# More on Modulus Based Iterative Method for Solving Implicit Complementarity
Problem

Bharat kumara,1, Deepmalaa,2 and A.K. Das b,3
_a Mathematics Discipline,_
_PDPM-Indian Institute of Information Technology, Design and Manufacturing,_
_Jabalpur - 482005 (MP), India_
_b Indian Statistical Institute, 203 B.T. Road, _
_Kolkata - 700108, India_
_1<EMAIL_ADDRESS>, 2Email<EMAIL_ADDRESS>
_3 Email<EMAIL_ADDRESS>
###### Abstract

This article presents a class of modified new modulus-based iterative methods
to process the large and sparse implicit complementarity problem (ICP). By
using two positive diagonal matrices, we formulate a fixed-point equation
which is equivalent to an ICP and based on a fixed-point equation, an
iterative method is presented to solve the ICP. We provide some convergence
conditions for the proposed methods when the system matrix is a $P$-matrix or
an $H_{+}$-matrix.

Keywords. Implicit complementarity problem, $H_{+}$-matrix, $P$-matrix, matrix
splitting method, convergence.

Mathematics Subject Classification. 90C33, 65F10, 65F50.

##  1 Introduction

The implicit complementarity problem (ICP) was introduced by Bensoussan et al.
in 1973. For detail, see [2], [3]. The linear complementarity problem (LCP) is
a special case of the ICP. The ICP is used in many fields, such as engineering
and economics, scientific computing, stochastic optimal control problems and
convex cones. The ICP has been discussed and studied in some literature. For
details, see [35], [22], [5], [29], [14], [4].
Consider matrix $A\in\mathbb{R}^{n\times n}$ and vector $q\in\mathbb{R}^{n}$.
The implicit complementarity problem denoted as ICP$(q,A,\psi)$ is to find the
solution $z\in\mathbb{R}^{n}$ to the following system:

$p(z)=Az+q\geq 0,~{}~{}~{}r(z)=z-\psi(z)\geq 0,~{}~{}~{}p(z)^{T}r(z)=0,$ (1)

where $\psi(z)$ is a mapping from $\mathbb{R}^{n}$ into $\mathbb{R}^{n}$ such
that $r(z)$ is invertible. When we put $\psi(z)=0$ in Equation (1), the
ICP$(q,A,\psi)$ reduces to the linear complementarity problem (LCP). For more
details on LCP and its applications, see [6], [30], [7], [9], [10], [27],
[28], [31], [33], [32], [26], [8], [23].
In recent decades to solve complementarity problems, many methods have been
proposed by considering some equivalent problems [37], [36], [38], such as
fixed point approaches [34], projection-type methods [43], smooth and
nonsmooth Newton methods [24], [40], linearization methods [16], domain
decomposition methods [1], matrix multisplitting methods [18], [12], [13],
[11], [39], [25] and inexact alternating direction methods [19], [20], [44],
[45]. In 2016, Hong and Li [21] demonstrated numerically the superiority of
the modulus-based matrix splitting (MMS) method over the projection fixed-
point method and the Newton method when solving the ICP$(q,A,\psi)$. Shilang
and Liang [42] presented a family of new modulus-based matrix splitting (NMMS)
iteration methods to solve the ICP$(q,A,\psi)$ in 2022. The NMMS method is not
only different from the inner-outer iteration methods. In addition, the NMMS
method is simple and efficient in its implementation. Motivated by the work of
Shilang and Liang, we propose a class of modified new modulus-based iterative
methods based on new matrix splitting for solving ICP$(q,A,\psi)$ which covers
the NMMS method.

This article is presented as follows: Some useful definitions, notations and
well-known lemmas are given in Section 2 which are required for the remaining
sections of this work. In Section 3, a family of modified new modulus-based
matrix splitting methods is constructed using the new equivalent fixed-point
form of the ICP$(q,A,\psi)$. In Section 4, we establish some convergence
criteria for the proposed methods. Section 5 contains the conclusion of the
article.

##  2 Preliminaries

In this section, we provide some basic notations, definitions and lemmas that
will be used throughout the article.
Let $A=({a}_{ij})\in\mathbb{R}^{n\times n}$,
$B=({b}_{ij})\in\mathbb{R}^{n\times n}$. We use $A\geq(>)B$ to denote
$a_{ij}\geq(>)b_{ij}$ $\forall$ $i,j$ and $A^{T}$ denotes the transpose of the
matrix $A$. Let $A=0\in\mathbb{R}^{n\times n}$ defined as
$a_{ij}=0~{}\forall~{}i,j$ and $\lvert A\rvert=(c_{ij})$ defined by
$c_{ij}=\lvert{a}_{ij}\rvert$ $\forall~{}i,j$.

######  Definition 2.1.

[15] Suppose $A=(a_{ij})\in\mathbb{R}^{n\times n}$. Then its comparison matrix
$\langle A\rangle=(\langle a_{ij}\rangle)$ is defined by $\langle
a_{ij}\rangle$ = $\lvert{a}_{ij}\rvert$ if $i=j$ and $-|{a}_{ij}\rvert$ if
$i\neq j$ for $i,j=1,2,\ldots,n$.

######  Definition 2.2.

[17] Suppose $A\in\mathbb{R}^{n\times n}$. Then $A$ is said to be

  1. 1.

a $Z$-matrix if all of its off-diagonal elements are nonpositive.

  2. 2.

an $M$-matrix if $A^{-1}\geq 0$ as well as $Z$-matrix.

  3. 3.

an $H$-matrix, if $\langle A\rangle$ is an $M$-matrix.

  4. 4.

an $H_{+}$-matrix if it is an $H$-matrix as well as ${a}_{ii}>0$ for
$~{}i=1,2,\ldots,n$.

######  Definition 2.3.

[42] Let $A=(a_{ij})\in\mathbb{R}^{n\times n}$. Then $A$ is said to be a
strictly diagonally dominant (sdd) matrix if $\lvert a_{ii}\rvert>\sum_{j\neq
i}\lvert a_{ij}\rvert$ for $i,j=1,2,\ldots,n.$

######  Lemma 2.1.

[41] Let $x,y\in\mathbb{R}^{n}.$ Then $x\geq 0$, $y\geq 0$ and $x^{T}y=0$ if
and only if $x+y=\lvert x-y\rvert.$

######  Lemma 2.2.

[42] Let $A\in\mathbb{R}^{n\times n}$ be a sdd matrix. Then

$\|A^{-1}E\|_{\infty}\leq\max_{1\leq i\leq n}\frac{(\lvert E\rvert
e)_{i}}{(\langle A\rangle e)_{i}},~{}~{}\mbox{for
all}~{}E\in\mathbb{R}^{n\times n},$

where $e=(1,1,\ldots,1)^{T}$.

######  Lemma 2.3.

[42] Let $A\in\mathbb{R}^{n\times n}$ be a nonsingular $M$-matrix, then there
exists a positive diagonal matrix $U$ such that $AU$ is an sdd matrix.

##  3 Main results

In this section, we present a class of modified new modulus-based splitting
methods for solving ICP$(q,A,\psi)$. First, we construct a new equivalent
expression of the ICP$(q,A,\psi)$ using the Lemma 2.1. The details are as
follows.

######  Theorem 3.1.

Let $A\in\mathbb{R}^{n\times n}$ and $q\in\mathbb{R}^{n}$. Suppose $\phi_{1}$,
$\phi_{2}\in\mathbb{R}^{n\times n}$ are two positive diagonal matrices. Then
the ICP$(q,A,\psi)$ is equivalent to

$(\phi_{1}A+\phi_{2})z=\lvert{(\phi_{1}A-\phi_{2})z+\phi_{1}q+\phi_{2}\psi(z)}\rvert-\phi_{1}q+\phi_{2}\psi(z).$
(2)

###### Proof.

The ICP$(q,A,\psi)$ can be rewritten as

$\phi_{1}(Az+q)\geq 0,~{}~{}~{}~{}~{}~{}\phi_{2}(z-\psi(z))\geq
0,~{}~{}~{}~{}~{}~{}~{}(\phi_{2}(z-\psi(z))^{T}(\phi_{1}(Az+q))=0.$

Let $x=\phi_{1}(Az+q)$ and $y=\phi_{2}(z-\psi(z))$. Then by using the Lemma
2.1, we write

$\phi_{1}(Az+q)+\phi_{2}(z-\psi(z))=\lvert\phi_{1}(Az+q)-\phi_{2}(z-\psi(z))\rvert.$

This implies that

$(\phi_{1}A+\phi_{2})z=\lvert{(\phi_{1}A-\phi_{2})z+\phi_{1}q+\phi_{2}\psi(z)}\rvert-\phi_{1}q+\phi_{2}\psi(z).$

∎

According to the statement in the Theorem 3.1, we are able to find the
solution to ICP$(q,A,\psi)$ with the use of an implicit fixed-point Equation
(2). Now we provide an iterative method for solving the ICP$(q,A,\psi)$ by
using a new matrix splitting of the system matrix $A$. Suppose
$A=(M_{1}+\phi)-(N_{1}+\phi)$ as a matrix splitting of
$A\in\mathbb{R}^{n\times n}$, where $\phi$ is a diagonal matrix of order $n$.
Now from Equation (2), we obtain

$\begin{split}(\phi_{1}M_{1}+\phi_{1}\phi+\phi_{2})z&=(\phi_{1}N_{1}+\phi_{1}\phi)z+\lvert{(\phi_{1}A-\phi_{2})z+\phi_{1}q+\phi_{2}\psi(z)}\rvert\\\
&-\phi_{1}q+\phi_{2}\psi(z).\end{split}$ (3)

Let
$\phi_{1}M_{1}=M_{\phi_{1}},~{}\phi_{1}N_{1}=N_{\phi_{1}},~{}\phi_{1}A=A_{\phi_{1}}$
and $\phi_{1}\phi=\phi_{\phi_{1}}$. Then the Equation (3) can be rewritten as

$\begin{split}(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})z&=(N_{\phi_{1}}+\phi_{\phi_{1}})z+\lvert(A_{\phi_{1}}-\phi_{2})z+\phi_{1}q+\phi_{2}\psi(z)\rvert\\\
&-\phi_{1}q+\phi_{2}\psi(z).\end{split}$ (4)

In order to solve the ICP$(q,A,\psi)$ from Equation (4) we establish a
modified new modulus-based matrix splitting iterative method which is known as
Method 3.1.

######  Method 3.1.

Let $A=(M_{1}+\phi)-(N_{1}+\phi)$ be a splitting of the matrix
$A\in\mathbb{R}^{n\times n}$ and $q\in\mathbb{R}^{n}$. Suppose that
$M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2}$ is a nonsingular matrix where $\phi$
is a diagonal matrix and $\phi_{1}$, $\phi_{2}$ are two positive diagonal
matrices. Then we use the following equation for Method 3.1 is

$\begin{split}z^{(k+1)}&=(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}[(N_{\phi_{1}}+\phi_{\phi_{1}})z^{(k)}+\lvert(A_{\phi_{1}}-\phi_{2})z^{(k)}+\phi_{1}q\\\
&+\phi_{2}\psi(z^{k})\rvert-\phi_{1}q+\phi_{2}\psi(z^{k})].\end{split}$ (5)

Let residual be the Euclidean norm of the error vector which is defined in
[42] as follows:

$Res(z^{(k)})=\|min(z^{(k)}-\psi(z^{k}),Az^{(k)}+q)\|_{2}.$

Consider a nonnegative initial vector $z^{(0)}\in\mathbb{R}^{n}$. For
$k=0,1,2,\ldots,$ the iterative process continues until the iterative sequence
$\\{z^{(k)}\\}_{k=0}^{+\infty}\subset\mathbb{R}^{n}$ converges. The iterative
process stops if $Res(z^{(k)})$ $<$ $\epsilon$. For computing
$z^{(k+1)}\in\mathbb{R}^{n}$ we use the following steps.
Step 1: Initialize a vector $z^{(0)}\in\mathbb{R}^{n}$, $\epsilon>0$ and set
$k=0$.
Step 2: Using the following scheme, generate the sequence $z^{(k)}$:

$\begin{split}z^{(k+1)}&=(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}[(N_{\phi_{1}}+\phi_{\phi_{1}})z^{(k)}+\lvert(A_{\phi_{1}}-\phi_{2})z^{(k)}+\phi_{1}q\\\
&+\phi_{2}\psi(z^{k})\rvert-\phi_{1}q+\phi_{2}\psi(z^{k})].\end{split}$

Step 3: If $Res(z^{(k)})$ $<$ $\epsilon$ then stop otherwise set $k=k+1$ and
return to step 2.

######  Remark 3.1.

If we put $\phi_{1}=I$, where $I$ is an identity matrix of order $n$ and
$\phi=0\in\mathbb{R}^{n\times n}$ in Equation $(\ref{eq2})$, then the result
of Theorem 3.1 reduces to the result [42].

Furthermore, Method 3.1 provides a general framework for solving
ICP$(q,A,\psi)$. We obtain a class of modified new modulus-based matrix
splitting iterative methods. In particular, we express the system matrix $A$
as $A=(M_{1}+\phi)-(N_{1}+\phi)$. Then

  1. 1.

when $M_{1}=A,~{}\phi_{1}=I,~{}\phi_{2}=I,~{}N_{1}=0$, from Equation (5) we
obtain

$(A+\phi+I)z^{(k+1)}=\phi
z^{(k)}+\lvert(A-I)z^{(k)}+q+\psi(z^{(k)})\rvert-q+\psi(z^{(k)}),$

this is known as the “modified new modulus-based iterative (MNMOD) method$"$.

  2. 2.

when $M_{1}=A,~{}\phi_{1}=I,~{}\phi_{2}=\theta_{1}I,~{}N_{1}=0$, from Equation
(5) we obtain

$(A+\phi+\theta_{1}I)z^{(k+1)}=\phi
z^{(k)}+\lvert(A-\theta_{1}I)z^{(k)}+q\psi(z^{(k)})\rvert-q+\psi(z^{(k)}),$

this is known as the “modified new modified modulus iterative (MNMMOD)
method$"$.

  3. 3.

when
$M_{1}=\frac{1}{\theta_{1}}(D_{1}-\theta_{2}L_{1}),~{}N_{1}=\frac{1}{\theta_{1}}[(1-\theta_{1})D_{1}+(\theta_{1}-\theta_{2})L_{1}+\theta_{1}U_{1}]$,
where $D_{1}=diag(A)$ and $L_{1}$, $U_{1}$ are the strictly lower, upper
triangular matrices of $A$, respectively. From Equation (5), we obtain

$\begin{split}((D_{\phi_{1}}-\theta_{2}L_{\phi_{1}})+\theta_{1}\phi_{\phi_{1}}+\theta_{1}\phi_{2})z^{(k+1)}&=((1-\theta_{1})D_{\phi_{1}}+(\theta_{1}-\theta_{2})L_{\phi_{1}}\\\
&+\theta_{1}U_{\phi_{1}}+\theta_{1}\phi_{\phi_{1}})z^{(k)}+\theta_{1}(\lvert(A_{\phi_{1}}\\\
&-\phi_{2})z^{(k)}+\phi_{1}q+\phi_{2}\psi(z^{(k)})\rvert\\\
&-\phi_{1}q+\phi_{2}\psi(z^{(k)})),\end{split}$

this is known as the “modified new modulus-based accelerated over relaxation
iterative (MNMAOR) method$"$.
The MNMAOR method clearly converts into the modified new modulus-based
successive overrelaxation (MNMSOR) method when $(\theta_{1},\theta_{2})$ takes
the values $(\theta_{1},\theta_{1})$, modified new modulus-based Gauss-Seidel
(MNMGS) method when $(\theta_{1},\theta_{2})$ takes the values $(1,1)$ and
modified new modulus-based Jacobi (MNMJ) method when $(\theta_{1},\theta_{2})$
takes the values $(1,0)$.

##  4 Convergence analysis

In this section, we discuss the convergence analysis for the Method 3.1.
First, we discuss the convergence condition when the system matrix $A$ is a
$P$-matrix.

######  Theorem 4.1.

Let $A=(M_{1}+\phi)-(N_{1}+\phi)$ be a splitting of the $P$-matrix
$A\in\mathbb{R}^{n\times n}$ and $M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2}$ be
nonsingular matrix, where $\phi$ is a diagonal matrix and $\phi_{1}$,
$\phi_{2}$ are positive diagonal matrices. Let $\rho(L)<1$, where
$L=\lvert(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}\rvert(\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G)$ and $\lvert\psi(x)-\psi(y)\rvert\leq
G\lvert x-y\rvert$ $\forall~{}x,y\in\mathbb{R}^{n}$, where $G$ is a
nonnegative matrix of order $n$. Then the iterative sequence
$\\{z^{(k)}\\}_{k=0}^{+\infty}\subset\mathbb{R}^{n}$ generated by Method 3.1
converges to a unique solution $z^{*}\in\mathbb{R}^{n}$ for any nonnegative
initial vector $z^{(0)}\in\mathbb{R}^{n}$.

###### Proof.

Let $z^{*}$ be a solution of the ICP$(q,A,\psi)$. From Equation(4) we obtain

$\begin{split}(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})z^{*}&=(N_{\phi_{1}}+\phi_{\phi_{1}})z^{*}+\lvert(A_{\phi_{1}}-\phi_{2})z^{*}+\phi_{1}q+\phi_{2}\psi(z^{*})\rvert\\\
&-\phi_{1}q+\phi_{2}\psi(z^{*}).\end{split}$ (6)

From Equation(4) and Equation (6), we obtain

$\begin{split}(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})(z^{(k+1)}-z^{*})&=(N_{\phi_{1}}+\phi_{\phi_{1}})(z^{(k)}-z^{*})+(\lvert(A_{\phi_{1}}-\phi_{2})z^{(k)}+\phi_{1}q\\\
&+\phi_{2}\psi(z^{(k)})\rvert-\lvert(A_{\phi_{1}}-\phi_{2})z^{*}+\phi_{1}q+\phi_{2}\psi(z^{*})\rvert)\\\
&+(\phi_{2}\psi(z^{(k)})-\phi_{2}\psi(z^{*})).\end{split}$

By applying both side modulus, it follows that

$\begin{split}&\leq\lvert(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}\rvert\lvert[(N_{\phi_{1}}+\phi_{\phi_{1}})(z^{(k)}-z^{*})\rvert+(\lvert(A_{\phi_{1}}-\phi_{2})z^{(k)}\\\
&+\phi_{1}q+\phi_{2}\psi(z^{(k)})\rvert-\lvert(A_{\phi_{1}}-\phi_{2})z^{*}+\phi_{1}q+\phi_{2}\psi(z^{*})\rvert)+\phi_{2}\psi(z^{(k)})\\\
&-\phi_{2}\psi(z^{*})]\rvert\\\
&\leq\lvert(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}\rvert[\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G]\lvert z^{(k)}-z^{*}\rvert\\\ &\leq
L\lvert z^{(k)}-z^{*}\rvert\end{split}$

where $L=\lvert(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}\rvert(\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G)$. Since $\rho(L)<1$, then

$\lvert z^{(k+1)}-z^{*}\rvert<\lvert z^{(k)}-z^{*}\rvert.$

Hence, for any nonnegative initial vector $z^{(0)}\in\mathbb{R}^{n}$ the
iterative sequence $\\{z^{(k)}\\}_{k=0}^{+\infty}\subset\mathbb{R}^{n}$
generated by Method 3.1 converges to a unique solution
$z^{*}\in\mathbb{R}^{n}$. ∎

In the following results, we discuss some convergence conditions for Method
3.1 when system matrix $A$ is an $H_{+}$-matrix.

######  Theorem 4.2.

Let $A=(M_{1}+\phi)-(N_{1}+\phi)$ be an $H$-splitting of the $H_{+}$-matrix
$A\in\mathbb{R}^{n\times n}$. Let $\phi_{2}\geq D_{\phi_{1}}$, where
$D_{\phi_{1}}=\phi_{1}D_{1}$ and $\phi_{1}$, $\phi_{2}$ are positive diagonal
matrices. Let $\phi=(\phi_{ij})$, $\phi_{1}=(w_{ij})$, $\phi_{2}=(\xi_{ij})$,
$M_{1}=(m_{ij})$, $N_{1}=(n_{ij})\in R^{n\times n}$ and $G=(g_{ij})\in
R^{n\times n}$ with $g_{ii}\leq\frac{w_{ii}\lvert
n_{ii}+\phi_{ii}\rvert-(w_{ii}n_{ii}+\lvert
w_{ii}\phi_{ii}\rvert)}{2\xi_{ii}}$ and $g_{ij}\leq\frac{\lvert
w_{ij}m_{ij}\rvert+\lvert w_{ij}n_{ij}\rvert-\lvert
w_{ij}a_{ij}\rvert}{2\xi_{ii}}$ for $i,j=1,2,\ldots,n$. Then the iterative
sequence $\\{z^{(k)}\\}_{k=0}^{+\infty}\subset\mathbb{R}^{n}$ generated by
Method 3.1 converges to a unique solution $z^{*}\in\mathbb{R}^{n}$ for any
nonnegative initial vector $z^{(0)}\in\mathbb{R}^{n}$.

###### Proof.

Since $\langle M_{1}+\phi\rangle-\lvert N_{1}+\phi\rvert$ is an $M$-matrix and

$\langle M_{1}+\phi\rangle-\lvert N_{1}+\phi\rvert\leq\langle
M_{1}+\phi\rangle,$

matrix $\langle M_{1}+\phi\rangle$ is an $M$-matrix and
$\phi_{2}\geq\phi_{1}D_{1}$, we can obtain that
$M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2}$ is an $H_{+}$-matrix and

$\lvert(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}\rvert\leq(\langle
M_{\phi_{1}}\rangle+\phi_{\phi_{1}}+\phi_{2})^{-1}.$

Now we show that matrix $\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G$ is a nonsingular $M$-matrix. First, we
calculate

for $i=j$,
$\lvert w_{ii}m_{ii}\rvert+\lvert w_{ii}\phi_{ii}\rvert+\xi_{ii}-\lvert
w_{ii}n_{ii}+w_{ii}\phi_{ii}\rvert-\lvert
w_{ii}a_{ii}-\xi_{ii}\rvert-2\xi_{ii}g_{ii}$
$\geq 2\lvert w_{ii}m_{ii}+w_{ii}\phi_{ii}\rvert-2\lvert
w_{ii}n_{ii}+w_{ii}\phi_{ii}\rvert$
$\geq 2(\lvert w_{ii}m_{ii}+w_{ii}\phi_{ii}\rvert-\lvert
w_{ii}n_{ii}+w_{ii}\phi_{ii}\rvert)$

and for $i\neq j$,
$-\lvert w_{ij}m_{ij}\rvert-\lvert w_{ij}n_{ij}\rvert-\lvert
w_{ij}a_{ij}\rvert-2w_{ij}g_{ij}$
$\geq-\lvert w_{ij}m_{ij}\rvert-\lvert w_{ij}n_{ij}\rvert-\lvert
w_{ij}a_{ij}\rvert-(\lvert w_{ij}m_{ij}\rvert+\lvert w_{ij}n_{ij}\rvert-\lvert
w_{ij}a_{ij}\rvert)$
$\geq-(\lvert w_{ij}m_{ij}\rvert+\lvert w_{ij}n_{ij}\rvert).$
It follows that
$\langle M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G\geq 2(\langle M_{1}+\phi\rangle-\lvert
N_{1}+\phi\rvert).$
We have $\langle M_{1}+\phi\rangle-\lvert N_{1}+\phi\rvert$ is an $M$-matrix.
Now we can obtain $\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G$ is a nonsingular $M$-matrix.
From Theorem 4.1, we obtain

$\begin{split}\lvert
z^{(k+1)}-z^{*}\rvert&=\lvert(M_{\phi_{1}}+\phi_{\phi_{1}}+\phi_{2})^{-1}\rvert(\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G)\lvert z^{(k)}-z^{*}\rvert\\\
&\leq(\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2})^{-1}(\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G)\lvert z^{(k)}-z^{*}\rvert\end{split}$

Let $\bar{L}=\bar{H}^{-1}\bar{F}$, where $\bar{H}=\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}$ and $\bar{F}=\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G$.
We now have
$\bar{H}-\bar{F}=\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G$ and $\bar{H}$ are $M$-matrices and
$\bar{F}\geq 0$.
It follows that, the splitting $\bar{H}-\bar{F}$ is an $M$-splitting.
Here, we are able to determine $\rho(\bar{L})<1$. According to Theorem 4.1,
the iterative sequence $\\{z^{(k)}\\}_{k=0}^{+\infty}\subset\mathbb{R}^{n}$
generated by Method 3.1 converges to a unique solution
$z^{*}\in\mathbb{R}^{n}$ for any nonnegative initial vector
$z^{(0)}\in\mathbb{R}^{n}$. ∎

######  Theorem 4.3.

Let $A=(M_{1}+\phi)-(N_{1}+\phi)=D_{1}-L_{1}-U_{1}=D_{1}-B$ be a splitting of
the $H_{+}$-matrix $A\in\mathbb{R}^{n\times n}$, where $B=L_{1}+U_{1}$ and
$\rho(D_{\phi_{1}}^{-1}\lvert B_{\phi_{1}}\rvert+\phi_{2}G)<1$. Let
$\phi_{2}\geq D_{\phi_{1}}$, where $G$ is a nonnegative matrix, $\phi$ is a
diagonal matrix and $\phi_{1}$ , $\phi_{2}$ are positive diagonal matrices. If
the parameters $\theta_{1}$, $\theta_{2}$ satisfy

$0~{}\leq~{}\max\\{\theta_{1},\theta_{2}\\}\rho(D_{\phi_{1}}^{-1}\lvert
B_{\phi_{1}}\rvert+\phi_{2}G)<\min\\{1,\theta_{1}\\},$ (7)

then the iterative sequence
$\\{z^{(k)}\\}_{k=0}^{+\infty}\subset\mathbb{R}^{n}$ generated by MNMAOR
method converges to a unique solution $z^{*}\in\mathbb{R}^{n}$ for any
nonnegative initial vector $z^{(0)}\in\mathbb{R}^{n}$.

###### Proof.

Following the proof of the Theorem 4.2, we have

$\bar{H}-\bar{F}=\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G.$

Let $M_{1}=\frac{1}{\theta_{1}}(D_{1}-\theta_{2}L_{1})$ and
$N_{1}=\frac{1}{\theta_{1}}((1-\theta_{1})D_{1}+(\theta_{1}-\theta_{2})L_{1}+\theta_{1}U_{1}).$
Then

$\begin{split}\bar{H}&=\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}\\\
&=\langle\frac{1}{\theta_{1}}(D_{\phi_{1}}-\theta_{2}L_{\phi_{1}})\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}\\\
&=\frac{1}{\theta_{1}}[(D_{\phi_{1}}-\theta_{2}\lvert
L_{\phi_{1}}\rvert)+\theta_{1}\lvert\phi_{\phi_{1}}\rvert+\theta_{1}\phi_{2}]\end{split}$

and

$\begin{split}\bar{F}=\frac{1}{\theta_{1}}[\lvert(1-\theta_{1})D_{\phi_{1}}+(\theta_{1}-\theta_{2})L_{\phi_{1}}+\theta_{1}U_{\phi_{1}}+\theta_{1}\phi_{\phi_{1}}\rvert+\theta_{1}\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\theta_{1}\phi_{2}G].\end{split}$

Now we compute

$\begin{split}\theta_{1}(\bar{H}-\bar{F})&=[(D_{\phi_{1}}-\theta_{2}\lvert
L_{\phi_{1}}\rvert)+\theta_{1}\lvert\phi_{\phi_{1}}\rvert+\theta_{1}\phi_{2}]-[\lvert(1-\theta_{1})D_{\phi_{1}}\\\
&+(\theta_{1}-\theta_{2})L_{\phi_{1}}+\theta_{1}U_{\phi_{1}}+\theta_{1}\phi_{\phi_{1}}\rvert+\theta_{1}\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\theta_{1}\phi_{2}G]\\\ &\geq(1+\theta_{1}-\lvert
1-\theta_{1}\rvert)D_{\phi_{1}}-\lvert\theta_{1}B_{\phi_{1}}-\theta_{2}L_{1}\rvert\\\
&-\theta_{1}\lvert B_{\phi_{1}}\rvert-\theta_{2}\lvert
L_{\phi_{1}}\rvert-2\theta_{1}\phi_{2}G.\end{split}$

Since

$(1+\theta_{1}-\lvert 1-\theta_{1}\rvert)=2\min\\{1,\theta_{1}\\}$

and

$\begin{split}&\lvert\theta_{1}B_{\phi_{1}}-\theta_{2}L_{\phi_{1}}\rvert+\theta_{1}\lvert
B_{\phi_{1}}\rvert+\theta_{2}\lvert L_{\phi_{1}}\rvert\\\
&\leq\theta_{1}\lvert B_{\phi_{1}}\rvert+\theta_{2}\lvert
L_{\phi_{1}}\rvert+\theta_{1}\lvert B_{\phi_{1}}\rvert+\theta_{2}\lvert
L_{\phi_{1}}\rvert\\\ &\leq 2\max\\{\theta_{1},\theta_{2}\\}\lvert
B_{\phi_{1}}\rvert.\end{split}$

At this point,

$\begin{split}\theta_{1}(\bar{H}-\bar{F})&\geq
2\min\\{1,\theta_{1}\\}D_{\phi_{1}}-2\max\\{\theta_{1},\theta_{2}\\}\lvert
B_{\phi_{1}}\rvert-2\theta_{1}\phi_{2}G\\\ &\geq
2D_{\phi_{1}}(\min\\{1,\theta_{1}\\}I-\max\\{\theta_{1},\theta_{2}\\}D^{-1}_{\phi_{1}}(\lvert
B_{\phi_{1}}\rvert+\phi_{2}G).\end{split}$

Since $\rho(D^{-1}_{\phi_{1}}(\lvert B_{\phi_{1}}\rvert+\phi_{2}G)<1$. It is
easy to verify that $\theta_{1}(\bar{H}-\bar{F})$ an $M$-matrix. Further, we
obtain that matrix $\bar{H}-\bar{F}$ is an $M$-matrix. ∎

######  Theorem 4.4.

Let $A=(M_{1}+\phi)-(N_{1}+\phi)$ be an $H$-splitting with the $H_{+}$-matrix
$A\in\mathbb{R}^{n\times n}$, matrix $G$ a nonnegative matrix and $D=diag(A)$.
Then the Method 3.1 is convergent, provided that for some positive diagonal
matrix $U$ such that $(\langle M_{1}+\phi\rangle-(N_{1}+\phi))U$ is a strictly
diagonally dominant matrix, one of the following conditions is valid:
(1) $\phi_{2}\geq D_{\phi_{1}}$,

$(\langle M_{\phi_{1}}+\phi\rangle-\lvert
N_{\phi_{1}}+\phi\rvert-\phi_{2}G)Ue>0;$

(2) $0<\phi_{2}<D_{\phi_{1}}$,

$(\phi_{2}-D_{\phi_{1}}+\langle M_{\phi_{1}}+\phi\rangle-\lvert
N_{\phi_{1}}+\phi\rvert-\phi_{2}G)Ue>0.$

###### Proof.

Let $\bar{A}=\langle M_{1}+\phi\rangle-\lvert N_{1}+\phi\rvert$,
$\phi_{1}\bar{A}=\bar{A}_{\phi_{1}}$ and $A=(M_{1}+\phi)-(N_{1}+\phi)$ be an
$H$-splitting. Therefore $\bar{A}$ is a nonsingular $M$-matrix.

$0<\bar{A}Ue\leq\langle M_{1}+\phi\rangle Ue\leq(\langle
M_{1}+\phi\rangle+\phi_{2})Ue$

and

$0<\bar{A}Ue<\langle A\rangle Ue$

Now we calculate

$\begin{split}&(\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G)Ue\\\ &\geq(\langle
M_{\phi_{1}}+\phi_{\phi_{1}}\rangle-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G)Ue.\end{split}$ (8)

Here, we consider two cases
Case (1): when $\phi_{2}\geq D_{\phi_{1}}$,

$\begin{split}(\bar{A}_{\phi_{1}}+\phi_{2}-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}Q)Ue&=(\bar{A}_{\phi_{1}}+\phi_{2}-(\phi_{2}-\langle
A_{\phi_{1}}\rangle)-2\phi_{2}G)Ue\\\ &\geq
2(\bar{A}_{\phi_{1}}-\phi_{2}G)Ue\\\ &\geq 0.\end{split}$

Case (1): when $0<\phi_{2}\leq D_{\phi_{1}}$,

$\begin{split}(\bar{A}_{\phi_{1}}+\phi_{2}-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G)Ue&=(\bar{A}_{\phi_{1}}+\phi_{2}-(\lvert
A_{\phi_{1}}\rvert-\phi_{2})-2\phi_{2}G)Ue\\\
&=(\bar{A}_{\phi_{1}}+\phi_{2}-(2D_{\phi_{1}}-\langle
A_{\phi_{1}}\rangle-\phi_{2})-2\phi_{2}G)Ue\\\ &\geq
2(\bar{A}_{\phi_{1}}+\phi_{2}-D_{\phi_{1}}-\phi_{2}G)Ue\\\ &\geq
0.\end{split}$

Now from case (1), case (2) and Equation$(\ref{eq8})$, we obtain that

$(\langle M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2}-\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert-\lvert
A_{\phi_{1}}-\phi_{2}\rvert-2\phi_{2}G)Ue>0.$

Now from Lemma 2.2, we have

$\begin{split}\rho(\bar{L})&=\rho(U^{-1}\bar{L}U)\\\
&\leq\|U^{-1}\bar{L}U\|_{\infty}\\\ &=\|U^{-1}(\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2})^{-1}(\lvert
N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G)U\|_{\infty}\\\ &\leq\max_{1\leq i\leq
n}\frac{((\lvert N_{\phi_{1}}+\phi_{\phi_{1}}\rvert+\lvert
A_{\phi_{1}}-\phi_{2}\rvert+2\phi_{2}G)Ue)_{i}}{((\langle
M_{\phi_{1}}\rangle+\lvert\phi_{\phi_{1}}\rvert+\phi_{2})Ue)_{i}}\\\
&<1.\end{split}$

Hence, the Method 3.1 is convergent. ∎

##  5 Conclusion

We introduce a class of modified new modulus-based matrix splitting methods
based on a new matrix splitting approach for solving the implicit
complementarity problem ICP$(q,A,\psi)$ with parameter matrices $\phi_{1}$ and
$\phi_{2}$. These methods are able to process the large and sparse structure
of the matrix $A$. For the proposed methods, we establish some convergence
conditions when the system matrix is a $P$-matrix as well as some sufficient
convergence conditions when the system matrix is an $H_{+}$-matrix.

### Acknowledgments

The author, Bharat Kumar is thankful to the University Grants Commission
(UGC), Government of India, under the SRF fellowship, Ref. No.: 1068/(CSIR-UGC
NET DEC. 2017).

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