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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices

Deshu Sun / Feng Wang
Published Online: 2017-07-21 | DOI: https://doi.org/10.1515/math-2017-0080

Abstract

Some new error bounds for the linear complementarity problems are obtained when the involved matrices are weakly chained diagonally dominant B-matrices. Numerical examples are given to show the effectiveness of the proposed bounds.

Keywords: Error bounds; Linear complementarity problems; Weakly chained diagonally dominant matrices; B-matrices

MSC 2010: 90C33; 60G50; 65F35

1 Introduction

Given a real matrix M = [mij] ∈ Rn×n and qRn, the linear complementarity problem LCP(M, q) is to find a vector xRn satisfying (Mx+q)Tx=0,Mx+q0,x0, or to prove that no such vector x exists. The LCP(M, q) has many applications such as finding Nash equilibrium point of a bimatrix game, the network equilibrium problems and the free boundary problems for journal bearing etc, see [1-3].

As is known, the LCP(M, q) has a unique solution for any vector qRn if and only if M is a P-matrix [2]. Here, a matrix M is called a P-matrix if all its principal minors are positive [4].

For the LCP(M, q), one of the interesting problems is to estimate maxd[0,1]n||(ID+DM)1||, which can be used to bound the error ||xx|| [5], that is ||xx||maxd[0,1]n||(ID+DM)1||||r(x)||, where x is the solution of the LCP(M, q), r(x) = min{x, Mx + q}, D = diag(di) with 0 ≤ di ≤ 1, and the min operator r(x) denotes the component wise minimum of two vectors. If the matrix M for the LCP(M, q) satisfies certain structures, various bounds for maxd[0,1]n||(ID+DM)1|| can be derived, see [6-13].

Definition 1.1

([4]). A real matrix A = [aij] ∈ Rn×n is called a B-matrix if for any i, jN = {1, 2, ..., n}, kNaik>0,1n(kNaik)>aij,ji.

Definition 1.2

([14]). A complex matrix A = [aij] ∈ Cn×n is called a weakly chained diagonally dominant (wcdd) matrix if A is diagonally dominant, i.e., aiiri(A)=j=1,in|aij|,iN, and for each iJ(A), there is a sequence of nonzero elements of A of the form aii1, ai1i2, ..., airj with jJ(A) = {iN : aii > ri(A)} ≠ ∅.

Definition 1.3

([13]). A real matrix M = [mij] ∈ Rn×n is called a weakly chained diagonally dominant (wcdd) B-matrix if it can be written in form M = B+ + C with B+ a wcdd matrix whose diagonal entries are all positive.

When M is a B-matrix as a subclass of P-matrices, García-Esnaola et al. [9] provided an upper bound for maxd[0,1]n||(ID+DM)1||.

Theorem 1.4

([9]). Let M = [mij ] ∈ Rn×n be a B-matrix with the form M=B++C, where B+=[bij]=[m11r1+m1nr1+mn1r1+mnnrn+], and ri+=max{0,mij|ji}. Then maxd[0,1]n||(ID+DM)1||n1min{β,1},(1) where β=miniN{βi}andβi=biiji|bij|.

As shown in [12], the bound (1) will be inaccurate when the matrix M has very small value of miniN{biiji|bij|}, see [12, 13]. To improve the bound (1), Li et al. [12] gave the following bound for maxd[0,1]n||(ID+DM)1|| when M is a B-matrix.

Theorem 1.5

([12]). Let M = [mij] ∈ Rn×n be a B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. Then maxd[0,1]n||(ID+DM)1||i=1nn1min{β¯i,1}j=1i1(1+1β¯jk=j+1n|bjk|),(2) where β¯i=biij=i+1n|bij|li(B+),lk(B+)=maxkin{1|bii|j=k,in|bij|}and j=1i1(1+1β¯jk=j+1n|bjk|)=1ifi=1. Recently, Li et al. [13] gave the following bound for maxd[0,1]n||(ID+DM)1|| when M is a wcdd B-matrix.

Theorem 1.6

([13]). Let M = [mij] ∈ Rn×n be a wcdd B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. Then maxd[0,1]n||(ID+DM)1||i=1n(n1min{β~i,1}j=1i1bjjβ~j),(3) where β~i=biij=i+1n|bij|>0andj=1i1bjjβ~j=1ifi=1.

Since a B-matrix is a wcdd B-matrix [13], thus the bound (3) also holds for B-matrix M.

Now, some notations are given, which will be used later. Given a matrix A = [aij] ∈ Rn×n, let ui(A)=1|aii|j=1+1n|aij|,un(A)=0,lki(A)=1|akk|j=i,kn|akj|,li(A)=maxikn{lki(A)},ln(A)=0,tki(A)={|aki||akk|j=i+1,kn|akj|,aki00,aki=0,ti(A)={maxi+1kn{tki},1in10,i=n,bk(A)=maxk+1in{j=k,in|aij||aii|},bn(A)=1,pk(A)=maxk+1in{|aik|+j=k+1,in|aij|bk(A)|aii|},pn(A)=1. The rest of this paper is organized as follows: In Section 2, we present some new bounds for maxd[0,1]n||(ID+DM)1|| when M is a wcdd B-matrix. Numerical examples are presented to illustrate the corresponding theoretical results in Section 3.

2 Main results

In this section, some new upper bounds of maxd[0,1]n||(ID+DM)1|| for wcdd B-matrix M are provided. We first list some lemmas which will be used later.

Lemma 2.1

([15]). Let A = [aij] ∈ Rn×n be a wcdd M-matrix with Uk(A)pk(A) < 1 (∀kN). Then ||A1||max{i=1n(1aii(1ui(A)ti(A))j=1i1uj(A)1uj(A)tj(A)),i=1n[pi(A)aii(1ui(A)ti(A))j=1i1(1+uj(A)pj(A)1uj(A)tj(A))]}, where j=1i1uj(A)1uj(A)pj(A)=1,j=1i1(1+uj(A)pj(A)1uj(A)tj(A))=1,ifi=1.

Lemma 2.2

([12]). Let γ > 0 and η ≥ 0. Then for any a ∈ [0, 1], 11a+γa1min{γ,1} and ηa1a+γaηγ.

Theorem 2.3

Let M = [mij] ∈ Rn×n be a wcdd B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. If for each iN, β^i=biij=i+1n|bij|ti(B+)>0, then maxd[0,1]n||(1D+DM)1||(n1)max{i=1n1min{β^i,1}j=1i1(1β^jk=j+1n|bjk|),i=1npi(B+)min{β^i,1}j=1i1(1+pj(B+)β^jk=j+1n|bjk|)},(4) where j=1i1(1β^jk=j+1n|bjk|)=1,j=1i1(1+pj(B+)β^jk=j+1n|bjk|)=1,ifi=1.

Proof

Let MD = ID + DM. Then MD=ID+DM=ID+D(B++C)=BD++CD, where BD+=ID+DB+. From Theorem 2 in [13], we know that BD+ is a wcdd M-matrix with positive diagonal elements, CD = DC, and ||MD1||||(I+(BD+)1CD)1||||(BD+)1||(n1)||(BD+)1||.(5) By Lemma 2.1, we have ||(BD+)1||max{i=1n1(1di+biidi)(1ui(BD+)ti(BD+))j=1i1uj((BD+))1uj((BD+))tj(BD+),i=1npi(BD+)(1di+biidi)(1ui((BD+))ti(BD+))j=1i1(1+uj(BD+)pj(BD+)1uj(BD+)tj(BD+))}. By Lemma 2.2, we can easily get the following results: for each i, j, kN, ui(BD+)=j=i+1n|bij|di1di+biidij=i+1n|bij|bii=ui(B+),(6) tk(BD+)=maxi+1kn{|bki|dk1dk+bkkdkj=i+1,kn|bkj|dk}maxi+1kn{|bki|bkkj=i+1,kn|bkj|}=tk(B+),(7) bk(BD+)=maxk+1in{j=k,in|bij|di1di+biidi}maxk+1in{j=k,in|bij|bii}=bk(B+),(8) pk(BD+)=maxk+1in{|bik|di+j=k+1,in|bij|dibk(BD+)1di+biidi}maxk+1in{|bik|+j=k+1,in|bij|bk(BD+)bii}maxk+1in{|bik|+j=k+1,in|bij|bk(B+)bii}=pk(B+).(9) Furthermore, by (6), (7), (8) and (9), we have 1(1di+biidi)(1ui(BD+)ti(BD+))=11di+biidij=i+1n|bij|diti(BD+)1min{biij=i+1n|bij|ti(B+),1}=1min{β^i,1},(10) and ui(BD+)1ui(BD+)ti(BD+)=j=i+1n|bij|di1di+biidij=i+1n|bij|diti(BD+)j=i+1n|bij|biij=i+1n|bij|ti(B+)=1β^ij=i+1n|bij|.(11)

From (10) and (11), we obtain ||(BD+)1||maxi=1n1min{β^i,1}j=1i11β^jk=j+1n|bjk|,i=1npi(B+)min{β^i,1}j=1i11+pj(B+)β^jk=j+1n|bjk|.(12) Therefore, the result in (4) follows from (5) and (12). □

Since a B-matrix is a wcdd B-matrix, then by Theorem 2.3, we can obtain the following upper bound of maxd[0,1]n||(ID+DM)1|| for B-matrix M.

Corollary 2.4

Let M = [mij] ∈ Rn×n be a B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. Then maxd[0,1]n||(ID+DM)1||(n1)maxi=1n1min{β^i,1}j=1i11β^jk=j+1n|bjk|,i=1npi(B+)min{β^i,1}j=1i11+pj(B+)β^jk=j+1n|bjk|,(13) where β̂i is defined as in Theorem 2.3.

We next give a comparison of the bounds in (3) and (4) as follows.

Theorem 2.5

Let M = [mij] ∈ Rn×n be a wcdd B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. Let β̄i, β̃i and β̂i be defined as in Theorem 1.5, Theorem 1.6 and Theorem 2.3, respectively. Then (n1)maxi=1n1min{β^i,1}j=1i11β^jk=j+1n|bjk|,i=1npi(B+)min{β^i,1}j=1i11+pj(B+)k=j+1k=j+1n|bij|i=1nn1min{β¯i,1}j=1i11+1β¯jk=j+1n|bjk|i=1nn1min{β~i,1}j=1i1bjjβ~j.(14)

Proof

Since B+ is a wcdd matrix with positive diagonal elements, thus pi (B+) ≤ 1 (∀iN), and for any kN, i + 1 ≤ kn, 1 ≤ in − 1, if bki ≠ 0, then lki(B+)tki(B+)=|bki|+j=i+1,kn|bkj||bkk||bki||bkk|j=i+1,kn|bkj|=j=i+1,kn|bkj|(|bkk||bki|j=i+1,kn|bkj|)|bkk|(|bkk|j=i+1,kn|bkj|)0. If bki = 0, then lki (B+) ≥ tki (B+) = 0. Hence, for any iN, we have 0ti(B+)li(B+)<1.(15) By (15), for each iN, β¯i=biik=i+1n|bik|li(B+)biik=i+1n|bik|ti(B+)=β^i,(16) then by (16), for each iN, 1min{β^i,1}1min{β¯i,1},(17) and for j = 1, 2,...,n − 1, 1+1β^jk=j+1n|bjk|1+1β¯jk=j+1n|bjk|,(18) From (17) and (18), we have (n1)maxi=1n1min{β^i,1}j=1i11β^jk=j+1n|bjk|,i=1npi(B+)min{β^i,1}j=1i11+pj(B+)β^jk=j+1n|bjk|i=1nn1min{β¯i,1}j=1i11+1β¯jk=j+1n|bjk|.(19) Otherwise, note that β~i=biij=i+1n|bij|,β¯i=biij=i+1n|bij|li(B+), and lk(BD+)lk(B+)<1. Hence, for each iN, β̄iβ̃i, and 1min{β¯i,1}1min{β~i,1}.(20) In the meantime, for j = 1, 2,...,n − 1, 1+1β¯jk=j+1n|bjk|1+1β~jk=j+1n|bjk|=1β~jβ~j+k=j+1n|bjk|=bjjβ~j.(21) From (20) and (21), we obtain i=1nn1min{β¯i,1}j=1i11+1β¯jk=j+1n|bjk|i=1nn1min{β~i,1}j=1i1bjjβ~j.(22) The result in (14) follows from (19) and (22). □

Adopting the same procedure as in the proof of Theorem 2.4 in [9], we can provide the following new bound for the constant βp(M) when M is a P-matrix, where βp(M)=maxd[0,1]n||ID+DM)1D||p, D = diag(di) with 0 ≤ di ≤ 1 (iN), and ||•||p is the matrix norm induced by the vector norm for p ≥ 1. The constant is used to measure the sensitivity of the solution of the P-matrix linear complementarity problem [1].

Theorem 2.6

Let M = [mij] ∈ Rn×n be a wcdd B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. Then β(M)(n1)maxi=1n1min{β^i,1}j=1i11β^jk=j+1n|bjk|,i=1npi(B+)min{β^i,1}j=1i11+pj(B+)β^jk=j+1n|bjk|,(23) where β̃i > 0 is defined as in Theorem 2.3.

Corollary 2.7

Let M = [mij] ∈ Rn×n be a B-matrix with the form M = B+ + C, where B+ = [bij] is defined as in Theorem 1.4. Then β(M)(n1)maxi=1n1min{β^i,1}j=1i11β^jk=j+1n|bjk|,i=1npi(B+)min{β^i,1}j=1i11+pj(B+)β^jk=j+1n|bjk|.(24)

3 Numerical examples

In this section, we present numerical examples to illustrate the advantages of our derived results.

Example 3.1

Consider the family of B-matrices in [12]: Mk=[1.50.10.800.51.70.1kk+10.70.40.50.70.61.80.70.81.8], where k ≥ l. Then Mk=Bk++Ck, where Bk+=[10.800.8010.1kk+10.80.10.101000.10.11]. By Theorem 1.4, we have maxd[0,1]4||ID+DMk)1||30(k+1)+,ifk. By Theorem 1.5, we get maxd[0,1]4||ID+DMk)1||<15.2675. By Corollary 1 of [13], we have maxd[0,1]4||ID+DMk)1||i=143min{β~i,1}j=1i1bjjβ~j15.2675. By Corollary 2.4, we have maxd[0,1]4||ID+DMk)1||<9.6467.

Example 3.2

Consider the wcdd B-matrix in [13]: M=[1.50.10.50.40.21.50.10.40.40.50.50.11.50.10.81.8]. Thus M = B+ + C, where B+=[10.600.40.310.60.40.101000.40.41]. By Theorem 1.6, we get maxd[0,1]4||ID+DM)1||41.1111. By Theorem 2.3, we obtain maxd[0,1]4||ID+DM)1||21.6667. This example shows that the bound in Theorem 2.3 is sharper than that in Theorem 1.6.

Acknowledgement

This work was 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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About the article

Received: 2016-07-28

Accepted: 2017-05-08

Published Online: 2017-07-21


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 978–986, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0080.

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