Abstract
In this paper, new error bounds for the linear complementarity problem are obtained when the involved matrix is a weakly chained diagonally dominant B-matrix. The proposed error bounds are better than some existing results. The advantages of the results obtained are illustrated by numerical examples.
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1 Introduction
A linear complementarity problem (LCP) is to find a vector \(x\in \mathbb{R}^{n\times1}\) such that
where \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) and \(q\in \mathbb{R}^{n\times1}\). The LCP has various applications in the free boundary problems for journal bearing, the contact problem, and the Nash equilibrium point of a bimatrix game [1–3].
The LCP has a unique solution for any \(q\in \mathbb{R}^{n\times1}\) if and only if M is a P-matrix [4]. In [5], Chen et al. gave the following error bound for the LCP when M is a P-matrix:
where \(x^{*}\) is the solution of the LCP, \(r(x)=\min\{x, Mx+q\}\), \(D=\operatorname{diag}(d_{i})\) with \(0\leq d_{i}\leq1\), and the min operator \(r(x)\) denotes the componentwise minimum of two vectors. If M satisfies special structures, then some bounds of \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) can be derived [6–11].
Definition 1
[4]
A matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called a B-matrix if for any \(i, j\in\mathbb{N}=\{1,2,\ldots,n\}\),
Definition 2
[12]
A matrix \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) is called a weakly chained diagonally dominant (wcdd) matrix if A is diagonally dominant, i.e.,
and for each \(i\notin J(A)=\{ i\in\mathbb{N}: \vert a_{ii}\vert >r_{i}(A)\}\neq\emptyset\), there is a sequence of nonzero elements of A of the form \(a_{ii_{1}}, a_{i_{1}i_{2}}, \ldots, a_{i_{r} j}\) with \(j\in J(A)\).
Definition 3
[13]
A matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called a weakly chained diagonally dominant (wcdd) B-matrix if it can be written in the form \(M=B^{+}+C\) with \(B^{+}\) a wcdd matrix whose diagonal entries are all positive.
García-Esnaola et al. [8] gave the upper bound for \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) when M is a B-matrix: Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a B-matrix with the form
where
and \(r_{i}^{+}=\max\{0, m_{ij}\vert j\neq i\}\). Then
where \(\beta= \min_{i\in\mathbb{N}}\{\beta_{i}\}\) and \(\beta_{i}=b_{ii}-\sum_{j\neq i} \vert b_{ij}\vert \).
To improve the bound in (2), Li et al. [14] presented the following result: Let \(M=[m_{ij}]\in \mathbb{R}^{n\times n}\) be a B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Then
where \(\bar{\beta}_{i}=b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert l_{i}(B^{+})\), \(l_{k}(B^{+})=\max_{k\leq i\leq n} \{ \frac{1}{\vert b_{ii}\vert } \sum_{j=k, \neq i}^{n} \vert b_{ij}\vert \}\) and
Recently, when M is a weakly chained diagonally dominant (wcdd) B-matrix, Li et al. [13] gave a bound for \(\max_{d\in[0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty }\): Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Then
where \(\tilde{\beta}_{i}=b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert >0\) and \(\prod_{j=1}^{i-1}\frac{b_{jj}}{\tilde{\beta}_{j}}=1\) if \(i=1\).
This bound in (4) holds when M is a B-matrix since a B-matrix is a weakly chained diagonally dominant B-matrix [13].
Now, some notation is given, which will be used in the sequel. Let \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\). For \(i, j, k \in \mathbb{N}\), denote
The rest of this paper is organized as follows: In Section 2, we present some new bounds for \(\max_{d\in[0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert_{\infty}\) when M is a wcdd B-matrix. Numerical examples are given to verify the corresponding results in Section 3.
2 Main results
In this section, some new upper bounds for \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) are provided when M is a wcdd B-matrix. Firstly, several lemmas, which will be used later, are given.
Lemma 1
[13]
Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}\) is defined as (1). Then
where \(B^{+}_{D}=I-D+DB^{+}\) and \(C_{D}=DC\).
Lemma 2
[15]
Let \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd M-matrix with \(u_{k}(A)p_{k}(A)<1\) (\(\forall k\in\mathbb{N}\)). Then
where
Lemma 3
[14]
Let \(\gamma>0\) and \(\eta\geq0\). Then, for any \(x\in[0,1]\),
Theorem 1
Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). If, for each \(i\in\mathbb{N}\),
then
where
Proof
Let \(M_{D}=I-D+DM\). Then
where \(B_{D}^{+}=I-D+DB^{+}\). Similar to the proof of Theorem 2 in [13], we see that \(B_{D}^{+}\) is a wcdd M-matrix with positive diagonal elements and \(C_{D}=DC\), and, by Lemma 1,
By Lemma 2, we have
By Lemma 3, we can easily get the following results: for each \(i , j, k\in\mathbb{N}\),
and
Furthermore, by Lemma 3, we have
and
By (7), (8), and (9), we obtain
Therefore, the result in (5) holds from (6) and (10). □
Since a B-matrix is also a wcdd B-matrix, then by Theorem 1, we find the following result.
Corollary 1
Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Then
where \(\hat{\beta}_{i}\) is defined as in Theorem 1.
We next give a comparison of the bounds in (4) and (5) as follows.
Theorem 2
Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Let \(\bar{\beta}_{i}\), \(\tilde{\beta}_{i}\), and \(\hat{\beta}_{i}\) be defined as in (3), (4), and (5), respectively. Then
Proof
Since \(B^{+}\) is a wcdd matrix with positive diagonal elements, for any \(i\in\mathbb{N}\),
By (13), for each \(i\in\mathbb{N}\),
The result in (12) follows by (13) and (14). □
Remark 1
3 Numerical examples
In this section, we present numerical examples to illustrate the advantages of our derived results.
Example 1
Consider the family of B-matrices in [14]:
where \(k\geq1\). Then \(M_{k}=B_{k}^{+}+C_{k}\), where
By (2), we have
It is obvious that
By (3), we get
By Theorem 7 of [11], we have
By Corollary 1 of [13], we have
By (11), we obtain
In these two cases, the bounds in (2) are equal to 60 (\(k=1\)) and 90 (\(k=2\)), respectively.
Example 2
Consider the wcdd B-matrix in [13]:
Then \(M=B^{+}+C\), where
By (4), we get
By (5), we have
4 Conclusions
In this paper, we present some new upper bounds for \(\max_{d\in[0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) when M is a weakly chained diagonally dominant B-matrix, which improve some existing results. A numerical example shows that the given bounds are efficient.
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Acknowledgements
The author is grateful to the referees for their useful and constructive suggestions. This work is supported by the National Natural Science Foundation of China (11361074, 11501141), the Foundation of Science and Technology Department of Guizhou Province ([2015]7206), the Natural Science Programs of Education Department of Guizhou Province ([2015]420), and the Research Foundation of Guizhou Minzu University (16yjsxm002, 16yjsxm040).
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Wang, F. Error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices. J Inequal Appl 2017, 33 (2017). https://doi.org/10.1186/s13660-017-1303-5
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DOI: https://doi.org/10.1186/s13660-017-1303-5