CKV-type B -matrices and error bounds for linear complementarity problems

: In this paper, we introduce a new subclass of P -matrices called Cvetkovi´c-Kosti´c-Varga type B -matrices (CKV-type B -matrices), which contains DZ-type-B -matrices as a special case, and present an inﬁnity norm bound for the inverse of CKV-type B -matrices. Based on this bound, we also give an error bound for linear complementarity problems of CKV-type B -matrices. It is proved that the new error bound is better than that provided by Li et al. [24] for DZ-type-B -matrices, and than that provided by M. Garc´ıa-Esnaola and J.M. Pe˜na [10] for B -matrices in some cases. Numerical examples demonstrate the e ﬀ ectiveness of the obtained results.


Introduction
Given an n × n real matrix A and a vector q ∈ R n , the linear complementarity problem is to find a vector x ∈ R n satisfying x ≥ 0, Ax + q ≥ 0, (Ax + q) T x = 0 (1. 1) or to show that no such vector x exists. We denote the problem (1.1) and its solution by LCP(A, q) and x * , respectively. The LCP(A, q), as one of the fundamental problems in optimization and mathematical programming, has various applications in the quadratic programming, the optimal stopping, the Nash equilibrium point of a bimatrix game, the network equilibrium problem, the contact problem, and the free boundary problem for journal bearing, for details, see [1,2,26].
It is well known that the LCP(A, q) has a unique solution x * for any q ∈ R n if and only if A is a P-matrix [2]. Here, a real square matrix A is called a P-matrix if all its principal minors are positive. For this case, an important topic in the study of the LCP(A, q) concerns the bound of ||x − x * || ∞ , since it can be used as termination criteria for iterative algorithms and can be used to measure the sensitivity of the solution of LCP(A, q) in response to a small perturbation, e.g., [18,19,28,34]. When the matrix A is a P-matrix, Chen and Xiang [3] gave the following error bound for the LCP(A, q): where D = diag(d i ) with 0 ≤ d i ≤ 1 for each i ∈ N := {1, . . . , n}, d = (d 1 , d 2 , . . . , d n ) T ∈ [0, 1] n , and r(x) = min{x, Ax + q} in which the min operator denotes the componentwise minimum of two vectors. Furthermore, to avoid the high-cost computations of the inverse matrix in (1.2), some easily computable bounds for the LCP(A, q) were derived for the different subclass of P-matrices, such as, B-matrices [10,20], doubly B-matrices [6], S B-matrices [7,8], MB-matrices [4], B-Nekrasov matrices [11,21], weakly chained diagonally dominant B-matrices [22,32,35], B R π -matrices [12,13,27], and so on [9,14-17, 23, 36].
Recently, Li et al. [24] presented a new subclass of P-matrices called Dashnic-Zusmanovich type B-matrices (DZ-type-B-matrices), and provided an error bound for the LCP(A, q) when A is a DZ-type-B-matrix.
[24] Let A = [a i j ] ∈ C n×n be written in the form and r + i := max{0, a i j | j i}. Then, A is called a DZ-type-B-matrix if B + is a DZ-type matrix with all positive diagonal entries.
Very recently, Cvetković et al. [5] proposed a new subclass of H-matrices called CKV-type matrices, which generalizes CKV matrices (also known as Σ-SDD matrices in the literature) and DZ-type matrices.
Motivated by the definition of DZ-type-B-matrices, two meaningful questions naturally arise: can we get a more general subclass of P-matrices using CKV-type matrices, and can we obtain a sharper error bound than the bound (1.4) for the linear complementarity problem of DZ-type-B-matrices? To answer these questions, in Section 2, we present a new class of matrices: CKV-type B-matrices, and prove that it is a subclass of P-matrices containing DZ-type-B-matrices and S B-matrices. Meanwhile, we give an upper bound for the infinity norm for the inverse of CKV-type B-matrices. In Section 3, we give an error bound for the LCP(A, q) when A is a CKV-type B-matrix, consequently, for the LCP(A, q) when A is a DZ-type-B-matrix, and some comparisons with other results are also discussed. Finally, in Section 4, numerical examples are given to illustrate the corresponding theoretical results.

CKV-type B-matrices
Using CKV-type matrices, we first give the definition of CKV-type B-matrices. To show that a CKV-type B-matrix is a P-matrix, we recall the following results. is a real nonsingular M-matrix and P is a nonnegative matrix with rank(P)=1, then A + P is a P-matrix.
Proof. Let A be written in the form A = B + + C as shown in (1.3). It follows from (1.3) and Definition 2.1 that B + is a Z-matrix (all non-diagonal entries are non-positive [1]) with positive diagonal entries and C is a nonnegative matrix of rank 1. By Lemma 2.1, we know that B + is a nonsingular H-matrix, and thus the conclusion follows from Lemma 2.2.
As shown in [5,33], the relations of strictly diagonally dominant (SDD) matrices, doubly strictly diagonally dominant (DSDD) matrices, S -strictly diagonally dominant (S -SDD) matrices, DZ-type matrices, and CKV-type matrices are: According to [24] and the above relations, we give a figure to illustrate the relations among Bmatrices, DZ-type-B-matrices, DB-matrices, S B-matrices, CKV-type B-matrices. Here, the notions of B-matrices, DB-matrices, and of S B-matrices are listed as follows.  Next, we give a sufficient and necessary condition for a CKV-type B-matrix. Before that, a lemma is needed.
given by Definition 1.3 is not empty for all i ∈ N. Especially, if A is an SDD matrix, then for all i ∈ N, all proper subsets S containing i belong to S i (A).

and let I be the identity matrix, then A is a CKV-type B-matrix if and only if I
Proof. Sufficiency is clearly established. We next show the necessity.
It follows thatr (2.1) and (2.2), we easily obtain that Since A is a CKV-type B-matrix, then B + = [b i j ] is a CKV-type matrix with positive diagonal entries. Thus, by Lemma 2.3, it follows that for each i ∈ N, there exists S ∈ S i (B + ), which implies that and that for all j ∈ S , if d i 0 and d j 0, then and if d i = 0 or d j = 0, then These mean that S ∈ S i (B + ) for each i ∈ N. Therefore, from Definition 1.3, B + is a CKV-type matrix with positive diagonal entries, and consequently, A = I − D + DA is a CKV-type B-matrix from Definition 2.1.
In the following, we give an infinity norm bound for the inverse of CKV-type B-matrices. First, two lemmas are listed.
where S i (A) is given by Definition 1.3, and .
where S i (B + ) is defined as in Definition 1.3, and .
Proof. Since A is a CKV-type B-matrix, so B + is a CKV-type matrix with positive diagonal entries and also a Z-matrix. By Corollary 4 of [31], we know that B + is an M-matrix and thus (B + ) −1 is nonnegative. Hence, from A = B + + C in which B + and C are given by (1.3), we have which implies that Note that C = (r + 1 , . . . , r + n ) T e is nonnegative. Therefore, (B + ) −1 C can be written as (p 1 , . . . , p n ) T e, where p i ≥ 0 for all i ∈ N. By Lemma 2.5, we get (2.5) Since B + is a CKV-type matrix, it follows from Lemma 2.4 that Hence, from (2.4), (2.5), and (2.6), the conclusion follows.

Error bounds for the linear complementarity problem
Based on Theorem 2.1, we give in this section an upper bound of max d∈[0,1] n ||(I − D + DA) −1 || ∞ when A is a CKV-type B-matrix, and give some comparisons with other results. Before that, a useful lemma is needed.  where S i (B + ) is defined as in Definition 1.3, and .
Since a DZ-type-B-matrix is a CKV-type B-matrix, the bound (3.1) can also be used to estimate max d∈[0,1] n ||(I − D + DA) −1 || ∞ when A is a DZ-type-B-matrix. The following theorem provides that the bound (3.1) is better than the bound (1.4) in Theorem 1.1 (Theorem 6 of [24]).
It is easy to see that j ∈ γ i (B + ) is equivalent to S = N \ { j} ∈ S i (B + ). Therefore, for each i ∈ N, This completes the proof.
Particularly, for B-matrices, as an important subclass of CKV-type B-matrices, we next show that the bound (3.1) is better than that given by García-Esnaola and Peña in [10] in some cases. where α S i j (B + ) is defined as in Theorem 3.1. Proof. By the fact that a B-matrix is a CKV-type B-matrix, we know that (3.1) holds directly. We now prove that (3.5) and (3.6) hold. For each i ∈ N, S ∈ S i (B + ), and j ∈ S , if b ii − r S i (B + ) ≤ 1 and b j j − r S j (B + ) ≤ 1, then from Remark 3.1 that . . .
The proof is complete.
Remark from Theorem 3.4 that we can take the minimum of bounds (3.1) and (3.4) to estimate the error bound for the LCP(A, q) with A being a B-matrix, that is,

Numerical examples
In this section, three examples are given to show the advantage of the bound (3.1) in Theorem 3.1.
Example 4.1. Consider the following matrix Obviously, B + = A and C = 0. It is easy to verify that B + is not a DZ-type matrix and an S -SDD matrix, consequently, not a SDD matrix and a DSDD matrix. Hence, A is not a DZ-type-B-matrix and an S B-matrix, and thus not a B-matrix and a DB-matrix. So we cannot use the error bounds in [6-8, 10, 20, 24] Note that r + i := max{0, a i j | j i} = 0 for i = 1, 2, 3, 4. Hence, B + = A and C = 0. By calculations, we have that A is a DZ-type-B-matrix, and thus it is a CKV-type B-matrix. By the bound

Conclusions
In this paper, on the basis of the class of CKV-type matrices, a new subclass of P-matrices: CKVtype B-matrices, containing B-matrices, DB-matrices, S B-matrices as well as DZ-type-B-matrices, is introduced, and an upper bound for the infinity norm for the inverse of CKV-type B-matrices is provided. Then, by this bound, an error bound for the corresponding LCP(A, q) is given. We also proved that the new error bound is sharper than those of [10] and [24] in some cases, and give numerical examples to show the advantage of our results.