New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

Zhiwu Hou; Xia Jing; Lei Gao

doi:10.1515/math-2019-0127

Artikel Open Access

New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

Zhiwu Hou , Xia Jing und Lei Gao

Veröffentlicht/Copyright: 31. Dezember 2019

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Abstract

A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

Keywords: linear complementarity problems; error bounds; Σ-SDD matrices; SB-matrices

MSC 2010: 15A48; 65G50; 90C31; 90C33

1 Introduction

Let ℝⁿ be the n dimensional real vector space, and ℂ^n×n (ℝ^n×n) be the set of all n×n complex (real) matrices. The linear complementarity problem often arises from the various scientific computing, economics and engineering areas such as quadratic programs, Nash equilibrium points for bimatrix games, network equilibrium problems, contact problems, and free boundary problems for journal bearing, etc. for more details, see [1, Che2, 3. Here, the linear complementarity problem (LCP) is to find a vector x ∈ ℝⁿ such that

x≥0, Mx+q≥0, (Mx+q)Tx=0 (1)

or to show that no such vector x exists, where M = [m_ij] ∈ ℝ^n×n and q ∈ ℝⁿ. We denote the problem (1) and its solution by LCP(M, q) and x^*, respectively. A real square matrix M is called a P-matrix if all its principal minors are positive, and the LCP(M, q) has a unique solution for any q ∈ ℝⁿ if and only if M is a P-matrix [3].

An important topic for the LCP(M, q) is to estimate the upper bounds for error ||x–x^*||∞, since these bounds can often be used in convergence analysis of iterative algorithms [4]. For M being a P-matrix, Chen and Xiang present the following computable upper bound for ||x–x^*||∞ [5]:

||x−x∗||∞≤maxd∈[0,1]n||(I−D+DM)−1||∞||r(x)||∞, (2)

where D = diag(d_i) with 0 ≤ d_i ≤ 1 for each i ∈ N, d = [d₁, d₂, …, d_n]^T ∈ [0, 1]ⁿ, and r(x) = min{x, Mx+q denotes the componentwise minimum of two vectors. Moreover, to avoid the high-cost computations of the inverse matrix from (2), several easily computable bounds for LCPs were derived for the different subclass ofP-matrices, such as positively diagonal Nekrasov matrices [6, 7], S-Nekrasov matrices [8, 9], QN-matrices [10, 11], S-QN-matrices [12], B-matrices [13], 14, 15], DB-matrices [16], SB-matrices [17, 18] MB-matrices [2], B-Nekrasov matrices [7, 19, 20], BπR -matrices [21, 22], Dashnic-Zusmanovich type matrices [23], and weakly chained diagonally dominant B-matrices [24, 25, 26]. In [27]G, García-Esnaola and Peña present an error bound for the LCP(M, q) involved with Σ-SDD matrices, this bound involves a parameter and works only for Σ-SDD matrices but not strictly diagonally dominant matrices.

In this paper, we give a new error bound for linear complementarity problems when the involved matrices are Σ-SDD matrices, which is dependent only on the entries of the involved matrix. As an application, we provide a new error bound for linear complementarity problem with SB-matrices. Numerical examples are reported to show that the obtained bounds are better than those in [17], [18] and [27]Ga3} in some cases.

2 New error bounds for LCPs of Σ-SDD matrices

Let us first introduce some basic notations. A matrix M = [m_ij] ∈ ℝ^n×n is a Z-matrix if all its off-diagonal entries are nonpositive, and a nonsingular M-matrix if M is a Z-matrix with M^–1 being nonnegative [1]. Let N := {1, …, n} and S denotes a proper nonempty subset of N, S := N\S denotes its complement in N. For a given matrix M = [m_ij] ∈ ℂ^n×n, denote

ri(M)=∑j∈N∖{i}|mij|, riS(M)=∑j∈S∖{i}|mij|, and riS¯(M)=∑j∈S¯∖{i}|mij|,

where

[ri(M):=riS(M)+riS¯(M).

Definition 2.1

[1] A matrix M = [m_ij] ∈ ℂ^n×n is called an strictly diagonally dominant (SDD) matrix if |m_ii| > r_i(M) for all i ∈ N.

Definition 2.2

[27] A matrix M = [m_ij] ∈ ℂ^n×n is called a Σ-SDD matrix if there exists a nonempty subset S of N such that the following conditions hold:

|mii|>riS(M),for all i∈S,(|mii|−riS(M))(|mjj|−rjS¯(M))>riS¯(M)rjS(M),for all i∈S, j∈S¯.

Remark here that Σ-SDD matrices were usually called S-strictly diagonally dominant matrices in [28].

In [27], García-Esnaola and Peña provide the following error bound for the linear complementarity problem involved with Σ-SDD matrices.

Theorem 2.3

[27, Proposition 3.1] Suppose that A = [a_ij] ∈ ℂ^n×n is a Σ-SDD matrix with positive diagonal entries and A is not SDD, and W = diag(w_i) is a diagonal matrix such that AW is SDD, where w_i = γ(≠1) if i ∈ S and w_i = 1 if i ∈ S. For each i ∈ N, let

βi(γ)=aiiwi−∑j≠i|aij|wj

and

β(γ):=mini∈N{βi(γ)}.

If γ < 1, then

maxd∈[0,1]n||(I−D+DA)−1||∞≤f1(γ):=max1β(γ),1γ, (3)

and if γ > 1, then

maxd∈[0,1]n||(I−D+DA)−1||∞≤f2(γ):=maxγβ(γ),γ, (4)

where

(0<)γ∈IS:=maxi∈SriS¯(A)|aii|−riS(A),minj∈S¯|ajj|−rjS¯(A)rjS(A)

assuming that if rjS(A)=0, then |ajj|−rjS¯(A)rjS(A)=∞.

Recently, Wang et al. [29] proved that the infimum of error bounds (3) and (4) as a function of the parameter γ exists, and also can be determined.

Theorem 2.4

[29, Theorem 3] Let A = [a_ij] ∈ ℂ^n×n be Σ-SDD matrix and let S be any subset satisfying Definition 2.2. Suppose that A and W satisfy all assumptions of Theorem 2.3, and I_S ⊆(0, 1). Let

βi(γ)=(|aii|−riS(A))⋅γ−riS¯(A),if i∈S,(|aii|−riS¯(A))−riS(A)⋅γ,if i∈S¯,PS:={γ:βi(γ)=βj(γ),i,j∈N,i≠j,γ∈IS},

and

P~S:={γ:β(γ)=γ,γ∈IS}.

Then

maxd∈[0,1]n||(I−D+DA)−1||∞≤infγ∈ISf1(γ)=minγ∈(PS⋃P~S⋃{γ0,γ|PS|+1})f1(γ), (5)

where

γ0:=maxi∈SriS¯(A)|aii|−riS(A) and γ|PS|+1:=minj∈S¯|ajj|−rjS¯(A)rjS(A).

Theorem 2.5

[29, Theorem 4] Let A = [a_ij] ∈ ℂ^n×n be a Σ-SDD matrix and let S be any subset satisfying Definition 2.2. Suppose that A and W satisfy all assumptions of Theorem 2.3, and I_S ⊆ (1, +∞). Let

βi′(γ):=βi(γ)γ=(|aii|−riS(A))−riS¯(A)⋅1γ,if i∈S,(|aii|−riS¯(A))⋅1γ−riS(A),if i∈S¯,

PS′:={γ:βi′(γ)=βj′(γ),i,j∈N,i≠j,γ∈IS},

and

P~S′:={γ:β′(γ)=1γ,γ∈IS}.

Then

maxd∈[0,1]n||(I−D+DA)−1||∞≤infγ∈ISf2(γ)=minγ∈(PS′⋃P~S′⋃{γ0,γ|PS′|+1})f2(γ), (6)

where

γ0:=maxi∈SriS¯(A)|aii|−riS(A) and γ|PS′|+1:=minj∈S¯|ajj|−rjS¯(A)rjS(A).

Observe from Theorem 2.4 and Theorem 2.5 that if A is a large matrix, then the calculations of P_S and P~S′ (or PS′ and P~S′ ) in bounds (5) and (6) will be very complicated. On the other hand, for strictly diagonally dominant matrices, the bounds (5) and (6) become invalid. So it is interesting to find alternative error bounds depending only on the elements of the matrices for the LCP(M, q) when M is a Σ-SDD matrix. We next address this problem, before that some lemmas are listed.

Lemma 2.6

[14, Lemma 3] Let γ > 0 and η ≥ 0. Then for any x ∈ [0, 1],

11−x+γx≤1min{γ,1}

and

ηx1−x+γx≤ηγ.

Lemma 2.7

[30, Theorem 3] Let A = [a_ij] ∈ ℂ^n×n be a Σ-SDD matrix and S be a nonempty proper subset of N.

Then

||A−1||∞≤maxi∈S,j∈S¯maxρijS(A),ρjiS¯(A),

where

ρijS(A)=|aii|−riS(A)+rjS(A)(|aii|−riS(A))(|ajj|−rjS¯(A))−riS¯(A)rjS(A)

and

ρjiS¯(A)=|ajj|−rjS¯(A)+riS¯(A)(|aii|−riS(A))(|ajj|−rjS¯(A))−riS¯(A)rjS(A).

Lemma 2.8

Let S be a nonempty proper subset of N, and M = [m_ij] ∈ ℂ^n×n be a Σ-SDD matrix with m_ii > 0 for all i ∈ N. Let M_D = I–D+DM = [m̃_ij], where D = diag(d_i) with 0 ≤ d_i ≤ 1. Then M_D is a Σ-SDD matrix. Furthermore, for each i ∈ N,

riS(MD)=diriS(M), riS¯(MD)=diriS¯(M),

and

riS(MD)m~ii≤riS(M)mii,riS¯(MD)m~ii≤riS¯(M)mii.

Proof

Since

m~ij=1−di+dimij,i=j,dimij,i≠j, and di1−di+dimii≤1mii for ~all i, j∈N,

it follows that for each i ∈ N,

riS(MD)=∑j∈S∖{i}|m~ij|=∑j∈S∖{i}|dimij|=di∑j∈S∖{i}|mij|=diriS(M), (7)

and

riS¯(MD)=∑j∈S¯∖{i}|m~ij|=∑j∈S¯∖{i}|dimij|=di∑j∈S¯∖{i}|mij|=diriS¯(M). (8)

By (7) and (8), we have that for each i ∈ S,

m~ii−riS(MD)=1−di+dimii−diriS(M)≥di(mii−riS(M)) (9)

and

riS(MD)m~ii=diriS(M)1−di+miidi≤riS(M)mii<1.

Similarly, for each j ∈ S,

m~jj−rjS¯(MD)=1−dj+djmjj−djrjS¯(M)≥dj(mjj−rjS¯(M))

and

rjS¯(MD)m~jj=djrjS¯(M)1−dj+mjjdj≤rjS¯(M)mjj<1. (10)

Hence,

(m~ii−riS(MD))(m~jj−rjS¯(MD))≥di(mii−riS(M))⋅dj(mjj−rjS¯(M)), for all i∈S,j∈S¯.

If d_k = 0 for some k ∈ N, then from (9) and (10) we get

(m~ii−riS(MD))(m~jj−rjS¯(MD))>diriS¯(M)⋅djrjS(M)=riS¯(MD)⋅rjS(MD), for i=k or j=k. (11)

If 0 < d_k ≤ 1 for some k ∈ N, then from the fact that M is a Σ-SDD matrix we obtain

(m~ii−riS(MD))(m~jj−rjS¯(MD))≥di(mii−riS(M))⋅dj(mjj−rjS¯(M))>diriS¯(M)⋅djrjS(M)=riS¯(MD)⋅rjS(MD). (12)

Now the conclusion follows from (9), (10), (11) and (12).□

By Lemma 2.6, Lemma 2.7, and Lemma 2.8, we establish the first main result of this paper.

Theorem 2.9

Let M = [m_ij] ∈ ℂ^n×n be a Σ-SDD matrix with positive diagonal entries for some nonempty subset S ⊂ N. Then

maxd∈[0,1]n||(I−D+DM)−1||∞≤maxi∈S,j∈S¯maxχijS(M),χjiS¯(M), (13)

where

χijS(M):=mii−riS(M)mjj−rjS¯(M)minmjj−rjS¯(M),1+rjS(M)mii−riS(M)minmii−riS(M),1(mii−riS(M))(mjj−rjS¯(M))−riS¯(M)rjS(M)

and

χjiS¯(M):=mjj−rjS¯(M)mii−riS(M)minmii−riS(M),1+riS¯(M)mjj−rjS¯(M)minmjj−rjS¯(M),1(mii−riS(M))(mjj−rjS¯(M))−riS¯(M)rjS(M).

Proof

Let M_D = I – D + DM = [m̃_ij], where D = diag(d_i) with 0 ≤ d_i ≤ 1. Since M is a Σ-SDD matrix, then by Lemma 2.7 and Lemma 2.8 we have that M_D is a Σ-SDD matrix, and

||MD−1||∞≤maxi∈S,j∈S¯maxρijS(MD),ρjiS¯(MD), (14)

where

ρijS(MD)=|m~ii|−riS(MD)+rjS(MD)(|m~ii|−riS(MD))(|m~jj|−rjS¯(MD))−riS¯(MD)rjS(MD)

and

ρjiS¯(MD)=|m~jj|−rjS¯(MD)+riS¯(MD)(|m~ii|−riS(MD))(|m~jj|−rjS¯(MD))−riS¯(MD)rjS(MD).

Note that

m~ij=1−di+dimij,i=j,dimij,i≠j.

Then by Lemma 2.6 and Lemma 2.8, it follows that for each i ∈ S,

1m~ii−riS(MD)=11−di+miidi−diriS(M)≤1minmii−riS(M),1, (15)

and

riS¯(MD)|m~ii|−riS(MD)=diriS¯(M)1−di+dimii−diriS(M)≤riS¯(M)mii−riS(M). (16)

Analogously, for each j ∈ S, we have

1m~jj−rjS¯(MD)≤1minmjj−rjS¯(M),1, (17)

and

rjS(MD)|m~jj|−rjS¯(MD)=djrjS(M)1−dj+djmjj−djrjS¯(M)≤rjS(M)mjj−rjS¯(M). (18)

By (15), (16), (17) and (18), it follows that for each i ∈ S, j ∈ S,

ρijS(MD)=|m~ii|−riS(MD)+rjS(MD)(|m~ii|−riS(MD))(|m~jj|−rjS¯(MD))−riS¯(MD)rjS(MD)=1|m~jj|−rjS¯(MD)+rjS(MD)|m~ii|−riS(MD)|m~jj|−rjS¯(MD)1−riS¯(MD)rjS(MD)|m~ii|−riS(MD)|m~jj|−rjS¯(MD)≤1minmjj−rjS¯(M),1+rjS(M)minmii−riS(M),1⋅mjj−rjS¯(M)1−riS¯(M)⋅rjS(M)mii−riS(M)⋅mjj−rjS¯(M)=mii−riS(M)mjj−rjS¯(M)minmjj−rjS¯(M),1+rjS(M)mii−riS(M)minmii−riS(M),1(mii−riS(M))(mjj−rjS¯(M))−riS¯(M)rjS(M)=:χijS(M). (19)

In a similar way, we can prove that for each i ∈ S, j ∈ S,

ρjiS¯(MD)≤χjiS¯(M). (20)

The conclusion follows from (14), (19) and (20).□

Remark 2.10

Observe that bound (13) in Theorem 2.9 only depends on the elements of M, and it is easy to implement. For a set S with finite elements, we use |S| to denote the number of elements in the set S. From bound (13), we obtain the number of the basic arithmetic operations of bound (13) is |S|⋅|S| ⋅ (2n + 14) (requiring |S| ⋅ |S|⋅[2(n – 1) + 4] additions and 12 ⋅ |S| ⋅ |S| comparisons, multiplications and divisions of numbers). Furthermore, it follows from |S| < n and |S| < n that |S| ⋅ |S| ⋅ (2n + 14) < n²(2n + 14). Thus, the bound (13) of Theorem 2.9 can be performed in polynomial time.

By Theorem 2.9, we can easily obtain the following result.

Corollary 2.11

Let M = [m_ij] ∈ ℂ^n×n be a Σ-SDD matrix with positive diagonal entries for some nonempty subset S of N. For each i ∈ S, j ∈ S,

if m_ii – riS (M) ≤ 1 and m_jj – rjS¯ (M) ≤ 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤maxi∈S,j∈S¯maxχijS(M),χjiS¯(M),

where

χijS(M):=|mii|−riS(M)+rjS(M)(|mii|−riS(M))(|mjj|−rjS¯(M))−riS¯(M)rjS(M)

and

χjiS¯(M):=|mjj|−rjS¯(M)+riS¯(M)(|mii|−riS(M))(|mjj|−rjS¯(M))−riS¯(M)rjS(M);
if m_ii – riS (M) > 1 and m_jj – rjS¯ (M) ≤ 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤maxi∈S,j∈S¯maxχijS(M),χjiS¯(M),

where

χijS(M):=(1+rjS(M))(|mii|−riS(M))(|mii|−riS(M))(|mjj|−rjS¯(M))−riS¯(M)rjS(M)

and

χjiS¯(M):=mjj−rjS¯(M)mii−riS(M)+riS¯(M)(mii−riS(M))(mjj−rjS¯(M))−riS¯(M)rjS(M);
if m_ii – riS (M) ≤ 1 and m_jj – rjS¯ (M) > 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤maxi∈S,j∈S¯maxχijS(M),χjiS¯(M),

where

χijS(M):=mii−riS(M)mjj−rjS¯(M)+rjS(M)(mii−riS(M))(mjj−rjS¯(M))−riS¯(M)rjS(M)

and

χjiS¯(M):=(1+riS¯(M))(|mii|−rjS¯(M))(|mii|−riS(M))(|mjj|−rjS¯(M))−riS¯(M)rjS(M);
if m_ii – riS (M) > 1 and m_jj – rjS¯ (M) > 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤maxi∈S,j∈S¯maxχijS(M),χjiS¯(M),

where

χijS(M):=(|mii|−riS(M))(|mjj|−rjS¯(M)+rjS(M))(|mii|−riS(M))(|mjj|−rjS¯(M))−riS¯(M)rjS(M)

and

χjiS¯(M):=(|mii|−riS(M))(|mjj|−rjS¯(M)+riS¯(M))(|mii|−riS(M))(|mjj|−rjS¯(M))−riS¯(M)rjS(M).

Next, three examples are given to show the advantage of the bound (13) in Theorem 2.9. Before that, a well-known result which will be used later is given.

Theorem 2.12

[31, Remark 2.4] Let M = [m_ij] ∈ ℂ^n×n is an SDD matrix with positive diagonal entries. Then

maxd∈[0,1]n||(I−D+DM)−1||∞≤max1β,1,

where

β:=mini∈Nmii−∑j≠i|mij|.

Example 2.13

Consider the family of SDD matrices in [14], where

Mk=10−0.10−0.810−0.10−0.1kk+1−0.81−0.1−0.8−0.101 with k≥1.

Since M_k does not satisfy the assumptions of Theorem 2.3, Theorem 2.4, and Theorem 2.5, so we cannot use bounds (3), (4), (5), and (6) to estimate maxd∈[0,1]4||(I−D+DMk)−1||∞ . However, according to an SDD matrix is a Σ-SDD matrix for any nonempty S ⊆ N, taking the set S = {1, 2} and S = {3, 4}, by bound (13) of Theorem 2.9, we obtain

maxd∈[0,1]4||(I−D+DMk)−1||∞≤maxi∈S,j∈S¯maxχijS(Mk),χjiS¯(Mk)=1+0.1kk+10.18−0.1[0.1kk+1+0.8]<1109.

In fact,

χ13S(Mk)=1.8+0.1kk+10.9−0.1[0.1kk+1+0.8], χ14S(Mk)=19091, χ23S(Mk)=1+0.1kk+10.18−0.1[0.1kk+1+0.8], χ24S(Mk)=10;

and

χ31S¯(Mk)=10.9−0.1[0.1kk+1+0.8], χ41S¯(Mk)=11091, χ32S¯(Mk)=10.18−0.1[0.1kk+1+0.8], χ42S¯(Mk)=10.

In contrast, by Theorem 2.12 (Remark 2.4 of [31]), we have

maxd∈[0,1]4||(I−D+DMk)−1||∞≤max1β,1=10(k+1).

It is obvious that

10(k+1)→+∞, when k→+∞.

Since a Σ-SDD matrix is an S-Nekrasov matrix, the bound (2.14) of Theorem 2.2 in [8] for S-Nekrasov matrices can also be used to estimate maxd∈[0,1]n||(I−D+DM)−1||∞ when M is a Σ-SDD matrix. The following example shows that the bound (13) given in Theorem 2.9 is sharper than the bound (2.14) of Theorem 2.2 in [8].

Example 2.14

The LCP(M, q) has often been used to discuss formulation and solution of traffic equilibrium problems [32, 33]. Consider the matrix M ∈ ℝ^5×5 arising from a simple traffic network problem [32]:

M=100050015005002000200200010025.

It is easy to check that M is a Σ-SDD matrix for any nonempty S ⊆ N. Since M does not satisfy the assumptions of Theorem 2.3, Theorem 2.4, and Theorem 2.5, so we cannot use bounds (3), (4), (5), and (6) to estimate maxd∈[0,1]5||(I−D+DM)−1||∞ . However, taking S = {2, 3, 5}, by bound (13) of Theorem 2.9, we obtain

maxd∈[0,1]5||(I−D+DM)−1||∞≤1.

In contrast, bound (2.14) of Theorem 2.2 in [8] for S-Nekrasov matrices gives the following estimation:

maxd∈[0,1]5||(I−D+DM)−1||∞≤1.5526.

It is also shown by Figure 1, in which the first 1000 matrices D are given by the following MATLAB code, that 1 is the exact value of maxd∈[0,1]5||(I−D+DM)−1||∞ .

MATLAB code:for i=1:1000; D=diag(rand(5,1)); end.

$Figure 1 ||(I – D + DM)–1||∞ for the first 1000 matrices D generated by diag(rand(5, 1)).$

Figure 1

||(I – D + DM)^–1||_∞ for the first 1000 matrices D generated by diag(rand(5, 1)).

Example 2.15

Consider the following matrix

M=1−0.6−0.20−0.251−0.25−0.5−0.2−0.41−0.4−1.2001.

It is easy to verify that M is not an SDD matrix, but it is a Σ-SDD matrix for S = {1}. In fact, by calculation, we have

r1S(M)=0,r2S(M)=0.25,r3S(M)=0.2,r4S(M)=1.2,r1S¯(M)=0.8,r2S¯(M)=0.75,r3S¯(M)=0.8, and r4S¯(M)=0,

which satisfy the conditions of Definition 2.2. Then I_S = (4/5, 5/6). So by Theorem 2.3 we can get the bound (3) with γ ∈ I_S for maxd∈[0,1]4||(I−D+DM)−1||∞ , which is drawn in Figure 2, and its infimum can be determined by Theorem 2.4 in the following way. Since

β1(γ)=γ−0.8, β2(γ)=0.25(1−γ), β3(γ)=0.2(1−γ), β4(γ)=1−1.2γ,PS:={γ:βi(γ)=βj(γ),i,j∈N,i≠j,γ∈IS},

$Figure 2 The bounds (3) and (13) in Theorem 2.3 and Theorem 2.9.$

Figure 2

The bounds (3) and (13) in Theorem 2.3 and Theorem 2.9.

and

P~S:={γ:β(γ)=γ,γ∈IS},

it follows that P_S = {9/11},

β(γ)=β1(γ), 4/5>γ≤9/11,β4(γ), 9/11>γ≤5/6,

and P̃_S = ∅. Each β_i(γ) and γ are drawn in Figure 3. By the bound (5) of Theorem 2.4, we have

maxd∈[0,1]n||(I−D+DA)−1||∞≤infγ∈(4/5,5/6)f1(γ)=minγ∈(PS⋃P~S⋃{4/5,5/6})max1β(γ),1γ=55.

$Figure 3 βi(γ), β(γ), and γ.$

Figure 3

β_i(γ), β(γ), and γ.

In addition, by the bound (13) of Theorem 2.9, we can see that

maxd∈[0,1]n||(I−D+DM)−1||∞≤maxi∈S,j∈S¯maxχijS(M),χjiS¯(M)=55.

It should be pointed out that the bound (13) is computationally much easier than the bound in Theorem 2.4, because it only depends on the elements of the matrix M.

3 New error bounds for LCPs of SB-matrices

Based on Theorem 2.9, we in this section present a new error bound for linear complementarity problems associated with SB-matrices. For a real matrix M = [m_ij] ∈ ℝ^n×n, we can write it as

M=B+C, (21)

where

B=m11−r1m12−r1⋯m1n−r1m21−r2m22−r2⋯m2n−r2⋮⋮⋱⋮mn1−rnmn2−rn⋯mnn−rn, C=r1r1⋯r1r2r2⋯r2⋮⋮⋱⋮rnrn⋯rn

with

ri=max{0,mij|j≠i}.

Obviously, B is a Z-matrix and C is a nonnegative matrix of rank 1.

Let us recall the definition of SB-matrices which is proposed by Li et al. in [34] as a subclass of P-matrices.

Definition 3.1

A real matrix M = [m_ij] ∈ ℝ^n×n is called an S-strictly dominant B-matrix (SB-matrix) if it can be written in form (21) with B being a Σ-SDD matrix for some nonempty proper subset S of N whose diagonal entries are all positive.

By Theorem 2.9, we give an upper bound for maxd∈[0,1]n||(I−D+DM)−1||∞ when M is an SB-matrix.

Theorem 3.2

Let M ∈ ℝ^n×n be an SB-matrix for some nonempty subset S of N, and B = [b_ij] be the matrix of (21). Then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(n−1)⋅maxi∈S,j∈S¯maxχijS(B),χjiS¯(B), (22)

where

χijS(B):=bii−riS(B)bjj−rjS¯(B)minbjj−rjS¯(B),1+rjS(B)bii−riS(B)minbii−riS(B),1(bii−riS(B))(bjj−rjS¯(B))−riS¯(B)rjS(B)

and

χjiS¯(B):=bjj−rjS¯(B)bii−riS(B)minbii−riS(B),1+riS¯(B)bjj−rjS¯(B)minbjj−rjS¯(B),1(bii−riS(B))(bjj−rjS¯(B))−riS¯(B)rjS(B).

Proof

Let M_D = I – D + DM, where D = diag(d_i) with 0 ≤ d_i ≤ 1. Since M is an SB-matrix, then we can write M = B + C as in (21) where B is a Σ-SDD matrix with positive diagonal entries, which yields that

MD=I−D+DM=(I−D+DB)+DC=BD+CD,

where B_D = I – D + DB and C_D = DC. Note that B is a Σ-SDD matrix with positive diagonal entries. Then by Lemma 2.8, B_D is also a Σ-SDD with positive diagonal entries. Obviously, B_D is a Z-matrix. Hence, we have that B_D is a nonsingular M-matrix. Similarly to the proof of Theorem 2 in [19], we have

||MD−1||∞≤||(I+(BD)−1CD)−1||∞||(BD)−1||∞≤(n−1)||(BD)−1||∞. (23)

Next, we give an upper bound for ||(B_D)^–1||_∞. Since B is a Σ-SDD matrix, we have from Theorem 2.9 that

||(BD)−1||∞≤maxi∈S,j∈S¯maxρijS(BD),ρjiS¯(BD)≤maxi∈S,j∈S¯maxχijS(B),χjiS¯(B). (24)

The conclusion follows from (23) and (24).□

Corollary 3.3

Let M ∈ ℝ^n×n be an SB-matrix for some nonempty proper subset S of N, and B = [b_ij] be the matrix of (21). Then for i ∈ S, j ∈ S,

if b_ii – riS (B) ≤ 1 and b_jj – rjS¯ (B) ≤ 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(n−1)⋅maxi∈S,j∈S¯maxχijS(B),χjiS¯(B),

where

χijS(B):=|bii|−riS(B)+rjS(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B)

and

χjiS¯(B):=|bjj|−rjS¯(B)+riS¯(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B);
if b_ii – riS (B) > 1 and b_jj – rjS¯ (B) ≤ 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(n−1)⋅maxi∈S,j∈S¯maxχijS(B),χjiS¯(B),

where

χijS(B):=(1+rjS(B))(|bii|−riS(B))(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B)

and

χjiS¯(B):=bjj−rjS¯(B)bii−riS(B)+riS¯(B)(bii−riS(B))(bjj−rjS¯(B))−riS¯(B)rjS(B);
if b_ii – riS (B) ≤ 1 and b_jj – rjS¯ (B) > 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(n−1)⋅maxi∈S,j∈S¯maxχijS(B),χjiS¯(B),

where

χijS(B):=bii−riS(B)bjj−rjS¯(B)+rjS(B)(bii−riS(B))(bjj−rjS¯(B))−riS¯(B)rjS(B)

and

χjiS¯(B):=(1+riS¯(B))(|bii|−rjS¯(B))(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B);
if b_ii – riS (B) > 1 and b_jj – rjS¯ (B) > 1, then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(n−1)⋅maxi∈S,j∈S¯maxχijS(B),χjiS¯(B),

where

χijS(B):=(|bii|−riS(B))(|bjj|−rjS¯(B)+rjS(B))(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B)

and

χjiS¯(B):=(|bii|−riS(B))(|bjj|−rjS¯(B)+riS¯(B))(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B).

To compare our error bound with those of [17] and [18], we first recall the following results given by Dai et al. of [17] and [18].

Theorem 3.4

[17, Theorem 3.1] Let M ∈ ℝ^n×n be an SB-matrix for some nonempty proper subset S of N, and B = [b_ij] be the matrix of (21). Then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(n−1)⋅max{η1,η2}, (25)

where

η1=maxmaxi∈S,j∈S¯|bjj|−rjS¯(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B),1+maxmaxi∈S,j∈S¯riS¯(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B),maxi∈SriS¯(B)|bii|−riS(B),

and

η2=maxmaxi∈S,j∈S¯|bii|−riS(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B),1+maxmaxi∈S,j∈S¯rjS(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B),maxj∈S¯rjS(B)|bjj|−rjS¯(B).

Theorem 3.5

[18, Theorem 2.4] Let M ∈ ℝ^n×n be an SB matrix for some nonempty proper subset S of N, B = [b_ij] be the matrix of (21), and W = diag(w_i) be a diagonal matrix such that BW is SDD, where w_i = 1 if i ∈ S and w_i = η if i ∈ S. Let N_S := card(S), N_S := card(S),

α:=maxNS+ηNS¯,1ηNS+NS¯,

and

ξi:=bii−∑j∈S∖{i}|bij|−η∑j∈S¯∖{i}|bij|, i∈S,ηbii−∑j∈S∖{i}|bij|−η∑j∈S¯∖{i}|bij|, i∈S¯,

where

(0<)η∈maxj∈S¯rjS(B)|bjj|−rjS¯(B),mini∈S|bii|−riS(B)riS¯(B)

assuming that if riS¯ (B) = 0, then |bii|−riS(B)riS¯(B) = ∞. Denote ξ := min_i{ξ_i}. Then

maxd∈[0,1]n||(I−D+DM)−1||∞≤(α−1)max{η,1}min{ξ,η,1}. (26)

By Corollary 3.3, we easily obtain the following comparison result, which shows that the bound (22) of Theorem 3.2 improves the bound (25) of Theorem 3.4 under certain conditions.

Theorem 3.6

Let M ∈ ℝ^n×n be an SB-matrix for some nonempty proper subset S of N, and B = [b_ij] be the matrix of (21). If b_ii – riS (B) ≤ 1 and b_jj – rjS¯ (B) ≤ 1 for i ∈ S, j ∈ S, then

maxi∈S,j∈S¯maxχijS(B),χjiS¯(B)≤max{η1,η2},

where χijS (B) and χjiS¯ (B) are given by Theorem 3.2, η₁ and η₂ are given by Theorem 3.4.

Proof

Since b_ii – riS (B) ≤ 1 and b_jj – rjS¯ (B) ≤ 1 for i ∈ S, j ∈ S, it follows from Corollary 3.3 that

χijS(B)=|bii|−riS(B)+rjS(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B),

and

χjiS¯(B)=|bjj|−rjS¯(B)+riS¯(B)(|bii|−riS(B))(|bjj|−rjS¯(B))−riS¯(B)rjS(B).

Obviously,

maxi∈S,j∈S¯χjiS¯(B)≤η1, maxi∈S,j∈S¯χijS(B)≤η2.

This completes the proof.□

Moreover, the following example shows that the bound (22) of Theorem 3.2 is sharper than the bound (26) of Theorem 3.5 in some cases.

Example 3.7

Consider the following matrix

M=1121212−25115−25−101−16343421401.

Obverse that M does not have one strictly diagonally dominant row in all rows, so it is not an H-matrix, consequently, not a Σ-SDD matrix. Furthermore, M can be written as M = B + C with

B=12000−35450−35−101−1600−94014 and C=1212121215151515000034343434.

Obviously, B is not SDD and so M is not a B-matrix. However, it is easy to check that B is a Σ-SDD matrix for S = {1}, which implies that M is an SB-matrix for S = {1}. Thus, by the bound (22) of Theorem 3.2, we can see that

maxd∈[0,1]4||(I−D+DM)−1||∞≤120.

In contrast, by Theorem 3.5 we can get the bound (26) involved with η ∈ (3, +∞) for maxd∈[0,1]4||(I−D+DM)−1||∞ , which is drawn in Figure 4. It can be seen from Figure 4 that the bound (22) in Theorem 3.2 is smaller than the bound (26) in Theorem 3.5 (Theorem 2.4 in [18]).

$Figure 4 The bounds (22) and (26) in Theorem 3.2 and Theorem 3.5.$

Figure 4

The bounds (22) and (26) in Theorem 3.2 and Theorem 3.5.

4 Conclusions

In this paper, for the linear complementarity problems with a Σ-SDD matrix M, we first give an alternative error bound for the LCP(M, q) which depends only on the entries of M. Then, by this new result, a new error bound for the LCP(M, q) with SB-matrices is provided. We also illustrate the results by numerical examples, where we improve bounds obtained in [17] and [18].

Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments, which help improve the quality of the paper. This work was supported by National Natural Science Foundation of China (31600299), the Scientific Research Programs Funded by Shaanxi Provincial Education Department (18JK1216, 18JK1217), the Science and Technology Project of Baoji (2017JH2-21), the Projects of Baoji University of Arts and Sciences (ZK2017095, ZK2017021), the Scientific Research Program Funded by Yunnan Provincial Education (2019J0910).

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Received: 2019-05-28

Accepted: 2019-10-22

Published Online: 2019-12-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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https://doi.org/10.1515/math-2019-0127

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linear complementarity problems; error bounds; Σ-SDD matrices; SB-matrices

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