The Generalized Horizontal Linear Complementarity Problem

Let M, N ∈ Rm×n,m ≥ n, be vertical block matrices and q ∈ Rm. We consider the generalized horizontal linear complementarity problem (GHLCP) associated with the pair {M,N}. Existence and uniqueness of solutions are characterized when the pair {M,N} satisfies new generalized column Wand column W0-properties.


Introduction
Given the vertical block matrices M and N of type (m 1 , m 2 , ..., m n ), the Generalized Horizontal Linear Complementarity Problem (GHLCP) is defined as follows: Find vectors w, z ∈ R n such that Mw − Nz = q w ≥ 0, z ≥ 0 w T z = 0 where M, N ∈ R m×n , m ≥ n, m = n i=1 m i , and q ∈ R m .We will denote this problem by GHLCP(M, N, q).Problem GHLCP(M, N, q) is related to the extended linear complementarity problem (LCP) studied by Mangasarian and Pang (1995); the extended horizontal linear complementarity problem (HLCP) studied by Sznajder and Gowda (1995), Schutter and Moor (1997); the generalized LCP studied by Ye (1993); and the generalized horizontal LCP of Xiu and Zhang (2001).Sznajder and Gowda (1995) extended the concept of W 0 -pair of Wilson (1971).They defined the column W 0 -and column W-properties for the set of (k + 1), k ≥ 1, square matrices, and gave some equivalent conditions for the column W 0 -and column W-properties.They also showed that the column W-properties are characterized by the unique solvability of the extended HLCP.
In Tütüncü and Todd (1995), properties of a square matrix pair are studied, when the associated matrices have the P-and P 0 -properties.They showed that a large class of matrix pairs including P 0 -pairs have the property of reducibility, and developed an algorithm that can be used to reduce an HLCP to a standard LCP.
Schutter and Moor (1997) considered matrices A, B ∈ R m×n .They provided an algorithm that can be used to solve the problem and demonstrated that the extended HLCP can be used to solve max-algebraic problems.In their work, Xiu and Zhang (2001) showed that the GHLCP(A, B, q), where A, B ∈ R m×n , m ≥ n, has a solution when the pair {A, B} satisfies the P o -and R 0 -properties.
In this paper, we generalize the concepts of column W 0 -property and column W-Property to the matrix pair {M, N}, where M, N ∈ R m×n are vertical block matrices of type {m 1 , . . ., m n } with m ≥ n.Using the column W 0 -and column W-Properties, existence and uniqueness of solutions are characterized for the GLCP(M, N, q).It has been pointed out in Xiu and Zhang (2001) that when m = n, the GHLCP(M, N, q) reduces to the classical horizontal linear complementarity problem HLCP(M, N, q); and when M = I m×m , the GHLCP(M, N, q) reduces to the vertical linear complementarity problem studied by Ebiefung (1999).
The HLCP and the extended HLCP have important applications.Eaves and Lemke (1981) showed that any piecewise linear system can be formulated as an HLCP.The extended HLCP has applications in structural mechanics, inventory theory, optimization, and statistics as reported in Kaneko (1978) and in Kaneko and Pang (1978).The extended LCP formulated by Schutter and Moor (1997) arose from their work in discrete event systems such as flexible manufacturing, subway traffic network, parallel processing systems, telecommunication systems, and logistic systems.
The rest of the paper is organized as follows: In Section 2, we provide preliminary definitions required for the development of the results.Existence and uniqueness of solutions are provided in Section 3, while concluding remarks are given in Section 4.

Preliminaries
The following definitions are needed for the rest of this paper.
Definition 1 (See Ebiefung, 1999) Let M be a vertical block matrix of type (m 1 , m 2 , . . ., m n ).An n × n submatrix of M is called a representative submatrix if its j-th row is drawn from the j-th block M j of M. A vertical block matrix M of type (m 1 , m 2 , . . ., m n ) is a P 0 (P) − matrix if each representative submatrix is a P 0 (P) Definition 3 (See Sznajder & Gowda, 1995) M has the column W 0 -property if one of the following holds: (i) The determinants of all column representative matrices of M are nonnegative and there is at least one such determinant which is positive.
(ii) The determinants of all column representative matrices of M are nonpositive and there is at least one such determinant which is negative.
Definition 4 (See Sznajder & Gowda, 1995) M has the column W-property if one of the following holds: (i) The determinants of all column representative matrices of M are positive.
(ii) The determinants of all column representative matrices of M are negative.
Definition 5 (See Ebiefung & Kostreva, 1993) Let U be a matrix of dimension n × m defined by: where for each j ( j = 1, 2, . . ., n) u j is a positive row vector of dimension 1 × m j .
In the next set of definitions, we consider the GHLCP(M, N, q) with M and N as vertical block matrices of dimension m × n and of type (m 1 , m 2 , . . ., m n ), where (m ≥ n).That is, M and N are partitioned row-wise into n blocks: where the j-th blocks M j and N j are of dimension m j × n and m = n j=1 m j .Definition 6 Column matrix: Let C = {M, N}, where M and N are vertical block matrices of order R m×n .The matrix R ∈ R m×n defined by is called a column representative matrix.The matrix R is also partitioned to conform with those of M and N.
Definition 7 Column W 0 -property: The pair C = {M, N} has the column W 0 -property if each column representative matrix is a vertical block P 0 -matrix.
Definition 8 Column W-property: The pair C = {M, N} has the column W-property if all column representative matrices are vertical block P-matrices.
Definition 9 Q-property: The pair C = {M, N} has the Q-property if the GHLCP(M, N, q) has a solution for each q ∈ R m .
Definition 10 Proper matrix: A matrix B defined by B = {B .j , j = 1, 2, . . ., n}, where B .j ∈ {M .j , −N .j }, is called a proper matrix of C = {M, N} if whenever (−N .j ) is a column of B, M .j is not a column of B.
Definition 11 Let B be a proper matrix of {M, N}.A vector x = (w K , z K ) ≥ 0 is a solution of the GHLCP(M, N, q) if Bx = q, where K ⊂ {1, 2, . . ., n} and K its complement such that z K ≥ 0, w K ≥ 0, z K = 0 and w K = 0.
Definition 12 A solution x = (w K , z K ) of the GHLCP(M, N, q) is said to be nondegenerate if it has no zero components.We call the GHLCP(M, N, q) nondegenerate if each solution is nondegenerate.
Definition 13 For each j ( j = 1, 2, . . ., n), let e j be an m j × 1 column vector with each component equal to 1.The matrix D defined by is a nonnegative vertical block matrix of type (m 1 , ..., m n ).We assume that D is partitioned to conform to those of M and N.
If M is a vertical block matrix, we assume that M + αD has the same partition as M for any α > 0. We now formulate the perturbed GHLCP(M(α), N(α), q): where M(α) = M + αD and N(α) = N + αD.Assume that a solution to Equation (2) exists for sufficiently small α > 0. Also assume that x(α) = (w(α) K , z(α) K ) is a solution of Equation ( 2) and that B(α) is the proper matrix corresponding to x(α).

Existence and Uniqueness of Solutions
The theory of existence and uniqueness of solutions for the GHLCP is presented in this section.Throughout the rest of this paper, the matrix U shall refer to the vertical block matrix defined in Definition 5, B the matrix defined in Definition 10, and D the matrix defined in Definition 13.
Theorem 1 If C = {M, N} has the column W 0 -property, then UC = {U M, UN} has the column W 0 -property.
Proof.Assume that C = {M, N} has the column W 0 -property.By definition, each column representative matrix is a vertical block P 0 -matrix.Let R * be a column representative matrix of UC.
where R is a column representative matrix of {M, N}.
Since R is a vertical block P 0 -matrix, R * is P 0 -matrix of order n, as proved by Ebiefung and Kostreva (1993).This is true for each column representative matrix of UC.Therefore, UC = {U M, UN} has the column W 0 -property.This completes the proof.
Theorem 2 If C = {M, N} has the column W-property, then UC = {U M, UN} has the column W-property.
Proof.The proof is similar to the proof of Theorem 1 and so it is omitted.
Theorem 3 Suppose that C = {M, N} has the column W-property.If the GHLCP(M, N, q) has a solution, then the solution is unique.
Now suppose that the GLCP(M, N, q) has two distinct solutions.By the above reasoning, these two solutions are also solutions to the HLCP(U M, UN, Uq).However, since C = {M, N} has the W-property, UC = {U M, UN}, where U M and UN are square matrices, has the W-property by Theorem 2. By Sznajder and Gowda (1995), the HLCP(U M, UN, Uq) has a unique solution.This contradicts the assumption that the GLCP(M, N, q) could have two or more distinct solutions.This completes the proof.
Proof.Assume that C = {M, N} has the column W 0 -property.Let R be a column representative matrix of C. The matrix {R + αD} is a vertical block P-matrix by Ebiefung and Kostreva (1993).This is true for each column representative matrix of C. Hence, C + αD = {M + αD, N + αD} has the column W-property for each sufficiently small α > 0. This completes the proof.
Theorem 5 Suppose that C = {M, N} has the column W 0 -property and that the GHLCP(M, N, q) has a solution with a proper matrix B, which is a vertical block P -matrix.Then there exists a sequence {x(α r )} of solutions of the perturbed GLCP(M(α), N(α), q) which converges to a solution of the GHLCP(M, N, q).
Proof.Suppose that the GHLCP(M, N, q) has a solution (w, z).Then Mw − Nz = q w ≥ 0, z ≥ 0 w T z = 0, implies that there exists a vector x ∈ R n such that Bx = q, where x = (w K , z K ), q ∈ R m , B ∈ R m×n a proper matrix of the pair {M, N}.Hence, where B = UB, q = Uq.Since B is a vertical block P-matrix, B is a P-matrix by Ebiefung and Kostreva (1993).
Theorem 6 Suppose that C = {M, N} has the column W 0 -property and that every proper matrix of C is a vertical block P 0 -matrix.If the GHLCP(M, N, q) has a nondegenerate solution with a proper matrix B, which is also a vertical block P-matrix, then the solution is unique.
Proof.Since the GHLCP(M, N, q) has a solution, there exists a proper matrix B and a vector x = (w K , z K ) such that Bx = q.As a result, B x = q where B = UB, q = Uq.Since the GHLCP(M, N, q) has a nondegenerate solution and B is a vertical block P-matrix, B is a P-matrix, and x = (B ) −1 q > 0 (11) For α > 0, let x(α) be a solution of the perturbed problem GLCP(M(α), N(α), q) and B(α) the corresponding proper matrix.Then where B (α) = UB + α(UD) = (B + αD ), and q = Uq.Since B is a P-matrix, B + αD is a P-matrix and for each α > 0.
Theorem 7 Suppose that C = {M, N} has the column W 0 -property.If the sequence {x(α r )} of solutions of the GLCP(M(α r ), N(α r ), q) is bounded as α r −→ 0, then the GHLCP(M, N, q) has a solution.
Proof.Since C = {M, N} has the column W 0 -property, (C + αD) has the column W-property.Assume that {x(α r )} = {(w(α r ) K , z(α r ) K )} is a bounded sequence of solutions of the GHLCP(M(α r ), N(α r ), q) and that {B(α r )} is the corresponding proper matrix solving the perturbed problem for each r ∈ {1, 2, . ..}.Then there exists a subsequence, say, {x(α r )} of {x(α r )} that is convergent.Assume that {x(α r )} converges to x.Let {B(α r )} be a subsequence of {B(α r )} corresponding to {x(α r )} and assume that {B(α r )} converges to B. It is easy to see that B is a proper matrix of {M, N }.