Error bounds for linear complementarity problems of weakly chained diagonally dominant B matrices

(1)

R E S E A R C H

Open Access

Error bounds for linear complementarity

problems of weakly chained diagonally

dominant

B-matrices

Feng Wang

*

*_{Correspondence:} [email protected]

College of Science, Guizhou Minzu University, Guiyang, Guizhou 550025, P.R. China

Abstract

In this paper, new error bounds for the linear complementarity problem are obtained when the involved matrix is a weakly chained diagonally dominantB-matrix. The proposed error bounds are better than some existing results. The advantages of the results obtained are illustrated by numerical examples.

MSC: 90C33; 60G50; 65F35

Keywords: error bound; linear complementarity problem; weakly chained diagonally dominant matrix;B-matrix

1 Introduction

A linear complementarity problem (LCP) is to ﬁnd a vectorx∈Rn×_{such that}

(Mx+q)Tx= , Mx+q≥, x≥,

whereM= [mij]∈Rn×nandq∈Rn×. TheLCPhas various applications in the free

bound-ary problems for journal bearing, the contact problem, and the Nash equilibrium point of a bimatrix game [–].

TheLCPhas a unique solution for anyq∈Rn×if and only ifMis aP-matrix []. In [], Chenet al.gave the following error bound for theLCPwhenMis aP-matrix:

x–x∗_∞≤ max

d∈[,]n(I–D+DM)

–

∞r(x)∞,

wherex∗is the solution of theLCP,r(x) =min{x,Mx+q},D=diag(di) with ≤di≤, and

the min operatorr(x) denotes the componentwise minimum of two vectors. IfMsatisﬁes special structures, then some bounds ofmax_d∈_[,]n(I–D+DM)–_∞can be derived [–

].

Deﬁnition ([]) A matrixM= [mij]∈Rn×n is called aB-matrix if for anyi,j∈N=

{, , . . . ,n},

k∈N

mik> ,



n

k∈N

mik

>mij, j=i.

(2)

Deﬁnition ([]) A matrixA= [aij]∈Rn×nis called a weakly chained diagonally

domi-nant (wcdd) matrix ifAis diagonally dominant,i.e.,

|aii| ≥ri(A) = n

j=,=i

|aij|, ∀i∈N,

and for eachi∈/J(A) ={i∈N:|aii|>ri(A)} =∅, there is a sequence of nonzero elements of

Aof the formaii,aii, . . . ,airjwithj∈J(A).

Deﬁnition ([]) A matrixM= [mij]∈Rn×nis called a weakly chained diagonally

dom-inant (wcdd)B-matrix if it can be written in the formM=B+₊_C_with_B+_a_wcdd_matrix

whose diagonal entries are all positive.

García-Esnaolaet al.[] gave the upper bound formax_d∈[,]n(I–D+DM)–_∞when Mis aB-matrix: LetM= [mij]∈Rn×nbe aB-matrix with the form

M=B++C, where

B+= [bij] = ⎡ ⎢ ⎢ ⎣

m–r+ · · · mn–r+

..

. ...

mn–rn+ · · · mnn–r+n ⎤ ⎥ ⎥

⎦, ()

andr_i+=max{,mij|j=i}. Then

max

d∈[,]n(I–D+DM)

–

∞≤

n– 

min{β, }, ()

whereβ=mini∈N{βi}andβi=bii–

j=i|bij|.

To improve the bound in (), Liet al.[] presented the following result: LetM= [mij]∈ Rn×n_{be a}_B_{-matrix with the form}_M₌_B+₊_C_{, where}_B+_{= [}_b

ij] is deﬁned as (). Then

max

d∈[,]n(I–D+DM)

–

∞≤ n

i=

n– 

min{ ¯βi, } i–

j=

 + _¯

βj n

k=j+

|bjk|

, ()

whereβ¯i=bii– n

j=i+|bij|li(B+),lk(B+) =maxk≤i≤n{_|b_ii_| n

j=k,=i|bij|}and i–

j=

 + _¯

βj n

k=j+

|bjk|

= , ifi= .

Recently, whenMis a weakly chained diagonally dominant (wcdd)B-matrix, Liet al.[] gave a bound formax_d∈_[,]n(I–D+DM)–_∞: LetM= [m_ij]∈Rn×nbe awcdd B-matrix

with the formM=B+₊_C_{, where}_B+_{= [}_b

ij] is deﬁned as (). Then

max

d∈[,]n(I–D+DM)

–

∞≤ n

i=

n– 

min{ ˜βi, } i–

j=

bjj

˜

βj

(3)

whereβ˜i=bii– n

j=i+|bij|>  and i–

j=

bjj

˜

βj =  ifi= .

This bound in () holds whenMis a B-matrix since aB-matrix is a weakly chained diagonally dominantB-matrix [].

Now, some notation is given, which will be used in the sequel. LetA= [aij]∈Rn×n. For

i,j,k∈N, denote

ui(A) =



|aii| n

j=i+

|aij|, un(A) = ,

bk(A) = max k+≤i≤n

n j=k,=i|aij|

|aii|

, bn(A) = ,

pk(A) = max k+≤i≤n

_|_a ik|+

n

j=k+,=i|aij|bk(A)

|aii|

, pn(A) = .

The rest of this paper is organized as follows: In Section , we present some new bounds formax_d∈[,]n(I–D+DM)–_∞whenMis awcdd B-matrix. Numerical examples are

given to verify the corresponding results in Section .

2 Main results

In this section, some new upper bounds formax_d∈[,]n(I–D+DM)–_∞are provided

whenMis awcdd B-matrix. Firstly, several lemmas, which will be used later, are given.

Lemma ([]) Let M= [mij]∈Rn×nbe a wcdd B-matrix with the form M=B++C,where

B+_{is deﬁned as}_()._Then

I+B+_D–CD –

∞≤n– ,

where B+

D=I–D+DB+and CD=DC.

Lemma ([]) Let A= [aij]∈Rn×nbe a wcdd M-matrix with uk(A)pk(A) <  (∀k∈N).

Then

A–_∞≤max _n

i=



aii( –ui(A)pi(A)) i–

j=

uj(A)

 –uj(A)pj(A)

,

n

i=

pi(A)

aii( –ui(A)pi(A)) i–

j=

  –uj(A)pj(A)

,

where

i–

j=

uj(A)

 –uj(A)pj(A)

= ,

i–

j=

  –uj(A)pj(A)

= , if i= .

Lemma ([]) Letγ > andη≥.Then,for any x∈[, ], 

 –x+γx≤



min{γ, },

ηx

 –x+γx≤

(4)

Theorem  Let M= [mij]∈Rn×nbe a wcdd B-matrix with the form M=B++C,where

B+_{= [}_b

ij]is deﬁned as().If,for each i∈N,

ˆ

βi=bii– n

j=i+

|bij|pi

B+> ,

then

max

d∈[,]n(I–D+DM)

– ∞ ≤max _n i=

n– 

min{ ˆβi, } i– j=  ˆ βj n

k=j+

|bjk|

,

n

i=

(n– )pi(B+) min{ ˆβi, }

i– j= bjj ˆ βj , () where i– j=  ˆ βj n

k=j+

|bjk| = , i– j= bjj ˆ βj

= , if i= .

Proof LetMD=I–D+DM. Then

MD=I–D+DM=I–D+D

B++C=B+_D+CD,

whereB+_D=I–D+DB+. Similar to the proof of Theorem  in [], we see thatB+_Dis awcdd M-matrix with positive diagonal elements andCD=DC, and, by Lemma ,

M–_D_∞≤I+B+_D–CD –

∞B+D –

∞≤(n– )B+D –

∞. ()

By Lemma , we have

B+_D–_∞≤max _n

i=



( –di+biidi)( –ui(B+D)pi(B+D)) i–

j=

uj((B+D))

 –uj((B+D))pj(B+D)

,

n

i=

pi(B+D)

( –di+biidi)( –ui((B+D))pi(B+D)) i–

j=

  –uj(B+D)pj(B+D)

.

By Lemma , we can easily get the following results: for eachi,j,k∈N,

bk

B+_D= max k+≤i≤n

n

j=k,=i|bij|di

 –di+biidi

≤ max

k+≤i≤n n

j=k,=i|bij|

bii

=bk

B+,

pk

B+_D= max k+≤i≤n

_|_b ik|di+

n

j=k+,=i|bij|dibk(B+D)

 –di+biidi

≤ max

k+≤i≤n

_|_b ik|+

n

j=k+,=i|bij|bk(B+D)

bii

≤ max

k+≤i≤n

_|_b ik|+

n

j=k+,=i|bij|bk(B+)

bii

=pk

(5)

and



( –di+biidi)( –ui(B+D)pi(B+D))

= 

 –di+biidi– n

j=i+|bij|dipi(B+D)

≤ 

min{bii– n

j=i+|bij|pi(B+), }

= 

min{ ˆβi, }

. ()

Furthermore, by Lemma , we have

ui(B+D)

 –ui(B+D)pi(B+D)

=

n

j=i+|bij|di

 –di+biidi– n

≤

n j=i+|bij|

bii– n

j=i+|bij|pi(B+)

= 

ˆ

βi n

j=i+

|bij| ()

and

  –ui(B+D)pi(B+D)

=  –di+biidi  –di+biidi–

n

≤  –di+biidi

bii– n

j=i+|bij|pi(B+)

=bii

ˆ

βi

. ()

By (), (), and (), we obtain

B+_D–_∞≤max _n

i=



min{ ˆβi, } i–

j=



ˆ

βj n

k=j+

|bjk|

,

n

i=

pi(B+) min{ ˆβi, }

i–

j=

bjj

ˆ

βj

. ()

Therefore, the result in () holds from () and (). Since a B-matrix is also awcdd B-matrix, then by Theorem , we ﬁnd the following result.

Corollary  Let M= [mij]∈Rn×nbe a B-matrix with the form M=B++C,where B+= [bij]

is deﬁned as().Then

max

d∈[,]n(I–D+DM)

–

∞

≤max _n

i=

n– 

j=



ˆ

βj n

k=j+

|bjk|

,

n

i=

i–

j=

bjj

ˆ

βj

, ()

(6)

We next give a comparison of the bounds in () and () as follows.

Theorem  Let M= [mij]∈Rn×nbe a wcdd B-matrix with the form M=B++C,where

B+_{= [}_b

ij]is deﬁned as().Letβ¯i,β˜i,andβˆibe deﬁned as in(), (),and(),respectively.

Then

max _n

i=

n– 

j=



ˆ

βj n

k=j+

|bjk|

,

n

i=

i–

j=

bjj

ˆ

βj

≤

n

i=

n– 

min{ ˜βi, } i–

j=

bjj

˜

βj

. ()

Proof SinceB+_{is a}_wcdd_{matrix with positive diagonal elements, for any}_i_∈_N_,

≤pi

B+≤, β˜i≤ ˆβi. ()

By (), for eachi∈N, 

ˆ

βi

≤ _˜

βi

, 

min{ ˆβi, }

≤ 

min{ ˜βi, }

. ()

The result in () follows by () and ().

Remark 

(i) Theorem  shows that the bound in () is better than that in ().

(ii) Whennis very large, one needs more computations to obtain these upper bounds by () than by ().

3 Numerical examples

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

Example  Consider the family ofB-matrices in []:

Mk= ⎡ ⎢ ⎢ ⎢ ⎣

. . . . –. . . . . –._kk_+ . .  . . .

⎤ ⎥ ⎥ ⎥ ⎦,

wherek≥. ThenMk=B+_k+Ck, where

B+_k=

⎡ ⎢ ⎢ ⎢ ⎣

  –.  –.   –.

 –. k

k+– .  –.

–. –.  

⎤ ⎥ ⎥ ⎥ ⎦.

By (), we have

max d∈[,]

(I–D+DMk)–_∞≤

 – 

(7)

It is obvious that

(k+ )→+∞, ifk→+∞.

By (), we get

max d∈[,]

(I–D+DMk)–_∞≤..

By Theorem  of [], we have

max

d∈[,](I–D+DMk)

–

∞≤..

By Corollary  of [], we have

max

d∈[,](I–D+DMk)

–

∞≤



i=



min{ ˜βi, } i–

j=

bjj

˜

βj

≈..

By (), we obtain

max

d∈[,](I–D+DMk)

–

∞≤..

In these two cases, the bounds in () are equal to  (k= ) and  (k= ), respectively.

Example  Consider thewcdd B-matrix in []:

M=

⎡ ⎢ ⎢ ⎢ ⎣

. . . . –. . . . . –. . . . . . .

⎤ ⎥ ⎥ ⎥ ⎦.

ThenM=B+₊_C_{, where}

B+=

⎡ ⎢ ⎢ ⎢ ⎣

 –. –.  –.   –.

 –.  –. –. –.  

⎤ ⎥ ⎥ ⎥ ⎦.

By (), we get

max

d∈[,](I–D+DM)

–

∞≤..

By (), we have

max d∈[,]

(8)

4 Conclusions

In this paper, we present some new upper bounds formax_d∈[,]n(I–D+DM)–_∞when Mis a weakly chained diagonally dominantB-matrix, which improve some existing re-sults. A numerical example shows that the given bounds are eﬃcient.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

Only the author contributed to this work. The author read and approved the ﬁnal manuscript.

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).

Received: 1 October 2016 Accepted: 16 January 2017

References

1. Chen, XJ, Xiang, SH: Perturbation bounds ofP-matrix linear complementarity problems. SIAM J. Optim.18, 1250-1265 (2007)

2. Cottle, RW, Pang, JS, Stone, RE: The Linear Complementarity Problem. Academic Press, San Diego (1992) 3. Murty, KG: Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1998) 4. Peña, JM: A class ofP-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM J. Matrix

Anal. Appl.22, 1027-1037 (2001)

5. Chen, XJ, Xiang, SH: Computation of error bounds forP-matrix linear complementarity problem. Math. Program.106, 513-525 (2006)

6. Chen, TT, Li, W, Wu, X, Vong, S: Error bounds for linear complementarity problems ofMB-matrices. Numer. Algorithms

70(2), 341-356 (2015)

7. Dai, PF, Lu, CJ, Li, YT: New error bounds for the linear complementarity problem with anSB-matrix. Numer. Algorithms

64, 741-757 (2013)

8. García-Esnaola, M, Peña, JM: Error bounds for linear complementarity problems forB-matrices. Appl. Math. Lett.22, 1071-1075 (2009)

9. García-Esnaola, M, Peña, JM: Error bounds for linear complementarity problems involvingBS_{-matrices. Appl. Math.}

Lett.25, 1379-1383 (2012)

10. García-Esnaola, M, Peña, JM:B-Nekrasov matrices and error bounds for linear complementarity problems. Numer. Algorithms72(2), 435-445 (2016)

11. Li, CQ, Gan, MT, Yang, SR: A new error bound for linear complementarity problems forB-matrices. Electron. J. Linear Algebra31, 476-484 (2016)

12. Shivakumar, PN, Chew, KH: A suﬃcient condition for nonvanishing of determinants. Proc. Am. Math. Soc.43(1), 63-66 (1974)

13. Li, CQ, Li, YT: Weakly chained diagonally dominantB-matrices and error bounds for linear complementarity problems. Numer. Algorithms73(4), 985-998 (2016)

14. Li, CQ, Li, YT: Note on error bounds for linear complementarity problems forB-matrices. Appl. Math. Lett.57, 108-113 (2016)

15. Huang, TZ, Zhu, Y: Estimation ofA–1

∞for weakly chained diagonally dominantM-matrices. Linear Algebra Appl.