Global error bounds of the extended vertical linear complementarity problems for Dashnic–Zusmanovich matrices and Dashnic–Zusmanovich-B matrices

Global error bounds of the extended vertical linear complementarity problems for Dashnic–Zusmanovich (DZ) matrices and Dashnic–Zusmanovich-B (DZ-B) matrices are presented, respectively. The obtained error bounds are sharper than those of Zhang et al. (Comput. Optim. Appl. 42(3):335–352, 2009) in some cases. Some numerical examples are given to illustrate the obtained results.


Introduction
The extended vertical linear complementarity problem (EVLCP) is to find a vector x ∈ R n such that r(x) := min(M 0 x + q 0 , M 1 x + q 1 , . . . , M k x + q k ), or to prove that there is no such vector x, where the min operator works componentwise for both vectors and matrices. It is denoted by EVLCP(M, q), where q = (q 0 , q 1 , . . . , q k ), q l ∈ R n , l = 0, 1, . . . , k, is a block vector and M = (M 0 , M 1 , . . . , M k ), M l ∈ R n×n , l = 0, 1, . . . , k, is a block matrix. When k = 1, M 0 = I, q 0 = 0, the EVLCP(M, q) comes back to linear complementarity problems (LCP), and when M 0 = I, q 0 = 0, the EVLCP(M, q) reduces to vertical linear complementarity problems (VLCP) [2]. The extended vertical linear complementarity problems are widely used in optimization theory, control theory, neural network model, convergence analysis, sensitive analysis, verification of the solutions, and so on, see [1,[3][4][5][6][7]. Many scholars are interested in the research on the error of the solution for the EVLCP(M, q) including the LCP case. Various results on the solution and its error bounds for the EVLCP(M, q) have appeared recently, see [8][9][10][11][12][13]. For example, Gowda and Sznajder [14] extended the sufficient and necessary condition for the existence and uniqueness of the solution from the LCP to the EVLCP. Afterwards, Sznajder and Gowda [15] provided some equivalent forms for the condition above. Xiu and Zhang [16] extended the error bound for the LCP given by [6] to the EVLCP. Zhang et al. [1] extended the error bound of the LCP given by Chen and Xiang [8] to the general EVLCP by the row rearrangement technique and provided some computable error bounds for two types of special block matrices. However, these error bounds generally can not be calculated accurately because they involve computing the inverse of matrices. In order to overcome this shortcoming, in this paper, we continue to explore the extended vertical linear complementarity problems, and we propose new error bounds for the other types of special block matrices, named DZ matrices [17,18] and DZ-B matrices [19], only relying on the elements of such matrices. The obtained results extend the corresponding results in [1]. The validity of new error bounds is theoretically guaranteed, and numerical examples show the validity of the new results.
The remainder of this paper is organized as follows. In Sect. 2, we recall some related definitions, theorems, lemmas, and notations, which will be used in the proof of this paper. In Sect. 3, we prove that each block in any row rearrangement of the block matrix

Preliminaries
In this section, we recall some theorems, definitions, lemmas, and notations. Given a ma- The first one is the existence and uniqueness condition of the solution for the EVLCP(M, q) given by Gowda and Sznajder [14].
where the min and max operators work componentwise for both vectors and matrices.
Using the row rearrangement technique, Zhang et al. [1] presented a sufficient and necessary condition for the block matrix M with the row W-property and proposed a global error bound for the EVLCP(M, q).
where A i. means the ith row of a given matrix A. This is also true for the block vectors q and q . Denote by R(M) and R(q) the set of all row rearrangements of M and q, respectively. x -

A global error bound for the EVLCP of Dashnic-Zusmanovich matrices
Dashnic-Zusmanovich matrix, as a subclass of the class of nonsingular P-matrix, was introduced by Dashnic and Zusmanovich [17] to upper bound for the infinity norm of its inverse matrix, whose definition and related conclusion are listed as follows.
Next, we will propose an upper bound for α ∞ (M) with each M l (l = 0, 1, . . . , k) being a DZ matrix. Before that, some useful propositions are provided below. Proposition 1 Let A = (a ij ) ∈ C n×n and B = (b ij ) ∈ C n×n be all DZ matrices with positive diagonal elements. If there exists an index i ∈ N such that, for any j ∈ N , j = i, a ij b ij ≥ 0 (or a ji b ji ≥ 0), and Proof Both A and B are DZ matrices, and there exists i ∈ N such that, for any j ∈ N , j = i, Note that d i ∈ [0, 1], then 1d i ≥ 0 and d i ≥ 0, and they are not equal to 0 at the same time. Let (I -D)A + DB = C = (c ij ), then Hence, we get that is, From Definition 2, the conclusion follows.
Based on Proposition 1, Lemma 1, and the fact that a DZ matrix is nonsingular, the block matrix composed of DZ matrices has the row W-property. Proof , .
Hence, it holds that Further, we get Therefore, we have where So, it holds that By Definition 1, we can regard M j , M l as two blocks in a row rearrangement of M = (M 0 , M 1 , . . . , M k ), and thus for t = j or t = l and for i ∈ N, there exists t i ∈ {0, 1, . . . , k} such that Similarly, we get Examples 1 and 2 show that the bound in Theorem 6 is sharper than that in Theorems 3 and 4 in some cases.

A global error bound for the EVLCP of Dashnic-Zusmanovich-B matrices
In 2020, Zhou et al. [19] introduced error bounds of the linear complementarity problems of Dashnic-Zusmanovich-B matrices, whose definition is listed below.
Next, we will present an upper bound for α ∞ (M) with each M l ∈ R n×n (l = 0, 1, . . . , k) being a DZ-B matrix. Before that, some useful results are presented as follows.

Proposition 3 Let A = (a ij ) ∈ R n×n and B = (b ij ) ∈ R n×n be all DZ-B matrices of the form
So, both (B + j ) and (B + l ) are DZ matrices with positive diagonal elements, and B + D is also a DZ matrix with positive diagonal entries by Proposition 1. Therefore, B + D is a nonsingular matrix, and that is, where the last equality holds because (I + (B + D ) -1 C D ) -1 ∞ ≤ (n -1), see Theorem 2.2 in [9].
In fact, since (B + j ) , (B + l ) , and B + D are all DZ Z-matrices with positive diagonal elements, by Theorem 5, we have that , .
Further, we get Examples 3 and 4 show that the bound in Theorem 7 is sharper than that in Theorems 3 and 4 in some cases.

Conclusions
In this paper, we present global error bounds for the extended vertical linear complementarity problems of DZ matrices and DZ-B matrices. These bounds are expressed in terms of elements of the matrices, so they can be checked easily. Numerical examples show the feasibility of new results. Finding computable global error bounds for the extended vertical linear complementarity problems of other matrices (S-SOB matrices, S-SOB-B matrices, weakly chained diagonally dominant B-matrices, SB-matrices, etc.) under some additional conditions is an interesting problem. It is worth studying in the future.