1 Introduction
The linear complementarity problem is to find a vector \(x\in \mathbb{R}^{n}\) such that
where \(M\in\mathbb{R}^{n\times n}\) and \(q\in\mathbb{R}^{n}\). We denote problem (1) and its solution by \(\operatorname{LCP}(M, q)\) and \(x^{*}\), respectively. The \(\operatorname{LCP}(M, q)\) often arises from the various scientific areas of computing, economics and engineering such as quadratic programs, optimal stopping, Nash equilibrium points for bimatrix games, network equilibrium problems, contact problems, and free boundary problems for journal bearing, etc. For more details, see [2‐4].
$$ x\geq0,\qquad Mx+q\geq0,\qquad (Mx+q)^{T}x=0, $$
(1)
An interesting problem for the \(\operatorname{LCP}(M, q)\) is to estimate
since it can often be used to bound the error \(\|x-x^{*}\|_{\infty}\) [5], that is,
where \(M_{D}=I-D+DM\), \(D=\operatorname{diag}(d_{i})\) with \(0\leq d_{i} \leq1\) for each \(i\in N\), \(d=[d_{1},d_{2},\ldots,d_{n}]^{T}\in[0,1]^{n}\), and \(r(x)=\min\{ x,Mx+q\}\) in which the min operator denotes the componentwise minimum of two vectors; for more details, see [1, 6‐14] and the references therein.
$$ \max_{d\in[0,1]^{n}}\big\| (I-D+DM)^{-1} \big\| _{\infty}, $$
(2)
$$ \big\| x-x^{*}\big\| _{\infty} \leq\max_{d\in[0,1]^{n}}\big\| M_{D}^{-1} \big\| _{\infty} \big\| r(x)\big\| _{\infty}, $$
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In [1], García-Esnaola and Peña provided an upper bound for (2) when M is a \(B^{S}\)-matrix as a subclass of P-matrices [15], which contains B-matrices. Here a matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called a B-matrix [16] if, for each \(i\in N=\{1,2,\ldots,n\}\),
and a matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called a \(B^{S}\)-matrix [15] if there exists a subset S, with \(2\leq \operatorname{card}(S)\leq n-2\), such that, for all \(i,j\in N\), \(t\in T(i)\setminus\{i\}\), and \(k\in K(j)\setminus\{j\}\),
where \(R_{i}^{S}=\frac{1}{n}\sum_{\substack{k\in S}}m_{ik}\), \(T(i):=\{t\in S | m_{it}>R_{i}^{S}\}\) and \(k(j):=\{k\in \overline{S} | m_{jk}>R_{j}^{\overline{S}}\}\) with \(\overline{S}=N\setminus\{S\}\).
$$ \sum_{\substack{k\in N}}m_{ik}>0,\quad \text{and}\quad \frac{1}{n} \biggl(\sum_{\substack{k\in N}}m_{ik} \biggr)>m_{ij} \quad\text{for any } j\in N \text{ and } j\neq i, $$
$$R_{i}^{S}>0, R_{j}^{\overline{S}}>0,\quad \text{and}\quad \bigl(m_{it}-R_{i}^{S} \bigr) \bigl(m_{jk}-R_{j}^{\overline{S}} \bigr)< R_{i}^{\overline{S}}R_{j}^{S}, $$
Theorem 1
([1, Theorem 2.8])
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix, and let
\(X=\operatorname{diag}(x_{1},x_{2},\ldots,x_{n})\)
with
such that
\(\tilde{M}:=MX\)
is a
B-matrix with the form
\(\tilde{M}=\tilde{B}^{+}+\tilde{C}\), where
and
\(\tilde{r}_{i}^{+}=\max\{ 0,m_{ij}x_{j} | j\neq i\}\). Then
where
\(\tilde{\beta}=\min_{i\in N}\{\tilde{\beta}_{i}\}\)
with
\(\tilde{\beta}_{i}=\tilde{b}_{ii}-\sum_{j\neq i}|\tilde{b}_{ij}|\), and
where max (min) is set to be −∞ (∞) if
\(K(j)\setminus \{j\}=\emptyset\) (\(T(i)\setminus\{i\}=\emptyset\)).
$$x_{i}=\left \{ \textstyle\begin{array}{l@{\quad}l} \gamma, &i\in S,\\ 1, &otherwise, \end{array}\displaystyle \right . $$
$$ \tilde{B}^{+} =[\tilde{b}_{ij}]= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} m_{11}x_{1}-\tilde{r}_{1}^{+} &\cdots &m_{1n}x_{n}-\tilde{r}_{1}^{+} \\ \vdots & &\vdots \\ m_{n1}x_{1}-\tilde{r}_{n}^{+} &\cdots &m_{nn}x_{n}-\tilde{r}_{n}^{+} \end{array}\displaystyle \right ],\qquad \tilde{C}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \tilde{r}_{1}^{+} &\cdots &\tilde{r}_{1}^{+} \\ \vdots & &\vdots \\ \tilde{r}_{n}^{+} &\cdots &\tilde{r}_{n}^{+} \end{array}\displaystyle \right ], $$
(3)
$$ \max_{d\in[0,1]^{n}}\big\| M_{D}^{-1} \big\| _{\infty} \leq \frac{(n-1)\max\{\gamma,1\}}{\min\{\tilde{\beta},\gamma,1\}}, $$
(4)
$$ (0< )\ \gamma\in \biggl(\max_{j\in N,k\in K(j)\setminus \{j\}}\frac{m_{jk}-R_{j}^{\overline{S}}}{R_{j}^{S}}, \min_{\substack{i\in N,t\in T(i)\setminus \{i\}}}\frac{R_{i}^{\overline{S}}}{m_{it}-R_{i}^{S}} \biggr), $$
(5)
Note that for some \(B^{S}\) matrices, β̃ can be very small, thus the error bound (4) can be very large (see examples in Section 3). Hence it is interesting to find an alternative bound for \(\operatorname{LCP}(M, q)\) to overcome this drawback. In this paper we provide a new upper bound for (2) and give a family of examples of \(B^{S}\)-matrices that are not B-matrices for which our bound is a small constant in contrast to bound (4) of [1], which can be arbitrarily large. Particularly, when the involved matrix is a B-matrix as a special class of \(B^{S}\)-matrices, the new bound is in line with that provided by Li et al. in [13].
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2 Main result
First, recall some definitions and lemmas which will be used later. A matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called: (1) a P-matrix if all its principal minors are positive; (2) a strictly diagonally dominant (SDD) matrix if \(|m_{ii}|>\sum_{j\neq i}^{n}|m_{ij}|\) for all \(i=1,2,\ldots,n\); (3) a nonsingular M-matrix if its inverse is nonnegative and all its off-diagonal entries are nonpositive [2].
Lemma 1
([1, Theorem 2.3])
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix. Then there exists a positive diagonal matrix
\(X=\operatorname{diag}(x_{1},x_{2},\ldots,x_{n})\)
with
such that
\(\tilde{M}:=MX\)
is a
B-matrix.
$$x_{i}=\left \{ \textstyle\begin{array}{l@{\quad}l} \gamma, &i\in S,\\ 1, &otherwise, \end{array}\displaystyle \right . $$
Lemma 2
([1, Lemma 2.4])
Let
\(M=[m_{ij}]\in \mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix, and let
X
be the diagonal matrix of Lemma
1
such that
\(\tilde{M}:=MX\)
is a
B-matrix with the form
\(\tilde{M}=\tilde{B}^{+}+\tilde{C}\), where
\(\tilde{B}^{+}=[\tilde{b}_{ij}]\)
is the matrix of (3). Then
\(\tilde{B}^{+}\)
is strictly diagonally dominant by rows with positive diagonal entries.
Lemma 3
([1, Lemma 2.6])
Let
\(M=[m_{ij}]\in \mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix that is not a
B-matrix, then there exist
\(k,i\in N\)
with
\(k\neq i\)
such that
Furthermore, if
\(k\in S\) (resp., \(k\in\overline{S}\)), then
\(\gamma<1\) (resp., \(\gamma>1\)), where the parameter
γ
satisfies (5).
$$ m_{ik}\geq\frac{1}{n}\sum _{j=1}^{n}m_{ij}. $$
(6)
Lemma 4
[17, Theorem 3.2] Let
\(A=[a_{ij}]\)
be an
\(n\times n\)
row strictly diagonally dominant
M-matrix. Then
where
\(u_{i}(A)=\frac{1}{|a_{ii}|}\sum_{j=i+1}^{n}|a_{ij}|\), \(l_{k}(A)=\max_{k\leq i\leq n} \{\frac{1}{|a_{ii}|}\sum_{\small\substack{j=k,\\j\neq i}}^{n}|a_{ij}| \}\), and
\(\prod_{j=1}^{i-1}\frac{1}{1-u_{j}(A)l_{j}(A)}=1\)
if
\(i=1\).
$$ \big\| A^{-1}\big\| _{\infty}\leq\sum_{i=1}^{n} \Biggl(\frac{1}{a_{ii}(1-u_{i}(A)l_{i}(A))}\prod_{j=1}^{i-1} \frac {1}{1-u_{j}(A)l_{j}(A)} \Biggr), $$
Lemma 5
([12, Lemma 3])
Let
\(\gamma> 0\)
and
\(\eta\geq0 \). Then, for any
\(x\in[0,1]\),
and
$$\frac{1}{1-x+\gamma x} \leq \frac{1}{\min\{\gamma,1\}} $$
$$\frac{\eta x}{1-x+\gamma x} \leq\frac{\eta}{\gamma}. $$
Lemma 6
([11, Lemma 5])
Let
\(A=[a_{ij}]\in\mathbb{R}^{n\times n}\)
with
\(a_{ii}>\sum_{j=i+1}^{n}|a_{ij}|\)
for each
\(i\in N\). Then, for any
\(x_{i}\in[0,1]\),
$$ \frac{1-x_{i}+a_{ii}x_{i}}{1-x_{i}+a_{ii}x_{i}-\sum_{j=i+1}^{n}|a_{ij}|x_{i}} \leq \frac{a_{ii}}{a_{ii}-\sum_{j=i+1}^{n}|a_{ij}|}. $$
Theorem 2
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix and
\(X=\operatorname{diag}(x_{1},x_{2},\ldots,x_{n})\)
with
such that
\(\tilde{M}:=MX\)
is a
B-matrix with the form
\(\tilde{M}=\tilde{B}^{+}+\tilde{C}\), where
\(\tilde{B}^{+}=[\tilde{b}_{ij}]\)
is the matrix of (3). Then
where
\(\widehat{\beta}_{i}=\tilde{b}_{ii}-\sum_{k=i+1}^{n}|\tilde {b}_{ik}|l_{i}(\tilde{B}^{+})\), and
\(\prod_{j=1}^{i-1}\frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}=1\)
if
\(i=1\).
$$ x_{i}=\left \{ \textstyle\begin{array}{l@{\quad}l} \gamma, &i\in S,\\ 1, &otherwise, \end{array}\displaystyle \right . $$
$$ \max_{d\in[0,1]^{n}}\big\| M_{D}^{-1} \big\| _{\infty} \leq \sum_{i=1}^{n} \frac{(n-1)\max\{\gamma,1\}}{\min\{\widehat{\beta }_{i},x_{i}\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}, $$
(7)
Proof
Since X is a positive diagonal matrix and \(\tilde{M}:=MX\), it is easy to get that \(M_{D}= I-D+DM=(X-DX+D\tilde{M})X^{-1}\). Let \(\tilde{M}_{D}=X-DX+D\tilde{M}\). Then
where \(\tilde{B}_{D}^{+}=X-DX+D\tilde{B}^{+}=[\hat{b}_{ij}]\) with
and \(\tilde{C}_{D}=D\tilde{C}\). By Lemma 2, \(\tilde{B}^{+}\) is strictly diagonally dominant by rows with positive diagonal entries. Similarly to the proof of Theorem 2.2 in [10], we can obtain that \(\tilde{B}_{D}^{+}\) is an SDD matrix with positive diagonal entries and that
$$\tilde{M}_{D}=X-DX+D\tilde{M}=X-DX+D\bigl(\tilde{B}^{+}+ \tilde{C}\bigr)=\tilde {B}_{D}^{+}+\tilde{C}_{D}, $$
$$\hat{b}_{ij}=\left \{ \textstyle\begin{array}{l@{\quad}l} x_{i}-d_{i}x_{i}+d_{i}\tilde{b}_{ij}, &i=j,\\ d_{i}\tilde{b}_{ij}, &i\neq j, \end{array}\displaystyle \right . $$
$$ \begin{aligned}[b] \big\| M_{D}^{-1}\big\| _{\infty}&\leq \big\| X^{-1}\big\| _{\infty}\cdot\big\| \tilde{M}_{D}^{-1}\big\| _{\infty}\\ &\leq \big\| X^{-1}\big\| _{\infty}\cdot\big\| \bigl(I + \bigl( \tilde{B}^{+}_{D}\bigr)^{-1}\tilde{C}_{D} \bigr)^{-1} \big\| _{\infty}\cdot\big\| \bigl(\tilde{B}^{+}_{D} \bigr)^{-1} \big\| _{\infty}\\ &\leq \max\{\gamma,1\}\cdot(n-1) \cdot\big\| \bigl(\tilde{B}^{+}_{D} \bigr)^{-1} \big\| _{\infty}.\end{aligned} $$
(8)
Next, we give an upper bound for \(\|(\tilde{B}^{+}_{D} )^{-1} \|_{\infty}\). Notice that \(\tilde{B}_{D}^{+}\) is an SDD
Z-matrix with positive diagonal entries, and thus \(\tilde{B}_{D}^{+}\) is an SDD
M-matrix. By Lemma 4, we have
where
By Lemma 5, we deduce for each \(k\in N\) that
and for each \(i\in N\) that
Furthermore, according to Lemma 6, it follows that for each \(j\in N\),
By (9) and (10), we derive
Now the conclusion follows from (8) and (11). □
$$ \big\| \bigl(\tilde{B}^{+}_{D} \bigr)^{-1} \big\| _{\infty}\leq \sum_{i=1}^{n} \Biggl(\frac{1}{(x_{i}-d_{i}x_{i}+d_{i}\tilde{b}_{ii})(1-u_{i}(\tilde{B}^{+}_{D})l_{i}(\tilde {B}^{+}_{D}))}\prod _{j=1}^{i-1}\frac{1}{1-u_{j}(\tilde{B}^{+}_{D})l_{j}(\tilde {B}^{+}_{D})} \Biggr), $$
$$ u_{i}\bigl(\tilde{B}^{+}_{D}\bigr)=\frac{\sum_{j=i+1}^{n}|\tilde {b}_{ij}|d_{i}}{x_{i}-d_{i}x_{i}+\tilde{b}_{ii}d_{i}}, \quad\text{and}\quad l_{k}\bigl(\tilde{B}^{+}_{D}\bigr)=\max _{k\leq i\leq n} \biggl\{ \frac{\sum_{\small\substack{j=k,\neq i}}^{n}|\tilde{b}_{ij}|d_{i}}{x_{i}-d_{i}x_{i}+\tilde{b}_{ii}d_{i}} \biggr\} . $$
$$ l_{k}\bigl(\tilde{B}^{+}_{D}\bigr)=\max_{k\leq i\leq n} \biggl\{ \frac{\frac{\sum_{\small\substack{j=k,\neq i}}^{n}|\tilde{b}_{ij}|}{x_{i}}d_{i}}{1-d_{i}+\frac{\tilde {b}_{ii}}{x_{i}}d_{i}} \biggr\} \leq \max_{k\leq i\leq n} \Biggl\{ \frac{1}{\tilde{b}_{ii}}\sum_{\small\substack{j=k,\neq i}}^{n}| \tilde{b}_{ij}| \Biggr\} =l_{k}\bigl(\tilde{B}^{+}\bigr)< 1, $$
$$ \begin{aligned}[b] \frac{1}{(x_{i}-d_{i}x_{i}+d_{i}\tilde{b}_{ii})(1-u_{i}(\tilde{B}^{+}_{D})l_{i}(\tilde {B}^{+}_{D}))}&= \frac{1}{x_{i}-d_{i}x_{i}+d_{i}\tilde{b}_{ii}-\sum_{j=i+1}^{n}|\tilde {b}_{ij}|d_{i}l_{i}(\tilde{B}^{+}_{D})} \\ &= \frac{\frac{1}{x_{i}}}{1-d_{i}+\frac{d_{i}}{x_{i}} (\tilde{b}_{ii}-\sum_{j=i+1}^{n}|\tilde{b}_{ij}|l_{i}(\tilde{B}^{+}_{D}) )} \\ &\leq \frac{1}{\min \{\tilde{b}_{ii}-\sum_{j=i+1}^{n}|\tilde{b}_{ij}| l_{i}(\tilde{B}^{+}),x_{i} \}} \\ &=\frac{1}{\min \{ \widehat{\beta}_{i},x_{i} \}}.\end{aligned} $$
(9)
$$ \begin{aligned}[b] \frac{1}{1-u_{j}(B^{+}_{D})l_{j}(B^{+}_{D})}&=\frac{1-d_{j}+\frac{\tilde {b}_{jj}}{x_{j}}d_{j}}{1-d_{j}+\frac{\tilde{b}_{jj}}{x_{j}}d_{j}-\frac{\sum_{k=j+1}^{n}|\tilde{b}_{jk}|}{x_{j}} d_{j} l_{j}(B^{+}_{D})} \\ &\leq\frac{\tilde{b}_{jj}}{\tilde{b}_{jj}-\sum_{k=j+1}^{n}|\tilde{b}_{jk}|l_{j}(\tilde{B}^{+})}=\frac{\tilde {b}_{jj}}{\widehat{\beta}_{j}}.\end{aligned} $$
(10)
$$ \big\| \bigl(\tilde{B}^{+}_{D} \bigr)^{-1} \big\| _{\infty}\leq \sum_{i=1}^{n} \frac{1}{\min\{\widehat{\beta}_{i},x_{i}\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}. $$
(11)
Remark here that when the matrix M is a B-matrix, then \(X=I\) and
which yields
This upper bound is consistent with that provided by Li et al. in [13]. Furthermore, for a \(B^{S}\)-matrix that is not a B-matrix, the following corollary can be obtained easily by Lemma 3 and Theorem 2.
$$ \tilde{B}^{+} =[\tilde{b}_{ij}]= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} m_{11}-r_{1}^{+} &\cdots &m_{1n}-r_{1}^{+} \\ \vdots &\ddots &\vdots \\ m_{n1}-r_{n}^{+} &\cdots &m_{nn}-r_{n}^{+} \end{array}\displaystyle \right ], $$
$$ \max_{d\in[0,1]^{n}}\big\| (I-D+DM)^{-1} \big\| _{\infty} \leq \sum_{i=1}^{n} \frac{n-1}{\min\{\widehat{\beta}_{i},1\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}. $$
Corollary 1
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix that is not a
B-matrix, and let
\(k,i\in N\)
with
\(k\neq i\)
such that
\(m_{ik}\geq\frac{1}{n}\sum_{j=1}^{n}m_{ij}\). If
\(k\in \overline{S}\), then
if
\(k\in S\), then
where
γ
satisfies (5).
$$ \max_{d\in[0,1]^{n}}\big\| M_{D}^{-1} \big\| _{\infty} \leq \sum_{i=1}^{n} \frac{(n-1)\gamma}{\min\{\widehat{\beta}_{i},1\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}; $$
(12)
$$ \max_{d\in[0,1]^{n}}\big\| M_{D}^{-1} \big\| _{\infty} \leq \sum_{i=1}^{n} \frac{n-1}{\min\{\widehat{\beta}_{i},\gamma\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}, $$
(13)
Example 1
Consider the family of \(B^{S}\)-matrices for \(S=\{1,2\}\):
where \(m\geq1\). Appropriate scaling matrices could be \(X=\operatorname{diag}\{\gamma, \gamma, 1, 1\}\), with \(\gamma\in(\frac{3.5}{3}, 1.5)\). So \(\tilde{M}_{m}:=M_{m}X\) can be written \(\tilde{M}=\tilde{B}_{m}^{+}+\tilde{C}_{m}\) as in (3), with
and
$$M_{m} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 &1 &1 &1.5 \\ -\frac{2m}{m+1} &2 &\frac{1}{m+1} &\frac{1}{m+1} \\ 1 &1 &2 &1 \\ 1 & 1 &1 & 2 \end{array}\displaystyle \right ], $$
$$ \tilde{B}_{m}^{+} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2\gamma-1.5 &\gamma-1.5 &-0.5 &0 \\ -\frac{2m}{m+1}\gamma-\frac{1}{m+1} & 2\gamma-\frac{1}{m+1} &0 &0 \\ 0 & 0 &2-\gamma &1-\gamma \\ 0 & 0 &1-\gamma &2-\gamma \end{array}\displaystyle \right ], $$
$$ \tilde{C}_{m}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1.5 &1.5 &1.5 &1.5 \\ \frac{1}{m+1} & \frac{1}{m+1} &\frac{1}{m+1} &\frac{1}{m+1} \\ \gamma & \gamma &\gamma &\gamma \\ \gamma & \gamma &\gamma &\gamma \end{array}\displaystyle \right ]. $$
By computations, we have \(\tilde{\beta}_{1}=3\gamma-3.5\), \(\tilde{\beta}_{2}=\frac{2(\gamma -1)}{m+1}\), \(\tilde{\beta}_{3}=\tilde{\beta}_{4}=3-2\gamma\), \(l_{1}({\tilde{B}^{+}})=\max \{\frac{2-\gamma}{2\gamma-1.5},\frac {2m\gamma+1}{2(m+1)\gamma-1},\frac{\gamma-1}{2-\gamma} \}\), \(\hat {\beta}_{1}=2\gamma-1.5-(2-\gamma)l_{1}({\tilde{B}^{+}})\), \(\hat{\beta}_{2}=2\gamma -\frac{1}{m+1}\), \(\hat{\beta}_{3}=\frac{3-2\gamma}{2-\gamma}\), and \(\hat{\beta}_{4}=2-\gamma\). Obviously, \(M_{m}\) satisfies \(m_{ik}\geq\frac{1}{4}\sum_{j=1}^{4}m_{ij}\) for \(i=1\) and \(k=4\) (∈S̅): \(1.5>1.375\), which implies that \(M_{m}\) is not a B-matrix. Then bound (12) in Corollary 1 is given by
which converges to a constant
with \(\gamma\in(\frac{3.5}{3}, 1.5)\) when \(m\rightarrow+\infty\). In contrast, bound (4) in Theorem 1, with the hypotheses that \(m\geq2\), is
and it can be arbitrarily large when \(m\rightarrow+\infty\).
$$ 3\gamma \biggl(\frac{1}{\min\{\widehat{\beta}_{1},\gamma\}}+\frac{1}{\min\{ \widehat{\beta}_{2},\gamma\}}\frac{\tilde{b}_{11}}{\widehat{\beta }_{1}}+ \frac{1}{\min\{\widehat{\beta}_{3},\gamma\}}\frac{\tilde {b}_{11}}{\widehat{\beta}_{1}}\frac{\tilde{b}_{22}}{\widehat{\beta }_{2}}+\frac{1}{\min\{\widehat{\beta}_{4},\gamma\}} \frac{\tilde {b}_{11}}{\widehat{\beta}_{1}}\frac{\tilde{b}_{22}}{\widehat{\beta }_{2}}\frac{\tilde{b}_{33}}{\widehat{\beta}_{3}} \biggr), $$
$$ 3\gamma \biggl(\frac{1}{3\gamma-3.5}+\frac{2\gamma -1.5}{(3\gamma-3.5)\gamma} +\frac{2(2-\gamma)(2\gamma-1.5)}{(3-2\gamma)(3\gamma-3.5)} \biggr) $$
$$ \frac{(4-1)\max\{\gamma,1\}}{\min\{\tilde{\beta},\gamma,1\}}=\frac {3\gamma}{2\gamma-1}(m+1) $$
In particular, if we choose \(\gamma=1.3\), then bound (4) and bound (7) for \(m=2, 20,30,\ldots, +\infty\) can be given as shown in Table 1.
Remark 1
From Example 1, it is easy to see that each bound (4) or (7) can work better than the other one. This means it is difficult to say in advance which one will work better. However, for a \(B^{S}\)-matrix M with \(\tilde{M}= \tilde{B}^{+}+ \tilde{C}\), where the diagonal dominance of \(\tilde{B}^{+}\) is weak (e.g., for a matrix \(M_{m}\) with a large number of m here), we can say that bound (7) is more effective to estimate \(\max_{d\in[0,1]^{n}}\|M_{D}^{-1}\|_{\infty}\) than bound (4). Therefore, in general case, for the \(\operatorname{LCP}(M, q)\) involved with a \(B^{S}\)-matrix, one can take the smallest of them:
$$ \max_{d\in[0,1]^{n}}\big\| M_{D}^{-1}\big\| _{\infty} \leq \min \Biggl\{ \frac{(n-1)\max\{\gamma,1\}}{\min\{\tilde{\beta},\gamma,1\}}, \sum_{i=1}^{n} \frac{(n-1)\max\{\gamma,1\}}{\min\{\hat{\beta}_{i},x_{i}\} }\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\hat{\beta}_{j}} \Biggr\} . $$
To measure the sensitivity of the solution of the P-matrix linear complementarity problem, Chen and Xiang in [5] introduced the following constant for a P-matrix M:
where \(\|\cdot\|_{p}\) is the matrix norm induced by the vector norm for \(p \geq1\).
$$ \beta_{P}(M)=\max_{d\in[0,1]^{n}}\big\| (I-D+DM)^{-1}D \big\| _{P}, $$
Similarly to the proof of Theorem 2.4 in [1], we can also give new perturbation bounds for \(B^{S}\)-matrices linear complementarity problems based on Theorem 2.
Theorem 3
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix and
\(\tilde{B}^{+}=[\tilde{b}_{ij}]\)
be the matrix given in Lemma
2. Then
where
\(\widehat{\beta}_{i}=\tilde{b}_{ii}-\sum_{k=i+1}^{n}|\tilde {b}_{ik}|l_{i}(\tilde{B}^{+})\), and
\(\prod_{j=1}^{i-1}\frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}=1\)
if
\(i=1\).
$$ \beta_{\infty}(M)\leq \sum_{i=1}^{n} \frac{(n-1)\max\{\gamma,1\}}{\min\{\widehat{\beta }_{i},x_{i}\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}, $$
Corollary 2
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
\(B^{S}\)-matrix that is not a
B-matrix, and let
\(k,i\in N\)
with
\(k\neq i\)
such that
\(m_{ik}\geq\frac{1}{n}\sum_{j=1}^{n}m_{ij}\). If
\(k\in \overline{S}\), then
if
\(k\in S\), then
where
γ
satisfies (5).
$$ \beta_{\infty}(M)\leq \sum_{i=1}^{n} \frac{(n-1)\gamma}{\min\{\widehat{\beta}_{i},1\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}; $$
$$ \beta_{\infty}(M)\leq \sum_{i=1}^{n} \frac{n-1}{\min\{\widehat{\beta}_{i},\gamma\}}\prod_{j=1}^{i-1} \frac{\tilde{b}_{jj}}{\widehat{\beta}_{j}}, $$
3 Conclusions
In this paper, we give an alternative bound for \(\max_{d\in [0,1]^{n}}\|(I-D+DM)^{-1}\|_{\infty}\) when M is a \(B^{S}\)-matrix, which improves that provided by García-Esnaola and Peña [1] in some cases. We also present new perturbation bounds of \(B^{S}\)-matrices linear complementarity problems.
Acknowledgements
The author is grateful to the two anonymous reviewers and the editor for their useful and constructive suggestions. The author also gives special thanks to Chaoqian Li for his discussion and comments during the preparation of this manuscript. This work is partly supported by the National Natural Science Foundation of China (31600299), Young Talent Fund of University Association for Science and Technology in Shaanxi, China (20160234), the Natural Science Foundation of Shaanxi province, China (2017JQ3020), and the key project of Baoji University of Arts and Sciences (ZK2017021).
Competing interests
The author declares that he has no competing interests.
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