A new improved error bound for linear complementarity problems for B

arXiv:1610.06305v1 [math.NA] 20 Oct 2016

A NEW IMPROVED ERROR BOUND FOR LINEAR COMPLEMENTARITY PROBLEMS FOR B-MATRICES∗ LEI GAO† AND CHAOQIAN LI‡

Abstract. A new error bound for the linear complementarity problem when the matrix involved is a B-matrix is presented, which improves the corresponding result in [C.Q. Li et al., A new error bound for linear complementarity problems for B-matrices. Electron. J. Linear Al., 31:476-484, 2016]. In addition some sufficient conditions such that the new bound is sharper than that in [M. Garc´ıa-Esnaola and J.M. Pe˜ na. Error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett., 22:1071-1075, 2009] are provided.

Key words. Error bound, Linear complementarity problem, B-matrix

AMS subject classifications. 15A48, 65G50, 90C31, 90C33

1. Introduction. Given an n × n real matrix M and q ∈ Rn , the linear complementarity problem (LCP) is to find a vector x ∈ Rn satisfying (1.1)

x > 0, M x + q > 0, (M x + q)T x = 0

or to show that no such vector x exists. We denote this problem (1.1) by LCP(M, q). The LCP(M, q) arises in many applications such as finding Nash equilibrium point of a bimatrix game, the network equilibrium problems, the contact problems and the free boundary problems for journal bearing etc, for details, see [1, 5, 18]. It is well-known that the LCP(M, q) has a unique solution for any vector q ∈ Rn if and only if M is a P -matrix [5]. Here a matrix M is called a P -matrix if all its principal minors are positive. For the LCP(M, q), one of the interesting problems is to estimate (1.2)

max ||(I − D + DM )−1 ||∞ ,

d∈[0,1]n

which can be used to bound the error ||x − x∗ ||∞ [4], that is, ||x − x∗ ||∞ 6 max n ||(I − D + DM )−1 ||∞ ||r(x)||∞ , d∈[0,1]

∗ Received by the editors on Month x, 201x. Accepted for publication on Month y, 201y Handling Editor: . † School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shannxi, 721007, P. R. China ([email protected]). ‡ School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, 650091, P.R. China ([email protected]).

1

2

L. GAO, C.Q. LI

where x∗ is the solution of the LCP(M, q), r(x) = min{x, M x + q}, D = diag(di ) with 0 6 di 6 1 for each i ∈ N , d = [d1 , d2 , ..., dn ]T ∈ [0, 1]n , and the min operator r(x) denotes the componentwise minimum of two vectors. When the matrix M for the LCP(M, q) belongs to P -matrices or some subclass of P -matrices, various bounds for (1.2) were proposed, e.g., see [2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14] and references therein. Recently, Garc´ıa-Esnaola and Pe˜ na in [10] provided an upper bound for (1.2) when M is a B-matrix as a subclass of P -matrices. Here, a matrix M = [mij ] ∈ Rn,n is called a B-matrix [6] if for each i ∈ N = {1, 2, . . . , n}, ! X 1 X mik > mij f or any j ∈ N and j 6= i. mik > 0, and n k∈N

k∈N

Theorem 1.1. [10, Theorem 2.2] Let M = [mij ] ∈ Rn,n be a B-matrix with the form M = B + + C,

(1.3) where



m11 − r1+  .. (1.4) B + = [bij ] =  . mn1 − rn+

  m1n − r1+   .. , C =  . · · · mnn − rn+ ···

r1+ .. . rn+

and ri+ = max{0, mij |j 6= i}. Then

max ||(I − D + DM )−1 ||∞ 6

(1.5)

d∈[0,1]n

where β = min{βi } and βi = bii − i∈N

P

 · · · r1+ ..  , .  · · · rn+

n−1 , min{β, 1}

|bij |.

j6=i

It is not difficult to see that the bound (1.5) will be inaccurate when the matrix P M has very small value of min{bii − |bij |}, for details, see [15, 16]. To conquer this i∈N

j6=i

problem, Li et al., in [17] gave the following bound for (1.2) when M is a B-matrix, which improves those provided by Li and Li in [15, 16]. Theorem 1.2. [17, Theorem 2.4] Let M = [mij ] ∈ Rn,n be a B-matrix with the form M = B + + C, where B + = [bij ] is the matrix of (4). Then n X

i−1 n − 1 Y bjj , d∈[0,1] min{β¯i , 1} j=1 β¯j i=1     n n i−1 P P Q bjj |bij |li (B + ) with lk (B + ) = max |b1ii | where β¯i = bii − = |bij | , and β¯j k≤i≤n   j=k, j=i+1 j=1

(1.6)

max n ||(I − D + DM )−1 ||∞ 6

j6=i

1 if i = 1.

A new improved error bound for linear complementarity problems for B-matrices

3

In this paper, we further improve error bounds on the LCP(M, q) when M belongs to B-matrices. The rest of this paper is organized as follows: In Section 2 we present a new error bound for (1.2), and then prove that this bound are better than those in Theorems 1 and 2. In Section 3, some numerical examples are given to illustrate our theoretical results obtained. 2. A new error bound for the LCP(M, q) of B-matrices. In this section, an upper bound for (1.2) is provided when M is a B-matrix. Firstly, some definitions, notation and lemmas which will be used later are given as follows. A matrix A = [aij ] ∈ C n,n is called a strictly diagonally dominant (SDD) matrix n P |bij | for all i = 1, 2, . . . , n. A matrix A = [aij ] is called a nonsingular if |aii | > j6=i

M -matrix if its inverse is nonnegative and all its off-diagonal entries are nonpositive [1]. In [6] it was proved that a B-matrix has positive diagonal elements, and a real matrix A is a B-matrix if and only if it can be written in form (1.3) with B + being a SDD matrix. Given a matrix A = [aij ] ∈ C n,n , let wij (A) = |aii | −

|aij | n P

, i 6= j, |aik |

k=j+1, k6=i

(2.1)

wi (A) = max{wij }, j6=i

n P

|aij | +

|aik |wk

k=j+1, k6=i

mij (A) =

, i 6= j.

|aii |

Lemma 2.1. [19, Theorem 14] Let A = [aij ] be an n × n row strictly diagonally dominant M -matrix. Then   ||A−1 ||∞ 6

where ui (A) =

1 |aii |

n X    i=1

n P

j=i+1

i−1 Y

1

aii −

n P

|aik |mki (A) j=1

k=i+1

 

1 k≤i≤n  |aii |

|aij |, lk (A) = max

if i = 1, and mki (A) is defined as in (2.1).

 1 , 1 − uj (A)lj (A) 

n P

j=k, j6=i

  i−1 Q |aij | ,  j=1

1 1−uj (A)lj (A)

Lemma 2.2. [15, Lemma 3] Let γ > 0 and η > 0. Then for any x ∈ [0, 1], 1 1 6 1 − x + γx min{γ, 1}

=1

4

L. GAO, C.Q. LI

and ηx η 6 . 1 − x + γx γ

Lemma 2.3. [16, Lemma 5] Let A = [aij ] with aii >

n P

|aij | for each i ∈ N .

j=i+1

Then for any xi ∈ [0, 1],

aii 1 − xi + aii xi . 6 n n P P |aij |xi |aij | 1 − xi + aii xi − aii − j=i+1

j=i+1

Lemmas 2.2 and 2.3 will be used in the proofs of the following lemma and of Theorem 2.5. Lemma 2.4. Let M = [mij ] ∈ Rn,n be a B-matrix with the form M = B + + C, + where B + = [bij ] is the matrix of (1.4). And let BD = I − D + DB + = [˜bij ] where D = diag(di ) with 0 6 di 6 1. Then             |bij | + wi (BD ) 6 max n j6=i   bii − P |bik |         k=j+1, k6=i

and

+ mij (BD )6m ˜ ij (B + ), + + where wi (BD ), mij (BD ) are defined as in (2.1), and                n   X    |b | 1  kh +    |b | · max |b | + m ˜ ij (B ) = ij n  ik h6=k   . P  bii      k=j+1,   |b |   b −  k6=i  kk l=h+1, kl   l6=k

Proof. Note that + [BD ]ij = ˜bij =

Since B + is SDD, bii −

n P

k=j+1, k6=i



1 − di + di bij , i = j, di bij , i= 6 j.

|bik | > 0. Hence, by Lemma 2.2 and (2.1), it follows

A new improved error bound for linear complementarity problems for B-matrices

5

that

 + + wi (BD ) = max wij (BD ) = max

     

     

|bij |di n P    1 − di + bii di − |bik |di        k=j+1, k6=i             |bij | 6 max . n j6=i   bii − P |bik |         k=j+1,

j6=i

j6=i

(2.2)

k6=i

Furthermore, it follows from (2.1), (2.2) and Lemma 2.2 that for each i 6= j (j < i 6 n) |bij | · di +

k=j+1, k6=i

+ mij (BD )=

6

n P

+ |bik | · di · wk (BD )

1 − di + bii · di



1  · |bij | + bii

n X

k=j+1, k6=i



+  |bik | · wk (BD )

     

        n  X    |b | 1  kh    |b | · max |b | + 6 ij n  ik h6=k   P  bii       k=j+1,  b − |b |    k6=i  kk l=h+1, kl   



l6=k

=m ˜ ij (B + ). The proof is completed.

By Lemmas 2.1, 2.2, 2.3 and 2.4, we give the following bound for (1.2) when M is a B-matrix. Theorem 2.5. Let M = [mij ] ∈ Rn,n be a B-matrix with the form M = B + + C, where B + = [bij ] is the matrix of (1.4). Then (2.3)

max n ||(I − D + DM )−1 ||∞ 6

d∈[0,1]

n X i=1

i−1 n − 1 Y bjj , min{βbi , 1} j=1 β¯j

n P |bik | · m ˜ ki (B + ) with m ˜ ki (B + ) is defined in lemma 2.4, β¯i is where βbi = bii − k=i+1

defined in Theorem 1.2, and

i−1 Q

j=1

bjj β¯j

= 1 if i = 1.

6

L. GAO, C.Q. LI

Proof. Let MD = I − D + DM . Then + MD = I − D + DM = I − D + D(B + + C) = BD + CD , + where BD = I−D+DB + = [˜bij ] and CD = DC. Similarly to the proof of Theorem 2.2 + in [10], we can obtain that BD is an SDD M -matrix with positive diagonal elements and that
−1 + −1 + −1 −1 + −1 ˜ (2.4) ||MD ||∞ ||(BD ) ||∞ 6 (n − 1)||(BD ) ||∞ . ||∞ 6 || I + (BD ) CD + −1 Next, we give an upper bound for ||(BD ) ||∞ . By Lemma 2.1, we have 

+ −1 ||(BD ) ||∞ 6

n X i=1

where

  

+ uj (BD )

=

 1  + + . + |bik | · di · mki (BD ) j=1 1 − uj (BD )lj (BD )

1 n P

1 − di + bii di −

k=i+1

n P

|bjk |dj

k=j+1

1 − dj + bjj dj

,

+ lk (BD )

= max

        

|bki | · dk +

n P

l=i+1, l6=k

n P

|bij |di

j=k, j6=i

k≤i≤n  1

and

+ mki (BD )=



i−1 Y

     

− di + bii di     

,

+ |bkl | · dk · wl (BD )

1 − dk + bkk · dk     |blh |dl + with wl (BD ) = max . n P h6=l     1−dl +bll dl − s=h+1, |bls |dl      

s6=l

By Lemmas 2.2 and 2.4, we can easily get that for each i ∈ N ,

1 − di + bii di −

1 n P

k=i+1

6 |bik | · di ·

+ mki (BD )

(

1 n P

min bii − 6

(

min bii − (2.5)

=

|bik | ·

k=i+1

n P

k=i+1

1 n o, min βbi , 1

+ mki (BD ), 1

1 |bik | · m ˜ ki

(B + ), 1

)

)

A new improved error bound for linear complementarity problems for B-matrices

7

and that for each k ∈ N ,   n P      |bij |di        n   1 X   j=k, j6=i + |bij | = lk (B + ) < 1. (2.6) lk (BD 6 max ) = max  k≤i≤n  k≤i≤n  1 − di + bii di    bii j=k,       j6=i  

Furthermore, according to Lemma 2.3 and (2.6), it follows that for each j ∈ N ,

(2.7)

1 1 − dj + bjj dj bjj 6 ¯ . n + + = P βj 1 − uj (BD )lj (BD ) + 1 − dj + bjj dj − |bjk | · dj · lj (BD ) k=j+1

By (2.5) and (2.7), we have (2.8)

n X



i−1 Y



1 bjj  1  n n o+ o . ¯ b b min β1 , 1 min βi , 1 j=1 βj i=2

+ −1 ||(BD ) ||∞ 6

The conclusion follows from (2.4) and (2.8).

The comparisons of the bounds in Theorems 1.2 and 2.5 are established as follows. Theorem 2.6. Let M = [mij ] ∈ Rn,n be a B-matrix with the form M = B + + C, where B + = [bij ] is the matrix of (1.4). Let β¯i and βbi be defined in Theorems 1.2 and 2.5, respectively. Then n X i=1

i−1 n i−1 X n − 1 Y bjj n − 1 Y bjj 6 . min{β¯i , 1} j=1 β¯j min{βbi , 1} j=1 β¯j i=1

Proof. Note that β¯i = bii −

n X

j=i+1

|bij |li (B + ), βbi = bii −

n X

|bik |m ˜ ki (B + ),

k=i+1

and B + is a SDD matrix, it follows that for each i 6= j (j < i 6 n)                n   X    |b | 1 kh +     |bij | + m ˜ ij (B ) = n |bik | · max  P   h6 = k bii     k=j+1,   |b |  b −  k6=i  kk l=h+1, kl   l6=k


bii −

k=i+1

n X

|bik |li (B + ) = β¯i ,

k=i+1

which implies that

This completes the proof.

1 1 6 . min{β¯i , 1} min{βbi , 1}

Remark here that when β¯i < 1 for all i ∈ N , then 1 1 , < min{β¯i , 1} min{βbi , 1}

which yields that n X i=1

n i−1 i−1 X n − 1 Y bjj n − 1 Y bjj < . min{β¯i , 1} j=1 β¯j min{βbi , 1} j=1 β¯j i=1

Next it is proved that the bound (2.3) given in Theorem 2.5 can improve the bound (1.5) in Theorem 1.1 (Theorem 2.2 in [10]) in some cases. Theorem 2.7. Let M = [mij ] ∈ Rn,n be a B-matrix with the form M = B + + C, where B + = [bij ] is the matrix of (1.4). Let β, β¯i and βbi be defined in Theorems 1.1, n i−1 P Q bjj 1.2 and 2.5, respectively, and let α = 1 + and βb = min{βbi }. If one of the β¯j i=2 j=1

i∈N

following conditions holds: (i) βb > 1 and α
1 and α
1. Then Mk = Bk+ + Ck , where   1 0 −0.1 0  −0.8 1 0 −0.1    B+ =  . k −0.1 k+1 − 0.8 1 −0.1   0 −0.8 −0.1 0 1

1 90k+91 By computations, we have β = 10(k+1) , β¯1 = β¯2 = 100k+100 , β¯3 = 0.99, β¯4 = 1, ˆ ˆ ˆ βˆ1 = 820k+828 900k+900 , β2 = 0.99, β3 = 1 and β4 = 1. Then it is easy to verify that Mk satisfies the condition (ii) of Theorem 2.7. Hence, by Theorem 1.1 (Theorem 2.2 in [10]), we have

max ||(I − D + DM )−1 ||∞ 6

d∈[0,1]n

4−1 = 30(k + 1). min{β, 1}

It is obvious that 30(k + 1) −→ +∞, when k −→ +∞. By Theorem 1.2, we have that for any k > 1, max ||(I − D + DM )−1 ||∞   1 1 1 1 1 1 63 ¯ + ¯ · ¯ + ¯ · ¯ ¯ + ¯ ¯ ¯ β1 β2 β1 β3 β1 β2 β1 β2 β3   2(100k + 100)2 100k + 100 (100k + 100)2 . =3 + + 90k + 91 (90k + 91)2 0.99(90k + 91)2 d∈[0,1]n

By Theorem 2.5, we have that for any k > 1, max ||(I − D + DM )−1 ||∞   1 1 1 1 1 + · ¯ + ¯ ¯ + ¯ ¯ ¯ 63 β1 β2 β1 β2 β3 βˆ1 βˆ2 β1 d∈[0,1]n

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L. GAO, C.Q. LI



900k + 900 (100k + 100) 1.99(100k + 100)2 + + 820k + 828 0.99(90k + 91) 0.99(90k + 91)2   2(100k + 100)2 100k + 100 (100k + 100)2 .