Convergence results of a matrix splitting algorithm for solving weakly nonlinear complementarity problems

Abstract

In this paper, we consider a class of weakly nonlinear complementarity problems (WNCP) with large sparse matrix. We present an accelerated modulus-based matrix splitting algorithm by reformulating the WNCP as implicit fixed point equations based on two splittings of the system matrixes. We show that, if the system matrix is a P-matrix, then under some mild conditions the sequence generated by the algorithm is convergent to the solution of WNCP.

1 Introduction

Consider the following weakly nonlinear complementarity problems, which is to find $z\in R^{n}$ such that

$$\begin{aligned} z\geq0,\quad Az+\Psi(z)\geq0, \quad z^{T}\bigl(Az+\Psi(z) \bigr)=0, \end{aligned}$$

(1)

abbreviated as WNCP, where $A=(a_{ij})\in R^{n\times n}$ is a given large, sparse and real matrix, $\Psi(z)$: $R^{n}\rightarrow R^{n}$ is a Lipschitz continuous nonlinear function, $z\geq0$ means $z_{i}\geq0$, $i=1,2,\ldots,n$, and T always denotes the transpose of a vector.

As is well known, the classical nonlinear complementarity problems (NCP) are very important and fundamental topics in optimization theory and they have been developed into a well-established and fruitful principle. See [1–3] for details as regards the basic theories, effective algorithms and important applications of NCP. Problem (1) is a special case of NCP, but it extends from linear complementarity problems. When $\Psi(z)=q$ is a constant vector, problem (1) reduces to a linear complementarity problem. Recently, lots of researchers [4–8] have paid close attention to feasible and efficient methods for solving linear complementarity problems. Especially, by reformulating linear complementarity problems as an implicit fixed point equation, Van Bokhoven [9] proposed a modulus iteration method, which is defined as the solution of system of linear equations at each iteration. In 2010, Bai [10] presented a modulus-based matrix splitting iteration method and showed the convergence when the system matrix is an $H_{+}$-matrix. Consequently, Zhang [11] proposed two-step modulus-based matrix splitting iteration methods and considered the convergence theory when the system matrix is an $H_{+}$-matrix. Based on the above work, for solving problem (1), in [12] Sun and Zeng proposed a modified semismooth Newton method with A being an M-matrix and $\Psi(z)$ being a continuously differentiable monotone diagonal function on $R^{n}$.

In this paper, for satisfying the requirements of the application, more details can be found in [13–16], we present an accelerated modulus-based matrix splitting algorithm for dealing with WNCP. The organization of this paper is as follows: Some necessary notations and definitions are introduced in Section 2. In Section 3, we establish a class of accelerated modulus-based matrix splitting iteration algorithms. In Section 4, the convergence conditions are considered.

2 Preliminaries

Some necessary notations, definitions and lemmas used in the sequel discussions are introduced in this section. For $B\in R^{n\times n}$, we write $B^{-1}$, $B^{T}$, $\rho(B)$ to denote the inverse, the transpose, the spectral radius of the matrix B, respectively. For $x\in R^{n}$, we write $\Vert x\Vert$, $\vert x\vert$ to denote the norm of the vector x, $\vert x\vert =(\vert x_{1}\vert ,\ldots, \vert x_{n}\vert )$, respectively. $\Vert A\Vert$ denotes any norm of matrix A. Especially, we use $\Vert \cdot \Vert _{2}$ to denote a spectral norm. $\lambda\in\lambda(A)$ denotes the eigenvalue of matrix A where $\lambda(A)$ is the set of all eigenvalues of matrix A.

Definition 1

[17]

If for any $x:=(x_{1},x_{2},\ldots,x_{n})\neq0$, there exists an index k, such that $x_{k}(Ax)_{k}=x_{k}(a_{k1}x_{1}+\cdots+a_{kn}x_{n})>0$, we call that matrix A is a P-matrix.

Definition 2

For the function $f(x): R^{n}\rightarrow R^{n}$, if for any x, $y\in R^{n}$, there exists a constant L such that

$$\bigl\Vert f(x)-f(y)\bigr\Vert \leq L\Vert x-y\Vert , $$

then we call f a Lipschitz continuous function on $R^{n}$, and L is called a Lipschitz constant.

Lemma 3

[18]

Let $A=(a_{ij})\in R^{n\times n}$ be a P-matrix, for any nonnegative diagonal matrix Ω, the matrix $A+\Omega$ is nonsingular.

3 Algorithm

Theorem 4

Let $M_{1}-N_{1}=M_{2}-N_{2}=A$ be two splittings of the matrix $A\in R^{n\times n}$ and Ω, Γ be $n\times n$ positive diagonal matrices, $\Omega_{1}$, $\Omega_{2}$ be $n\times n$ nonnegative diagonal matrices such that $\Omega=\Omega_{1}+\Omega_{2}$, then the following statements hold.

• (i) If z is a solution of problem (1), then $x=\frac {1}{2} (\Gamma^{-1}z-\Omega^{-1}(Az+\Psi(z)) )$ satisfies the implicit fixed point equation

  $$ (M_{1}\Gamma+\Omega_{1})x =(N_{1}\Gamma-\Omega_{2})x+(\Omega-M_{2} \Gamma)\vert x\vert +N_{2}\Gamma \vert x\vert -\Psi \bigl(\Gamma \bigl(\vert x\vert +x\bigr) \bigr). $$

  (2)

• (ii) If x satisfies the implicit fixed point equation (2), then

  $$z=\Gamma\bigl(\vert x\vert +x\bigr) $$

  is a solution of problem (1).

Proof

First we prove part (i). Since z is a solution of problem (1), we have $z\geq0$. So there exists $x\in R^{n}$ such that

$$z=\Gamma\bigl(\vert x\vert +x\bigr). $$

Define another nonnegative vector,

$$\upsilon=\Omega\bigl(\vert x\vert -x\bigr). $$

It is easy to see that $\upsilon\geq0$, $z^{T}\upsilon=0$, and $\upsilon =Az+\Psi(z)$ if and only if

$$\Omega\bigl(\vert x\vert -x\bigr)=A\Gamma\bigl(\vert x\vert +x\bigr)+ \Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr). $$

Replace A with $M_{1}-N_{1}$, we have

$$(M_{1}\Gamma+\Omega)x=N_{1}\Gamma x+ \bigl( \Omega-(M_{1}-N_{1})\Gamma \bigr)\vert x\vert -\Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr). $$

Taking $M_{1}-N_{1}=M_{2}-N_{2}$ and $\Omega=\Omega_{1}+\Omega_{2}$ into account, it follows that

$$(M_{1}\Gamma+\Omega_{1})x=(N_{1}\Gamma- \Omega_{2}) x+(\Omega-M_{2}\Gamma )\vert x\vert +N_{2}\Gamma \vert x\vert -\Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr). $$

This shows that equation (2) holds.

We now turn to prove part (ii). By some simple calculations, the implicit fixed point equation (2) is equivalent to

$$A\Gamma\bigl(\vert x\vert +x\bigr)+\Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr)=\Omega\bigl(\vert x\vert -x\bigr). $$

Set $z=\Gamma(\vert x\vert +x)$ and $\upsilon=\Omega(\vert x\vert -x)$. Evidently, it yields $z\geq0$, $\upsilon\geq0$, $z^{T}\upsilon=0$, and $\upsilon =Az+\Psi(z)$, which means that z is a solution of problem (1). This completes the proof. □

Note that the implicit fixed point equation (2) includes many parameters that are quite complicated to be determined in a computation. For solving this problem, we will give the simple formulation as follows. Subsequently, a matrix splitting iteration algorithm will be given for solving WNCP. Let $\Omega_{1}=\Omega$, $\Omega_{2}=0, \Gamma=\frac{1}{\gamma}I$, where $\gamma> 0$ is a real number, then the implicit fixed point equation reduces to

$$(M_{1}+\gamma\Omega)x=N_{1} x+(\gamma\Omega-M_{2}) \vert x\vert +N_{2}\vert x\vert -\gamma \Psi \biggl( \frac{(\vert x\vert +x)}{\gamma} \biggr). $$

In fact, γΩ in the above equation denotes a positive diagonal parameter matrix, which can be replaced by Ω for simplicity. That is, the above equation is essentially equivalent to

$$\begin{aligned} (M_{1}+\Omega)x=N_{1} x+( \Omega-M_{2})\vert x\vert +N_{2}\vert x\vert -\gamma \Psi \biggl(\frac {(\vert x\vert +x)}{\gamma} \biggr). \end{aligned}$$

(3)

In the rest of the paper, we will use the fixed point equation above to give the algorithm and convergence analysis for solving WNCP.

Algorithm 5

Step 1. Choose two splittings of the matrix $A\in R^{n\times n}$ satisfying $A=M_{1}-N_{1}=M_{2}-N_{2}$.

Step 2. Set $k=0$. Give an initial vector $x^{0}\in R^{n}$, compute $z^{0}=\frac{1}{\gamma}(\vert x^{0}\vert +x^{0})$.

Step 3. Choose $x^{k+1}$ such that

$$\begin{aligned} (M_{1}+\Omega)x^{k+1}=N_{1}x^{k}+( \Omega -M_{2})\bigl\vert x^{k}\bigr\vert +N_{2}\bigl\vert x^{k+1}\bigr\vert -\gamma\Psi \bigl(z^{k}\bigr), \end{aligned}$$

(4)

and set

$$z^{k+1}=\frac{1}{\gamma}\bigl(\bigl\vert x^{k+1}\bigr\vert +x^{k+1}\bigr),\quad k=k+1. $$

Here, $\Omega, \Gamma\in R^{n\times n}$ are positive diagonal matrices and γ is a positive constant.

Step 4. If the sequence $\{z^{k}\}_{0}^{+\infty}$ is convergent, stop. Otherwise, go to Step 3.

4 Convergence theorems

In this section, we will consider the conditions that ensure the convergence of $\{z^{k}\}_{0}^{+\infty}$ obtained by Algorithm 5.

Theorem 6

Let $A\in R^{n\times n}$ be a P-matrix, $M_{1}-N_{1}=M_{2}-N_{2}=A$ be two splittings of the matrix A with $M_{1}\in R^{n\times n}$ being a P-matrix. Assume that $\Omega\in R ^{n\times n}$ is a positive diagonal matrix, γ is a positive constant and $\Psi(z)$: $R^{n}\rightarrow R^{n}$ is a Lipschitz continuous function with the Lipschitz constant L. Set

$$\rho(\Omega)=2g(\Omega)+2q(\Omega)+f(\Omega), $$

where $g(\Omega)=\Vert (M_{1}+\Omega)^{-1}N_{2}\Vert$, $q(\Omega)=\Vert (M_{1}+\Omega)^{-1}N_{1}\Vert +L\Vert (M_{1}+\Omega)^{-1}\Vert$, $f(\Omega)=\Vert (M_{1}+\Omega)^{-1}(\Omega-M_{1})\Vert$. If the matrix Ω satisfies $\rho(\Omega)<1$, then for any initial vector $x^{0}\in R^{n}$, the iteration sequence $\{z^{k}\}^{+\infty }_{k=0}$ generated by Algorithm 5 converges to a solution $z^{*}\in R^{n}_{+}$ of problem (1).

Proof

Suppose that $z^{*}\in R^{n}_{+}$ is a solution of problem (1). Note that $\Gamma=\frac{1}{\gamma}I$, by Theorem 4 and (3), we see that $x^{*}=\frac{1}{2} (\gamma z^{*}-\Omega ^{-1} (A(z^{*})+\Psi(z^{*}) ) )$ is a solution of the equation

$$\begin{aligned} (M_{1}+\Omega)x^{*}=N_{1}x^{*}+( \Omega-M_{2})\bigl\vert x^{*}\bigr\vert +N_{2}\bigl\vert x^{*}\bigr\vert -\gamma \Psi \bigl(z^{*}\bigr) \end{aligned}$$

(5)

with $z^{*}=\frac{1}{\gamma}(\vert x^{*}\vert +x^{*})$. Let us consider (4) minus (5); we have

$$\begin{aligned} &(M_{1}+\Omega) \bigl(x^{k+1}-x^{*}\bigr) \\ ={}&N_{1}\bigl(x^{k}-x^{*}\bigr)+( \Omega-M_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ &{}+N_{2}\bigl(\bigl\vert x^{k+1}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-\gamma\bigl(\Psi\bigl(z^{k} \bigr)-\Psi\bigl(z^{*}\bigr)\bigr). \end{aligned}$$

Note that $M_{1}-N_{1}=M_{2}-N_{2}$, we have

$$\begin{aligned} &{}(\Omega-M_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ ={}&(\Omega-M_{2}+N_{2}-N_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ ={}&(\Omega-M_{1}+N_{1}-N_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ ={}&(\Omega-M_{1}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)+N_{1} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-N_{2} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr). \end{aligned}$$

(6)

Since $M_{1}$ is a P-matrix and Ω is a positive diagonal matrix, it follows from Lemma 3 that $M_{1}+\Omega$ is a nonsingular matrix. Hence, by (5) and (6), we have

$$\begin{aligned} &{}x^{k+1}-x^{*} \\ ={}&(M_{1}+\Omega)^{-1}N_{1} \bigl(x^{k}-x^{*}\bigr)+(M_{1}+ \Omega)^{-1}(\Omega -M_{1}) \bigl(\bigl\vert x^{k} \bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ &{}+(M_{1}+\Omega)^{-1}N_{1} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-(M_{1}+\Omega )^{-1}N_{2} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ &{}+(M_{1}+\Omega)^{-1}N_{2} \bigl(\bigl\vert x^{k+1}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-(M_{1}+\Omega )^{-1}\gamma \bigl(\Psi \bigl(z^{k}\bigr)-\Psi\bigl(z^{*}\bigr) \bigr). \end{aligned}$$

(7)

Taking the facts

$$\begin{aligned} \bigl\Vert z^{k}-z^{*}\bigr\Vert =& \biggl\Vert \frac{\vert x^{k}\vert +x^{k}}{\gamma}-\frac {\vert x^{*}\vert +x^{*}}{\gamma} \biggr\Vert \leq \frac{2}{\gamma}\bigl\Vert x^{k}-x^{*}\bigr\Vert , \end{aligned}$$

(8)

$\Psi(z)$ is a Lipschitz continuous function, and (8) into account, we have

$$\begin{aligned} \bigl\Vert \Psi\bigl(z^{k}\bigr)-\Psi \bigl(z^{*}\bigr)\bigr\Vert \leq L\bigl\Vert z^{k}-z^{*} \bigr\Vert \leq\frac{2L}{\gamma}\bigl\Vert x^{k}-x^{*} \bigr\Vert . \end{aligned}$$

(9)

Thereby, we derive from (7) and (9) that

$$\begin{aligned} &{} \bigl(1- \bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert \bigr)\bigl\Vert x^{k+1}-x^{*}\bigr\Vert \\ \leq{}& \bigl(2 \bigl(\bigl\Vert (M_{1}+\Omega)^{-1}N_{1} \bigr\Vert +L\bigl\Vert (M_{1}+\Omega)^{-1}\bigr\Vert \bigr) +\bigl\Vert (M_{1}+\Omega)^{-1}(\Omega-M_{1}) \bigr\Vert \\ &{}+\bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert \bigr)\bigl\Vert x^{k}-x^{*}\bigr\Vert , \end{aligned}$$

which is equivalent to

$$\bigl\Vert x^{k+1}-x^{*}\bigr\Vert \leq\frac{2q(\Omega)+f(\Omega)+g(\Omega)}{1-g(\Omega )} \bigl\Vert x^{k}-x^{*}\bigr\Vert $$

with $g(\Omega)<1$. The condition

$$\frac{2q(\Omega)+f(\Omega)+g(\Omega)}{1-g(\Omega)}< 1 $$

with $g(\Omega)<1$, which is equivalent to $\rho(\Omega)=2g(\Omega )+2q(\Omega)+f(\Omega)<1$, ensures that the limit $\lim_{k\rightarrow +\infty}x^{k}=x^{*}$ holds. These results complete the proof. □

Theorem 7

Let $A\in R^{n\times n}$ be a P-matrix, $M_{1}-N_{1}=M_{2}-N_{2}=A$ be two splittings of the matrix A with $M_{1}\in R^{n\times n}$ being a symmetric P-matrix. Suppose that $\Omega=\omega I\in R ^{n\times n}$ is a positive scalar matrix and ω is a positive constant. $\Psi (z): R^{n}\rightarrow R^{n}$ is a Lipschitz continuous function with the Lipschitz constant L. $\lambda_{\max}$ and $\lambda_{\min}$ to denote the largest and smallest eigenvalue of the matrix $M_{1}$, respectively. Let $\tau_{1}=\Vert M_{1}^{-1}N_{1}\Vert _{2}$ and $\tau_{2}=\Vert M_{1}^{-1}N_{2}\Vert _{2}$ satisfy $\tau_{1}+\tau_{2}<1$. If $\lambda _{\min}>L$, the choices of the parameters ω, $M_{1}, N_{1}, M_{2}, N_{2}$ satisfy either of the following conditions, then the iteration sequence $\{z_{k}\}_{k}^{+\infty}\subset R^{n}_{+}$ generated by Algorithm 5 converges to the unique solution $z^{*}\in R^{n}_{+}$ of WNCP for any initial vector $x^{0}\in R^{n}$.

• (i) When $0<\tau_{1}+\tau_{2}<\frac{\lambda _{\min}-L}{\lambda_{\max}}$,

  $$\omega=\sqrt{\lambda_{\max}\lambda_{\min}}. $$

• (ii) When $\frac{\lambda_{\min}-L}{\lambda_{\max}} <\tau _{1}+\tau_{2}<\frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}$,

  $$\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega< \frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}. $$

• (iii) When $\tau_{1}+\tau_{2}=\frac{\lambda _{\min}-L}{\lambda_{\max}}$,

  $$\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}. $$

Proof

We first give some formulations, which will be used in the proof. Since M is a symmetric P-matrix and $\tau_{1}+\tau_{2}<1$, by the definition of the spectral norm, we have

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}N_{1} \bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}M_{1}M_{1}^{-1}N_{1} \bigr\Vert _{2} \\ \leq&\bigl\Vert (M_{1}+\omega I)^{-1}M_{1} \bigr\Vert _{2}\bigl\Vert M_{1}^{-1}N_{1} \bigr\Vert _{2} \\ =&\max_{\lambda\in\lambda(M_{1})}\frac{\lambda\tau_{1}}{\omega+\lambda } \\ =&\frac{\lambda_{\max}\tau_{1}}{\omega+\lambda_{\max}}. \end{aligned}$$

(10)

Similarly, we have

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}M_{1}M_{1}^{-1}N_{1} \bigr\Vert _{2} \\ \leq&\frac{\lambda_{\max}\tau_{2}}{\omega+\lambda_{\max}} \end{aligned}$$

(11)

and

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}\bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}\bigr\Vert _{2} \\ =&\max_{\lambda\in\lambda(M_{1})}\frac{1}{\omega+\lambda} \\ =&\frac{1}{\omega+\lambda_{\min}}. \end{aligned}$$

(12)

In addition, from a simple calculating process, we have

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}( \Omega-M_{1})\bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}(\omega I-M_{1})\bigr\Vert _{2} \\ =&\max_{\lambda\in\lambda(M_{1})}\frac{\vert \omega-\lambda \vert }{\omega+\lambda } =\max\biggl\{ \frac{\vert \omega-\lambda_{\max} \vert }{\omega+\lambda_{\max}},\frac {\vert \omega-\lambda_{\min} \vert }{\omega+\lambda_{\min}}\biggr\} \\ =&\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\lambda_{\max}-\omega}{\lambda_{\max}+\omega}, & \omega \leq\sqrt{\lambda_{\max}\lambda_{\min}},\\ \frac{\omega-\lambda_{\min}}{\omega+\lambda_{\min}}, & \omega \geq\sqrt{\lambda_{\max}\lambda_{\min}}. \end{array}\displaystyle \right . \end{aligned}$$

(13)

As follows from (10)-(13), we have

$$\begin{aligned} \rho(\Omega) =&2g(\Omega)+2q(\Omega)+f(\Omega) \\ =&2 \bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert _{2}+2 \bigl(\bigl\Vert (M_{1}+\Omega )^{-1}N_{1}\bigr\Vert _{2} \\ &{}+L\bigl\Vert (M_{1}+\Omega)^{-1}\bigr\Vert _{2} \bigr)+\bigl\Vert (M_{1}+\Omega)^{-1}( \Omega -M_{1})\bigr\Vert _{2} \\ =&2\frac{\lambda_{\max}\tau_{2}}{\omega+\lambda_{\max}}+2\biggl(\frac{\lambda _{\max}\tau_{1}}{\omega+\lambda_{\max}} +\frac{L}{\omega+\lambda_{\min}}\biggr)+\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\lambda_{\max}-\omega}{\omega+\lambda_{\max}}, & \omega \leq\sqrt{\lambda_{\max}\lambda_{\min}},\\ \frac{\omega-\lambda_{\min}}{\omega+\lambda_{\min}}, & \omega \geq\sqrt{\lambda_{\max}\lambda_{\min}}. \end{array}\displaystyle \right . \end{aligned}$$

(14)

We then consider two cases.

(a) When $\omega\leq\sqrt{\lambda_{\max}\lambda_{\min}}$, by a simple calculation on (14), we see that ω, $\tau_{1}$, and $\tau_{2}$ satisfy $\rho(\Omega)<1$, which is equivalent to

$$\omega^{2}- \bigl((\tau_{1}+\tau_{2}) \lambda_{\max}+L-\lambda_{\min} \bigr)\omega- \bigl(( \tau_{1}+\tau_{2})\lambda_{\max} \lambda_{\min}+L\lambda _{\max} \bigr)>0. $$

Note that $\omega>0$ and $\omega\leq\sqrt{\lambda_{\max}\lambda_{\min}}$, then the solution of the above inequality is

$$\theta(\tau_{1},\tau_{2})< \omega\leq\sqrt{ \lambda_{\max}\lambda_{\min}}, $$

where

$$\begin{aligned} &{}\theta(\tau_{1},\tau_{2}) \\ =&\frac{\sqrt{ ((\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min} )^{2}+4(\tau_{1}+\tau_{2})\lambda_{\max}\lambda_{\min}+L\lambda _{\max}}}{2} \\ &{}+\frac{(\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}}{2}. \end{aligned}$$

Certainly, we have

$$\begin{aligned} \theta(\tau_{1},\tau_{2})< \sqrt{ \lambda_{\max}\lambda_{\min}}. \end{aligned}$$

(15)

Since $\lambda_{\min}>L$, by the definitions of $\tau_{1}$, $\tau_{2}$ and solving (15), we get

$$0< \tau_{1}+\tau_{2}< \frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

(b) When $\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}$, in a same way as (a), ω, $\tau_{1}$, and $\tau_{2}$ satisfying $\rho(\Omega)<1$, which is equivalent to

$$\begin{aligned} \bigl[(\tau_{1}+\tau_{2}) \lambda_{\max}+L-\lambda_{\min} \bigr]\omega+(\tau _{1}+\tau_{2}-1)\lambda_{\max}\lambda_{\min}+L \lambda_{\max}< 0. \end{aligned}$$

(16)

If $(\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}>0$, that is, $\tau _{1}+\tau_{2}>\frac{\lambda_{\min}-L}{\lambda_{\max}}$, then

$$\omega< \frac{[1-(\tau_{1}+\tau_{2})]\lambda_{\max}\lambda_{\min}-L\lambda _{\max}}{(\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}}. $$

Combined with $\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}$, we get

$$\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega< \frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}. $$

Naturally, we have

$$\sqrt{\lambda_{\max}\lambda_{\min}}< \frac{[1-(\tau_{1}+\tau_{2})]\lambda _{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau_{2})\lambda _{\max}+L-\lambda_{\min}}. $$

That is,

$$\tau_{1}+\tau_{2}< \frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

This, together with $\tau_{1}+\tau_{2}>\frac{\lambda_{\min}-L}{\lambda _{\max}}$, shows that we have

$$\frac{\lambda_{\min}-L}{\lambda_{\max}}< \tau_{1}+\tau_{2}< \frac{\lambda _{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

If $(\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}\leq0$, that is, $\tau_{1}+\tau_{2}\leq\frac{\lambda_{\min}-L}{\lambda_{\max}}$, then for any $\omega>0$ (16) holds. So

$$\omega\geq{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

Hence, from (a) and (b), we see that when $0<\tau_{1}+\tau_{2}<\frac {\lambda_{\min}-L}{\lambda_{\max}}, \omega=\sqrt{\lambda_{\max}\lambda _{\min}}$; when $\frac{\lambda_{\min}-L}{\lambda_{\max}} <\tau_{1}+\tau _{2}<\frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}$, $\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega<\frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}$; when $\tau_{1}+\tau_{2}=\frac{\lambda_{\min}-L}{\lambda_{\max}}$, $\omega \geq\sqrt{\lambda_{\max}\lambda_{\min}}$. The proof is completed. □

5 Results and discussion

This study focused on the weakly nonlinear complementarity problems with a large sparse matrix. We proposed an algorithm that is not only computationally more convenient to use but also faster than the modulus-based matrix splitting iteration methods and the convergence conditions are presented when the system matrix is a P-matrix.

Some scholars had already stressed the accelerated modulus-based matrix splitting iteration methods for linear complementarity problems and pointed out that the system matrix is either a positive definite matrix or an $H_{+}$-matrix. However, we suggest that the system matrix is a P-matrix, this is more adaptable but also a limitation. Notwithstanding its limitation, this study does suggest that WNCP can be solved faster.

6 Conclusions

In this paper, by reformulating the complementarity problem (1) as an implicit fixed point equation based on splittings of the system matrix A, we establish an accelerated modulus-based matrix splitting iteration algorithm and show the convergence analysis when the involved matrix of the WNCP is a P-matrix.