned} f'(\omega ) =\dfrac{2g(\omega )}{(\omega +\lambda _{\mathrm{max}})^2(\omega +\lambda _{\mathrm{min}})^2}. \end{aligned}$$

Now, we simplify (3.17). From (3.5) and (3.17), we have

$$\begin{aligned} g(\omega )= & {} (\lambda _{\mathrm{min}}-L)(\omega +\lambda _{\mathrm{max}})^2-\lambda _{\mathrm{max}}(\tau _1+\tau _2)(\omega +\lambda _{\mathrm{min}})^2\\= & {} \omega ^2 [\lambda _{\mathrm{min}}-L-\lambda _{\mathrm{max}}(\tau _1+\tau _2)]+2\omega [\lambda _{\mathrm{max}}(\lambda _{\mathrm{min}}-L)-\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}(\tau _1+\tau _2)]\\&+[\lambda _{\mathrm{max}}^2(\lambda _{\mathrm{min}}-L)-\lambda _{\mathrm{min}}^2\lambda _{\mathrm{max}}(\tau _1+\tau _2)]\\= & {} \omega ^2\lambda _{\mathrm{min}} [1-L_1-\kappa (\tau _1+\tau _2)]+\omega 2\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}[(1-L_1)-(\tau _1+\tau _2)]\\&+\lambda _{\mathrm{min}}^2\lambda _{\mathrm{max}}[\kappa (1-L_1)-(\tau _1+\tau _2)]\\= & {} a\omega ^2+b\omega +c, \end{aligned}$$

where $a=\lambda _{\mathrm{min}} [1-L_1-\kappa (\tau _1+\tau _2)]$, $b=2\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}[(1-L_1)-(\tau _1+\tau _2)]>0$, $c=\lambda _{\mathrm{min}}^2\lambda _{\mathrm{max}}[\kappa (1-L_1)-(\tau _1+\tau _2)]>0$. If quadratic term coefficient $a\ne 0$, then discriminant of $a\omega ^2+b\omega +c=0$ is

$$\begin{aligned} \Delta= & {} b^2-4ac\\= & {} 4\lambda _{\mathrm{min}}^2\lambda _{\mathrm{max}}^2[(1-L_1)-(\tau _1\!+\!\tau _2)]^2 \\&-4\lambda _{\mathrm{min}}^3\lambda _{\mathrm{max}}[\kappa (1-L_1)-(\tau _1\!+\!\tau _2)] [1-L_1-\kappa (\tau _1\!+\!\tau _2)]\\= & {} 4\lambda _{\mathrm{min}}^3\lambda _{\mathrm{max}}\kappa [(1-L_1)-(\tau _1\!+\!\tau _2)]^2\\&-4\lambda _{\mathrm{min}}^3\lambda _{\mathrm{max}}[\kappa (1-L_1)-(\tau _1\!+\!\tau _2)] [1-L_1-\kappa (\tau _1\!+\!\tau _2)]\\= & {} 4\lambda _{\mathrm{min}}^3\lambda _{\mathrm{max}}(\kappa -1)^2(1-L_1)(\tau _1+\tau _2)\\> & {} 0. \end{aligned}$$

Hence, there are two real roots of $a\omega ^2+b\omega +c=0$, that is,

$$\begin{aligned} \omega _{(1)}=\dfrac{-b-\sqrt{\Delta }}{2a},\quad \omega _{(2)} =\dfrac{-b+\sqrt{\Delta }}{2a},\quad \omega _{(1)}+\omega _{(2)}=-\dfrac{b}{a}. \end{aligned}$$

In the sequel, we are going to come up with the results by discussing three cases.

• (2i)  If $a>0$, that is,

  $$\begin{aligned} \tau _1+\tau _2<\dfrac{1-L_1}{\kappa }. \end{aligned}$$

  From the proof of Theorem 3.2, $f(\omega )<1$ for $\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}<\omega$ when $\frac{1-L_1}{\kappa }>\tau _1+\tau _2$. On the other hand, $\omega _{(1)}<0$, $\omega _{(1)}<\omega _{(2)}$, $\omega _{(1)}+\omega _{(2)}=-\frac{b}{a}<0$. As quadratic function $g(\omega )$ opens upward, if $\omega _{(2)}\ge 0$, then certainly $g(0)\le 0$, which obtains the contradiction with $g(0)=c>0$. Therefore, $\omega _{(2)}<0$ and $f(\omega )$ is strictly monotonic increasing in the interval $[\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}},+\infty )$. Then $f(\omega )$ attains the minimum vale at $\omega =\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$.

• (2ii)  If $a=0$, that is,

  $$\begin{aligned} \tau _1+\tau _2=\dfrac{1-L_1}{\kappa }. \end{aligned}$$

  Then $g(\omega )=b\omega +c>0$, which is always true for $\omega >\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$. On the other hand, from the proof of Theorem 3.2, we know that $f(\omega )<1$ for all $\omega >\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$. Therefore, $f(\omega )$ attains the minimum vale at $\omega =\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ in the interval $[\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}},+\infty )$.

• (2iii)  If $a<0$, that is,

  $$\begin{aligned} \tau _1+\tau _2>\dfrac{1-L_1}{\kappa }. \end{aligned}$$

From the proof of Theorem 3.2, $f(\omega )<1$ for $\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}\le \omega <\frac{[1-L_1-(\tau _1+\tau _2)]\lambda _{\mathrm{max}}}{(\tau _1+\tau _2)\kappa +L_1-1}$ when $\frac{1-L_1}{\kappa }<\tau _1+\tau _2<\frac{1-L_1}{\sqrt{\kappa }}$. By using the same analysis as the case (2i), we can conclude that $f(\omega )$ is strictly monotonic increasing in the interval $\Big \{\omega |\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}} \le \omega <\frac{[1-L_1-(\tau _1+\tau _2)]\lambda _{\mathrm{max}}}{(\tau _1+\tau _2) \kappa +L_1-1}\Big \}$, which means $\sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}}$ is the minimum value point of the function $f(\omega )$.

Sum up the above cases, we know that

$$\begin{aligned} \omega ^*=\mathrm {arg}\min _{\omega \in \mathcal {D}}f(\omega )= \sqrt{\lambda _{\mathrm{min}}\lambda _{\mathrm{max}}},\quad f(\omega ^*)= \dfrac{[1+2(\tau _1+\tau _2)]\sqrt{\kappa }+2L_1-1}{\sqrt{\kappa }+1}, \end{aligned}$$

which completes the proof. $\square$

Notice that Zhang (2011) researched the two-step modulus-based matrix splitting iteration method for linear complementarity problems. Xie and Xu (2016) proposed the two-step modulus-based matrix splitting iteration method for nonlinear complementarity problems. They proved that two-step modulus-based matrix splitting iteration method can achieve higher computing efficiency by utilizing the information contained in the system matrix. Based on Algorithm 2.1, we now present a two-step modulus-based matrix splitting iteration method for (1.1).

Algorithm 3.1

Let $A=M_1-N_1=M_2-N_2$ be two splittings of the matrix $A\in R^{n\times n}$, let $\Omega$ be an $n\times n$ positive diagonal matrix and $\gamma$ be a positive constant. Given an initial vector $x^0\in R^n$, compute $z^0=(|x^0|+x^0)/\gamma$. For $k=0,1,2,\ldots$, until the iteration sequence $\{z^k\}_{k=0}^{+\infty }$ is convergence, compute $x^{k+1}\in R^n$ by solving the linear system

$$\begin{aligned} \left\{ \begin{array}{l} (M_1+\Omega )x^{k+\frac{1}{2}}=N_1x^k+(\Omega -M_2)|x^k|+N_2|x^{k+\frac{1}{2}}|-\gamma \Psi (z^k)\\ (M_2+\Omega )x^{k+1}=N_2x^{k+\frac{1}{2}}+(\Omega -M_1)|x^{k+\frac{1}{2}}|+N_1|x^{k+1}|-\gamma \Psi (z^{k+\frac{1}{2}}) \end{array} \right. \end{aligned}$$

(3.18)

and set

$$\begin{aligned} z^{k+1}=\dfrac{1}{\gamma }\left( |x^{k+1}|+x^{k+1}\right) . \end{aligned}$$

Similar to the proof in Theorem 3.1, we can easily obtain the following convergence theorem:

Theorem 3.4

Let $A\in R^{n\times n}$ be a positive definite matrix, and $A=M_1-N_1=M_2-N_2$ be two splittings of the matrix A with $M_1,M_2\in R^{n\times n}$ being positive definite matrix. Assume that $\Omega \in R^{n\times n}$ is a positive diagonal matrix, $\gamma$ is a positive constant and $\Psi (z):R^n\rightarrow R^n$ is a Lipschitz continuous function with the Lipschitz constant L, that is, for any $z_1,z_2\in R^n$,

$$\begin{aligned} \Vert \Psi (z_1)-\Psi (z_2)\Vert \le L\Vert z_1-z_2\Vert \end{aligned}$$

holds. Let

$$\begin{aligned} \xi _1(\Omega )= & {} \Vert (\Omega +M_1)^{-1}N_1\Vert ,~ \xi _2(\Omega )=\Vert (\Omega +M_1)^{-1}N_2\Vert , \\ ~\xi _3(\Omega )= & {} \Vert (\Omega +M_1)^{-1}(\Omega -M_1)\Vert , \\ \xi _4(\Omega )= & {} L\Vert (\Omega +M_1)^{-1}\Vert ,~\xi (\Omega )=\xi _2(\Omega )+\xi _3(\Omega )+2\xi _1(\Omega )+2\xi _4(\Omega ) \end{aligned}$$

and

$$\begin{aligned} \eta _1(\Omega )= & {} \Vert (\Omega +M_2)^{-1}N_2\Vert ,~ \eta _2(\Omega )=\Vert (\Omega +M_2)^{-1}N_1\Vert ,\\ \eta _3(\Omega )= & {} \Vert (\Omega +M_2)^{-1}(\Omega -M_2)\Vert , \\ \eta _4(\Omega )= & {} L\Vert (\Omega +M_2)^{-1}\Vert ,~\eta (\Omega )=\xi _2(\Omega )+\eta _3(\Omega )+2\xi _1(\Omega )+2\xi _4(\Omega ). \end{aligned}$$

Let $\varrho (\Omega )=\dfrac{\xi (\Omega )+\eta (\Omega )}{[1-\xi _2(\Omega )][1-\eta _2(\Omega )]}$. If the parameter matrix $\Omega$ satisfies $\varrho (\Omega )<1$, then the iteration sequence $\{z^k\}_{k=0}^{+\infty }\subseteq R^n_+$ generated by Algorithm 3.1 converges to the solution $z^* \in R^n_+$ of the problem (1.1) for any initial vector $x^0\in R^n$.

Proof

The proof is the same as that of Theorem 3.1, so we omit here. $\square$

4 Convergence analysis for the case of $H_+$-matrix

In the following, we consider the convergence analysis of Algorithm 2.1 when the system matrix A is an $H_+$-matrix. To this end, we suppose that there exists a nonnegative matrix G such that

$$\begin{aligned} |\Psi (y)-\Psi (z)|\le G|y-z| \end{aligned}$$

(4.1)

holds for any $y,z\in R^n$.

We shall emphasize that the assumption of $\Psi$ satisfies (4.1) is the same as that in Hong and Li (2016). In Sun and Zeng (2011), the authors assume that $\Psi$ is continuously differentiable monotone. In Ma and Huang (2016), the authors assume that $\Psi$ is Lipschitz continuous diagonal function on $R^n$, that is, the ith component $\Psi _i$ of $\Psi$ is a function of the ith variable $z_i$ only:

$$\begin{aligned} \Psi (z)=(\Psi _1(z),\Psi _2(z),\ldots ,\Psi _n(z))^T =(\Psi _1(z_1),\Psi _2(z_2),\ldots ,\Psi _n(z_n))^T \end{aligned}$$

where $z=(z_1,z_2,\ldots ,z_n)^T$, and some functions $\Psi _i:R\rightarrow R$, for any $y,z\in R^n$ holds that

$$\begin{aligned} |\Psi _i(y_i)-\Psi _i(z_i)|\le l_i|y_i-z_i|,~i=1,2,\ldots ,n, \end{aligned}$$

where $l_i\ge 0$ are the Lipschitz constants. By contrast, the assumption here is much weaker.

4.1 Convergence analysis

In this subsection, we will give some convergence theorems.

Theorem 4.1

Let $A\in R^{n\times n}$ be an $H_+$-matrix, and $A=M_1-N_1=M_2-N_2$ be two H-compatible splittings of the matrix A, with $M_1=(m_{ij}^{(1)}),M_2=(m_{ij}^{(2)})\in R^{n\times n}$. Let $A=D-B$ be a splitting of A with D, $-B$ are the diagonal and the nondiagonal matrices, respectively. Assume that $\Omega \in R^{n\times n}$ is a positive diagonal matrix, and $\gamma$ is a positive constant. If parameter matrix $\Omega$ satisfies $\Omega \ge diag(M_2)$, $\langle A\rangle -G$ and $\Omega +M_1-|N_2|$ are M-matrix, then the iteration sequence $\{z^k\}_{k=0}^{+\infty }\subseteq R^n_+$ generated by Algorithm 2.1 converges to the solution $z^* \in R^n_+$ of the problem (1.1) for any initial vector $x^0\in R^n$.

Proof

First, we prove that $M_1+\Omega$ is a $H_+$-matrix. Since $A=M_1-N_1$ is an H-compatible splitting, i.e., $\langle A\rangle =\langle M_1\rangle -|N_1|$, which follows that $|m_{ii}^{(1)}|-|n_{ii}^{(1)}|>0$. Together with A is an $H_+$-matrix, we have $a_{ii}=m_{ii}^{(1)}-n_{ii}^{(1)}>0$; hence $m_{ii}^{(1)}>0$, $i=1,2,\ldots ,n$. As $\langle A\rangle =\langle M_1\rangle -|N_1|$ and $\Omega$ is a positive diagonal matrix, it holds that

$$\begin{aligned} \langle A\rangle \le \langle M_1\rangle \le diag(M_1). \end{aligned}$$

According to Lemma 2.1, $M_1$ is an $H_+$-matrix; hence $M_1+\Omega$ is an $H_+$-matrix and it holds from Lemma 2.3 that

$$\begin{aligned} |(M_1+\Omega )^{-1}|\le \langle M_1+\Omega \rangle ^{-1}=(\langle M_1\rangle +\Omega )^{-1}. \end{aligned}$$

Combining (3.13) and (4.1) yields that

$$\begin{aligned} |x^{k+1}-x^{*}|\le & {} |(M_1+\Omega )^{-1}|\big [|N_1||x^k-x^{*}|+|\Omega -M_2|||x^k|-|x^{*}||+|N_2|||x^{k+1}|-|x^{*}||\big ]\nonumber \\&+\gamma |(M_1+\Omega )^{-1}||\Psi (z^k)-\Psi (z^{*})|\nonumber \\\le & {} (\langle M_1\rangle +\Omega )^{-1}\big [|N_1|+|\Omega -M_2|+|N_2|\big ]|x^k-x^{*}|\nonumber \\&+(\langle M_1\rangle +\Omega )^{-1}|N_2||x^{k+1}-x^{*}|+\gamma (\langle M_1\rangle +\Omega )^{-1}G|z^k-z^{*}|. \end{aligned}$$

(4.2)

As $\gamma >0$, we have

$$\begin{aligned} |z^k-z^*|= & {} \Big |\dfrac{|x^{k}|+x^{k}}{\gamma }-\dfrac{|x^{*}|+x^{*}}{\gamma }\Big |\nonumber \\= & {} \dfrac{1}{\gamma }\big ||x^{k}|+x^{k}+|x^{*}|+x^{*}\big |\nonumber \\\le & {} \dfrac{1}{\gamma }\big [||x^{k}|-|x^{*}||+|x^{k}-x^{*}|\big ]\nonumber \\\le & {} \dfrac{2}{\gamma }|x^{k}-x^{*}|. \end{aligned}$$

(4.3)

Substituting (4.3) into (4.2), we have

$$\begin{aligned} |x^{k+1}-x^{*}|\le & {} (\langle M_1\rangle +\Omega )^{-1}\big [|N_1|+|\Omega -M_2|+2G\big ]|x^k-x^{*}|\nonumber \\&+(\langle M_1\rangle +\Omega )^{-1}|N_2||x^{k+1}-x^{*}|. \end{aligned}$$

(4.4)

By simple calculation, (4.4) can be rewritten as

$$\begin{aligned} {[}1-(\langle M_1\rangle +\Omega )^{-1}|N_2|]|x^{k+1}-x^{*}|\le & {} (\langle M_1\rangle +\Omega )^{-1}\big [|N_1|+|\Omega -M_2|+2G\big ]|x^k-x^{*}|. \end{aligned}$$

Since $\Omega +M_1-|N_2|$ is an M-matrix, $\langle M_1\rangle +\Omega$ is an M-matrix by Lemma 2.1; hence the splitting $\Omega +M_1-|N_2|$ is an M-splitting and $\rho ((\langle M_1\rangle +\Omega )^{-1}|N_2|)<1$; thus if $1-(\langle M_1\rangle +\Omega )^{-1}|N_2|$ is an $M-$matrix and its inverse is nonnegative, then

$$\begin{aligned} |x^{k+1}{-}x^{*}|\le & {} [1{-}(\langle M_1\rangle +\Omega )^{-1}|N_2|]^{-1}(\langle M_1\rangle +\Omega )^{-1}\big [|N_1|+|\Omega -M_2|+2G\big ]|x^k-x^{*}|\\\le & {} (\Omega +\langle M_1\rangle -|N_2|)^{-1}\big [|N_1|+|\Omega -M_2|+2G\big ]|x^k-x^{*}|. \end{aligned}$$

Let $\widetilde{A}=\widetilde{M}-\widetilde{N}$, $\widetilde{M}=\Omega +\langle M_1\rangle -|N_2|$, $\widetilde{N}=|N_1|+|\Omega -M_2|+2G$, by some calculation, we immediately have

$$\begin{aligned} \widetilde{A}= & {} \widetilde{M}-\widetilde{N}\\= & {} \Omega +\langle M_1\rangle -|N_2|-|N_1|-|\Omega -M_2|-2G\\= & {} \Omega -|\Omega -\mathrm{diag}(M_2)|+|M_2|-\mathrm{diag}(M_2)+\langle A\rangle -|N_2|-2G\\= & {} \Omega -\mathrm{diag}(M_2)-|\Omega -\mathrm{diag}(M_2)|+2\langle A\rangle -2G \end{aligned}$$

If $\Omega \ge \mathrm{diag}(M_2)$ and $\langle A\rangle -G$ is an M-matrix, the splitting $\widetilde{A}=\widetilde{M}-\widetilde{N}$ is an M-splitting; thus $\rho (\widetilde{M}^{-1}\widetilde{N})<1$ and hence $\lim \nolimits _{k\rightarrow \infty }x^k=x^*$, which completes the proof. $\square$

Analogously, the convergence theorem for AMAOR iteration methods can be established as follows:

Theorem 4.2

Let $A\in R^{n\times n}$ be an $H_+$-matrix with $A=D-B$ being a splitting of A, where D, $-B$ are the diagonal and the nondiagonal part of A, respectively. Assume that $\rho :=\rho (\langle A\rangle ^{-1}G)<1$, $\gamma$ is a positive constant, and $\Omega \in R^{n\times n}$ is a positive diagonal matrix satisfying $\Omega \ge D$. Then for any initial vector, the $\mathrm {AMAOR}$ iteration method is convergent if the parameter $\alpha$ and $\beta$ satisfying

$$\begin{aligned} \alpha<\beta <\dfrac{1}{\rho },~~\beta \in [0,\alpha ]\cup [\alpha ,\alpha \theta _{\alpha }), \end{aligned}$$

where $\theta _{\alpha }\in [1,+\infty )$ such that

$$\begin{aligned} \rho (D^{-1}(\theta |L|+|U|+G))=\dfrac{\alpha +1-|1-\alpha |}{2\alpha } \end{aligned}$$

(4.5)

Proof

From the proof of Theorem 4.1, the AMAOR splitting $A=M_1-N_1=M_2-N_2$ with

$$\begin{aligned} M_1=\dfrac{1}{\alpha }(D-\beta L), \quad N_1=\dfrac{1}{\alpha }[(1-\alpha )D+(\alpha -\beta )L+\alpha U],\quad M_2=D-U~\mathrm {and}~N_2=L \end{aligned}$$

is convergence when $\widetilde{M}$ is an M-matrix, $\widetilde{N}\ge 0$ and $\widetilde{A}=\widetilde{M}-\widetilde{N}$ is an M-matrix. By some calculation, we have

$$\begin{aligned} \widetilde{M}= & {} \Omega +\langle M_1\rangle -|N_2|=\Omega + \dfrac{D}{\alpha }-\dfrac{\beta }{\alpha }|L|-|L|\nonumber \\= & {} \Omega + \dfrac{D}{\alpha }-\dfrac{\alpha +\beta }{\alpha }|L| \end{aligned}$$

(4.6)

and

$$\begin{aligned} \widetilde{N}= & {} |N_1|+|\Omega -M_2|+2G=\Big |\dfrac{1}{\alpha }[(1-\alpha )D+(\alpha -\beta )L+\alpha U]\Big |+|\Omega -D+U|+2G\nonumber \\= & {} |\Omega -D|+\dfrac{|1-\alpha |}{\alpha }D+\dfrac{ |\alpha -\beta |}{\alpha }|L|+2|U|+2G. \end{aligned}$$

(4.7)

Hence

$$\begin{aligned} \widetilde{A}= & {} \widetilde{M}-\widetilde{N}\\= & {} [\Omega +\langle M_1\rangle -|N_2|]-[|N_1|+|\Omega -M_2|+2G]\\= & {} \Big [\Omega + \dfrac{D}{\alpha }-\dfrac{\alpha +\beta }{\alpha }|L|\Big ]-\Big [|\Omega -D|+\dfrac{|1-\alpha |}{\alpha }D+\dfrac{ |\alpha -\beta |}{\alpha }|L|+2|U|+2G\Big ]\\= & {} \dfrac{\alpha +1-|1-\alpha |}{\alpha }D-\dfrac{\alpha +\beta +|\alpha -\beta |}{\alpha }|L|-2|U|-2G, \end{aligned}$$

where the last equality uses the condition $\Omega \ge D$. Since $\widetilde{M}\ge \widetilde{A}$ and $\widetilde{M}$ is a Z-matrix, $\widetilde{M}$ is an M-matrix if $\widetilde{A}$ is an M-matrix by Lemma 2.1. And the sufficient conditions for $\widetilde{A}$ to be an M-matrix are $\alpha +1-|1-\alpha |>0$, $\alpha >0$ and

$$\begin{aligned} \rho (D^{-1}(\theta |L|+|U|+G))<\dfrac{\alpha +1-|1-\alpha |}{2\alpha },\quad \mathrm {where}\quad \theta =\dfrac{\alpha +\beta +|\alpha -\beta |}{2\alpha }. \end{aligned}$$

(4.8)

On the other hand, note that

$$\begin{aligned} \dfrac{\alpha +1-|1-\alpha |}{2\alpha }=\left\{ \begin{array}{cc} 1, &{}\quad \mathrm {for}~\alpha \in (0,1]; \\ \dfrac{1}{\alpha } &{}\quad \mathrm {for}~\alpha \in [1,+\infty ) , \end{array} \right. \end{aligned}$$

with the maximum value 1 at $\alpha =1$. Since $\rho :=\rho (\langle A\rangle ^{-1}G)<1$, it can be easily verified that $(\alpha +1-|1-\alpha |)/(2\alpha )\in (\rho ,1]$ if $\alpha \in (0,1/\rho )$. Hence, for any fixed $\alpha$, there exists $\theta _{\alpha }\in [1,+\infty )$ such that (4.5) is valid. Therefore, for $\alpha<\beta <\dfrac{1}{\rho }$, $\beta \in [0,\alpha ]\cup [\alpha ,\alpha \theta _{\alpha })$, the inequality (4.8) is true. $\square$

Similar to the proof of Theorem 4.1, we can easily obtain the following convergence theorem:

Theorem 4.3

Let $A\in R^{n\times n}$ be an $H_+$-matrix, and $A=M_1-N_1=M_2-N_2$ be two H-compatible splittings of the matrix A, with $M_1=(m_{ij}^{(1)}),M_2=(m_{ij}^{(2)})\in R^{n\times n}$. Let $A=D-B$ be a splitting of A with D, $-B$ are the diagonal and the nondiagonal matrices, respectively. Assume that $\Omega \in R^{n\times n}$ is a positive diagonal matrix, $\gamma$ is a positive constant. If parameter matrix $\Omega$ satisfies $\Omega \ge diag(M_1)$, $\Omega \ge diag(M_2)$, $\langle A\rangle -G$ and $\Omega +M_1-|N_2|$, $\Omega +M_2-|N_1|$ are M-matrix, then the iteration sequence $\{z^k\}_{k=0}^{+\infty }\subseteq R^n_+$ generated by Algorithm 3.1 converges to the solution $z^* \in R^n_+$ of the problem (1.1) for any initial vector $x^0\in R^n$.

Proof

The proof is similar to that of Theorem 4.1, so we omit here. $\square$

4.2 The optimal parameter of AMAOR method

In this subsection, we will discuss the optimal possible AMAOR method by minimizing the associated spectral radius of the iteration matrix $\rho (\widetilde{M}^{-1}\widetilde{N})$. First, we review the following lemma, which can be seen in Marek and Szyld (1990):

Lemma 4.1

(Marek and Szyld 1990) Let $A_i=M_i-N_i$ be weak nonnegative splittings with $L_i=M_i^{-1}N_i$ and $\rho (L_i)<1$, $i=1,2$. Let $x_i\ge 0$ be such that $L_ix_i=\rho (L_i)x_i$, $i=1,2$. Let $A_2^{-1}\ge 0$ and $A_2^{-1}\ge A_1^{-1}$. If either $N_2x_1\ge N_1x_1\ge 0$ or $N_2x_2\ge N_1x_2\ge 0$ with $x_2>0$, then $\rho (L_1)\le \rho (L_2)$. Moreover, if $A_2^{-1}>0$ and if $N_1\ne N_2$, then $\rho (L_1)<\rho (L_2)$.

Theorem 4.4

Under the assumption of Theorem 4.2 and for any fixed $\Omega =\omega D\ge D$, the spectral radius of the $\mathrm {AMAOR}$ iteration matrix is a decreasing function for $\beta \in [0,\alpha ]$, and an increasing function for $\beta \in [\alpha ,\alpha \theta _{\alpha })$. Hence, the optimal $\mathrm {AMAOR}$ method is the $\mathrm {AMSOR}$ method.

Proof

• 1.  If $0\le \beta _2\le \beta _1\le \alpha$, from (4.6) and (4.7), we know that the AMAOR methods corresponding to two $\beta '$s are

  $$\begin{aligned} \widetilde{M}_{\omega ,i}=(\omega + \dfrac{1}{\alpha })D-\dfrac{\alpha +\beta _i}{\alpha }|L|,~\widetilde{N}_{\omega ,i} =(\omega -1)D+\dfrac{|1-\alpha |}{\alpha }D+\dfrac{ \alpha -\beta _i}{\alpha }|L|+2|U|+2G \end{aligned}$$

  and hence

  $$\begin{aligned} \widetilde{A}_{\omega ,i}=D+\dfrac{1-|1-\alpha |}{\alpha }D-2|B|-2G. \end{aligned}$$

  Since $\widetilde{A}_{\omega ,i}$ is irrelevant with $\beta _i$, $\widetilde{N}_{\omega ,2}\ge \widetilde{N}_{\omega ,1}$, it then follows from Lemma 4.1 that $\rho (\widetilde{M}_{\omega ,2}^{-1}\widetilde{N}_{\omega ,2})\ge \rho (\widetilde{M}_{\omega ,1}^{-1}\widetilde{N}_{\omega ,1})$. Hence, the spectral radius of the AMAOR iteration matrix $\rho (\widetilde{M}_{\omega }^{-1}\widetilde{N}_{\omega })$ is a decreasing function for $\beta \in [0,\alpha ]$.

• 2.  If $\alpha \le \beta _2\le \beta _1<\alpha \theta _{\alpha }$, then from (4.6) and (4.7), we have

  $$\begin{aligned} \widetilde{M}_{\omega ,i}=(\omega + \dfrac{1}{\alpha })D-\dfrac{\alpha +\beta _i}{\alpha }|L|,~\widetilde{N}_{\omega ,i} =(\omega -1)D+\dfrac{|1-\alpha |}{\alpha }D+\dfrac{ \beta _i-\alpha }{\alpha }|L|+2|U|+2G \end{aligned}$$

  and hence

  $$\begin{aligned} \widetilde{A}_{\omega ,i}=D+\dfrac{1-|1-\alpha |}{\alpha }D-\dfrac{2\beta _i}{\alpha }|L|-2|U|-2G. \end{aligned}$$

  From Theorem 4.2, we know that $\widetilde{M}_{\omega ,i}-\widetilde{N}_{\omega ,i}$ are M-splittings of nonsingular M-matrices, with $\widetilde{M}_{\omega ,i}^{-1}\widetilde{N}_{\omega ,i}\ge 0$ and $\rho (\widetilde{M}_{\omega ,i}^{-1}\widetilde{N}_{\omega ,i})<1$, $i=1,2$. Let $\delta =2\alpha /(\alpha +1-|1-\alpha |)$ and $\vartheta _i=\beta _i/\alpha$. Then

  $$\begin{aligned} \widetilde{A}_{\omega ,i}^{-1}= & {} \left( \dfrac{2}{\delta }D-2\vartheta _i|L|-2|U|-2G\right) ^{-1}= \dfrac{\delta }{2}[I-\delta D^{-1}(\vartheta _i|L|+|U|+G)]^{-1}D^{-1}\\= & {} \dfrac{\delta }{2}\Big [I+\delta D^{-1}(\vartheta _i|L|+|U|+G)+(\delta D^{-1}(\vartheta _i|L|+|U|+G))^2+\cdots \Big ]D^{-1}\\\ge & {} 0. \end{aligned}$$

  Hence, $\widetilde{A}_{\omega ,1}^{-1}\ge \widetilde{A}_{\omega ,2}^{-1}$. Since $\widetilde{N}_{\omega ,1}^{-1}\ge \widetilde{N}_{\omega ,2}^{-1}\ge 0$, then $\widetilde{N}_{\omega ,1}^{-1}x_1\ge \widetilde{N}_{\omega ,2}^{-1}x_1\ge 0$, where $x_1$ is the eigenvector associated with $\widetilde{M}_{\omega ,1}^{-1}\widetilde{N}_{\omega ,1}$. It then follows from Lemma 4.1 that $\rho (\widetilde{M}_{\omega ,2}^{-1}\widetilde{N}_{\omega ,2})\le \rho (\widetilde{M}_{\omega ,1}^{-1}\widetilde{N}_{\omega ,1})$. Hence, the spectral radius of the AMAOR iteration matrix $\rho (\widetilde{M}_{\omega }^{-1}\widetilde{N}_{\omega })$ is a increasing function for $\beta \in [\alpha ,\alpha \theta _{\alpha })$.

From cases 1 and 2, we can conclude that the optimal AMAOR method is the AMSOR method. The proof is completed.

Theorem 4.5

Under the assumption of Theorem 4.2 and for any fixed $\Omega =\omega D\ge D$, the spectral radius of the $\mathrm {AMAOR}$ iteration matrix is an increasing function for $\omega \in [1,+\infty )$. Hence, the optimal parameter of $\mathrm {AMAOR}$ method is $\omega ^*=1$.

Proof

Since $\Omega =\omega D\ge D$, then $\omega \in [1,\infty )$. If $\omega _1\ge \omega _2\ge 1$, then from (4.6) and (4.7), we have

$$\begin{aligned} \widetilde{M}_{\omega ,i}=\left( \omega _i+ \dfrac{1}{\alpha }\right) D-\dfrac{\alpha +\beta }{\alpha }|L|,\quad \widetilde{N}_{\omega ,i} =(\omega _i-1)D+\dfrac{|1-\alpha |}{\alpha }D+\dfrac{ \alpha -\beta }{\alpha }|L|+2|U|+2,G \end{aligned}$$

and hence

$$\begin{aligned} \widetilde{A}_{\omega ,i}=\left( D+\dfrac{1-|1-\alpha |}{\alpha }\right) D-2\theta |L|-2|U|-2G. \end{aligned}$$

Since $\widetilde{A}_{\omega ,i}$ is irrelevant with $\omega _i$, $\widetilde{N}_{\omega ,1}\ge \widetilde{N}_{\omega ,2}$, it then follows from Lemma 4.1 that $\rho (\widetilde{M}_{\omega ,1}^{-1}\widetilde{N}_{\omega ,1})\ge \rho (\widetilde{M}_{\omega ,2}^{-1}\widetilde{N}_{\omega ,2})$. Hence, the spectral radius of the AMAOR iteration matrix $\rho (\widetilde{M}_{\omega }^{-1}\widetilde{N}_{\omega })$ is an increasing function for $\omega \in [1,+\infty ]$. Therefore, we can conclude that the optimal parameter is $\omega ^*=1$. $\square$

5 Numerical experiments

In this section, we represent some numerical examples to demonstrate the effectiveness of accelerated modulus-based matrix splitting iteration methods from the aspects of iteration steps (denoted by ‘Iter’), elapsed CPU time in seconds (denoted by ‘CPU’) and the norm of absolute residual vectors (denoted by ‘Res’). Here, ‘Res’ is defined as

$$\begin{aligned} \mathrm{RES}(z^k):=\Vert \min (Az^k+\Psi (z^k),z^k)\Vert _2, \end{aligned}$$

where $z^k$ is the kth approximate solution to the problem (1.1), and the minimum is taken componentwise.

All of the tests were run on the Intel (R) Core (TM), where the CPU is 2.40 GHz and the memory is 8.0 GB, the programming language was MATLAB R2015a. The stopping criterion for all methods are $\mathrm{Res}(z^k)\le 10^{-5}$ or k reaches the maximal number of iteration, e.g., 5000.

5.1 Comparison of Algorithm 2.1 with modulus-based method in Ma and Huang (2016)

In this subsection, we compare our method with modulus-based matrix splitting methods (Ma and Huang 2016). In addition, all initial vectors are chosen to be $x^0=(1,1,1,\ldots ,1)^T\in R^n$, and $\gamma =1$, $\Omega =\theta D$ are chosen for both accelerated modulus-based matrix splitting methods and modulus-based matrix splitting methods.

For convenience, let $A=D-L-U$ with D, $-L$ and $-U$ being the diagonal, the strictly lower triangular and the strictly upper-triangular matrices of A, then

$$\begin{aligned} M_1=\dfrac{1}{\alpha }(D-\beta L), \quad N_1=\dfrac{1}{\alpha }[(1-\alpha )D+(\alpha -\beta )L+\alpha U] \end{aligned}$$

Algorithm 2.1 of Ma and Huang (2016) reduces to modulus-based accelerated overrelaxation (MAOR) iteration method

$$\begin{aligned} (D+\alpha \Omega -\beta L)x^{k+1}=[(1-\alpha )D+(\alpha -\beta )L+\alpha U]x^k+\alpha (\Omega -A)|x^k|-\alpha \gamma \Psi (z^k). \end{aligned}$$

It also gives modulus-based successive overrelaxation (MSOR) iteration method, modulus-based Gauss-Seidel (MGS) iteration method and modulus-based Jacobi (MJ) iteration method when $\alpha =\beta$, $\alpha =\beta =1$ and $\alpha =1, \beta =0$, respectively.

In Table 1, the abbreviations of testing methods are listed.

Table 1 Abbreviations of testing methods

Example 5.1

We consider the nonlinear complementarity problem (1.1), which is also considered in Xia and Li (2015), for which $A\in R^{n\times n}$ is given as

$$\begin{aligned} A=\left( \begin{array}{cccccc} B &{} -I &{} O &{} \cdots &{} O &{} O \\ -I &{} B &{} -I &{} \cdots &{} O &{} O \\ O &{} -I &{} B &{} \cdots &{} O &{} O \\ \vdots &{} \vdots &{} &{} \ddots &{} \vdots &{} \vdots \\ O &{} O &{} \cdots &{} \cdots &{} B &{} -I \\ O &{} O &{} \cdots &{} \cdots &{} -I &{} B \\ \end{array} \right) ,\quad \Psi (z)=\left( \begin{array}{c} z_1/(1+z_1)\\ z_2/(1+z_2)\\ z_3/(1+z_3)\\ \vdots \\ z_{n-1}/(1+z_{n-1})\\ z_n/(1+z_n)\\ \end{array} \right) \end{aligned}$$

where $B=\mathrm{tridiag}(-1, 4 ,-1)\in R^{m\times m}$, $I\in R^{m\times m}$ is a unit matrix and $n=m^2$. It is clear that $A\in R^{n\times n}$ is symmetric positive definite matrix.

In Table 2, the iteration steps, the CPU time and the residual norms for the modulus-based matrix splitting iterative methods and the accelerated modulus-based matrix splitting iterative methods for Example 5.1 are listed. Here, $\Omega =3 D$.

Table 2 Numerical comparison of the testing methods for Example 5.1

Example 5.2

We consider the nonlinear complementarity problem (1.1), which is also considered in Xia and Li (2015), for which $A\in R^{n\times n}$ is given as

$$\begin{aligned} A=\left( \begin{array}{cccccc} B &{} -0.5I &{} O &{} \cdots &{} O &{} O \\ -1.5I &{} B &{} -0.5I &{} \cdots &{} O &{} O \\ O &{} -1.5I &{} B &{} \cdots &{} O &{} O \\ \vdots &{} \vdots &{} &{} \ddots &{} \vdots &{} \vdots \\ O &{} O &{} \cdots &{} \cdots &{} B &{} -0.5I \\ O &{} O &{} \cdots &{} \cdots &{} -1.5I &{} B \\ \end{array} \right) ,\quad \Psi (z)=\left( \begin{array}{l} \arctan (z_1) \\ \arctan (z_2) \\ \arctan (z_3) \\ \vdots \\ \arctan (z_{n-1}) \\ \arctan (z_n) \\ \end{array} \right) \end{aligned}$$

where $B=\mathrm{tridiag}(-1.5, 4 ,-0.5)\in R^{m\times m}$, $I\in R^{m\times m}$ is a unit matrix and $n=m^2$.

In Table 3, the iteration steps, the CPU time and the residual norms for the modulus-based matrix splitting iterative methods and the accelerated modulus-based matrix splitting iterative methods for Example 5.2 are listed. Here, $\Omega =5 D$.

Table 3 Numerical comparison of the testing methods for Example 5.2

From Tables 2 and 3, we can easily see that the iteration steps and CPU time of six methods increase with the increasing of the problem size $n=m^2$. Moreover, it is observed that the accelerated modulus-based Jacobi, Gauss–Seidel and SOR methods require less iteration steps and CPU time than modulus-based Jacobi, Gauss–Seidel and SOR methods, respectively. Among all these methods, accelerated modulus-based SOR use the least iteration steps and CPU time.

Table 4 Comparison of parameter $\alpha$ for Example 5.3

Table 5 Results of AMGS with different $\omega$ for Example 5.4

5.2 The optimal parameter of AMSOR method

In this subsection, we consider the optimal AMAOR method, that is, AMSOR method. First, we determine the optimal iteration parameter $\alpha$, which is obtained experimentally by minimizing the corresponding iteration steps. Moreover, we determine the optimal iteration parameter $\omega$ in the $\mathrm {AMGS}$ method to illustrate the conclusion of Sect. 4. Finally, we choose the initial vector as $x^0=(0,0,\ldots ,0)^T\in R^n$, and $\gamma =1$ in this subsection.

Example 5.3

We consider the nonlinear complementarity problem (1.1), for which $A\in R^{n\times n}$ is given as

$$\begin{aligned} A=\left( \begin{array}{cccccc} B &{} -0.5I &{} O &{} \cdots &{} O &{} O \\ -1.5I &{} B &{} -0.5I &{} \cdots &{} O &{} O \\ O &{} -1.5I &{} B &{} \cdots &{} O &{} O \\ \vdots &{} \vdots &{} &{} \ddots &{} \vdots &{} \vdots \\ O &{} O &{} \cdots &{} \cdots &{} B &{} -0.5I \\ O &{} O &{} \cdots &{} \cdots &{} -1.5I &{} B \\ \end{array} \right) , \end{aligned}$$

where $B=\mathrm{tridiag}(-1.5, 4 ,-0.5)\in R^{m\times m}$, $I\in R^{m\times m}$ is a unit matrix and $n=m^2$, $\Psi (z)=h(z)-Az^*$, here $h(z)=(\sqrt{z_1^2+0.01}, \ldots ,\sqrt{z_{n}^2+0.01})^T$, $z^*=(1,2,1,2,\ldots ,1,2,\ldots )^T$. In this example, we choose $\Omega =D+I$.

In Table 4, the number of iteration steps and the elapsed $\mathrm {CPU}$ time in seconds are listed for two methods when the parameter $\alpha$ varies from 0.8 to 1.3 for Example 5.3. From Table 4, it is observed that for Example 5.3 the optimal parameter $\alpha ^*=1.0$ for $\mathrm {MSOR}$ method and $\mathrm {AMSOR}$ method.

Example 5.4

We consider the nonlinear complementarity problem (1.1), for which $A\in R^{n\times n}$ is given as

$$\begin{aligned} A=\left( \begin{array}{ccccccc} &{} B &{} -I &{}-I&{} &{} &{} \\ &{} &{} B &{} -I&{}\ddots &{} &{} \\ &{} &{} &{} B&{}\ddots &{} -I &{} \\ &{} &{} &{} &{} \ddots &{}-I &{} \\ &{} &{} &{}&{} &{} B &{} \\ \end{array} \right) , \end{aligned}$$

where $B=\mathrm{tridiag}(-1, 4 ,-1)\in R^{m\times m}$, $I\in R^{m\times m}$ is a unit matrix and $n=m^2$, $\Psi (z)=h(z)-Az^*$, here $h(z)=(\arctan (z_1), \ldots ,\arctan (z_n))^T$, $z^*=(1,2,1,2,\ldots ,1,2,\ldots )^T$. In this example, we fix $\alpha =1$. That is, we consider the $\mathrm {AMGS}$ method.

In Table 5, we list the iteration steps and $\mathrm {CPU}$ time of $\mathrm {AMGS}$ method with iteration parameter $\omega =1,~2,~3,~4,~5$. From Table 5, it is further confirmed that the iteration steps and $\mathrm {CPU}$ time increase as problem size increases. At the same time, we find that the $\mathrm {AMGS}$ method with $\omega =1$ requires the least iteration steps and $\mathrm {CPU}$ time compared to other iteration parameters. This result verifies the analysis in Theorem 4.5.

6 Conclusions

The accelerated modulus-based matrix splitting iteration methods for the solution of a class of nonlinear complementarity problem is presented. The proposed method not only computationally more convenient to use because of storage requirement, but it is also faster than the modulus-based matrix splitting methods. We show their convergence by assuming that the system matrix is positive definite or the splitting of the system matrix are $H_+$-compatible splitting. Also, we discuss the optimal parameter. Furthermore, we give two-step accelerated modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. In addition, we present numerical examples, which demonstrate that accelerated modulus-based matrix splitting iteration method is efficient.