New global error bound for extended linear complementarity problems

For the extended linear complementarity problem (ELCP), by virtue of a new residual function, we establish a new type of global error bound under weaker conditions. Based on this, we respectively obtain new global error bounds for the vertical linear complementarity problem and the mixed linear complementarity problem. The obtained results presented in this paper supplement some recent corresponding results in the sense that they can provide some error bounds for a more general ELCP. Their feasibility is verified by some numerical experiments.


Introduction
Consider the extended linear complementarity problem (ELCP) of finding vector (x, y) ∈ R n × R n such that Mx -Ny ∈ K, x ≥ 0, y ≥ 0, x y = 0, where M, N ∈ R m×n and K = {Qz + q|z ∈ R l } with Q ∈ R m×l , q ∈ R m . The solution set of the ELCP is denoted by X * which is assumed to be nonempty throughout this paper.
The ELCP finds applications in various domains, such as engineering, economics, finance, and robust optimizations [1,2]. It was first considered by Mangasarian and Pang [1] and was further considered by Gowda [3] and Xiu et al. [4]. For more details on its development, see [5] and the references therein. It is well known that the global error bound plays an important role in theoretical analysis and numerical treatment of optimization problems such as variational inequalities and nonlinear complementarity problems [5][6][7][8][9][10][11][12][13][14]. The global error bound for the classical linear complementarity problems is well studied (see, e.g., [6,[15][16][17][18][19]). For a class of generalized linear complementarity problems, the global error bound was fully analyzed in [20][21][22]. Zhang and Xiu [4] presented an error bound for the ELCP with the column monotonicity and for the R 0 -ELCP. In this paper, we give a further consideration on this issue by establishing a global error bound estimation for the ELCP under a milder condition motivated by the work in [4].

Results and discussion
Here we are concerned with the global error bound on the distance between a given point in R 2n and the solution set of the ELCP in terms of some residual functions. This paper is a follow-up to [4], as in this paper we establish a new global error bound for the ELCP under weaker conditions than those used in [4].
Some error bounds for the ELCP have been presented in [4], and they hold under some stringent condition, that is, the underlying matrices M, N satisfy the column monotonicity with respect to K or R 0 -property. Furthermore, we can only get the error bound of any points in the set = {(x, y) ∈ R 2n |Mx -Ny ∈ K} by the results in [4]. Then the following two questions are posed naturally: Can the conditions imposed on the matrices M, N in [4] be relaxed or removed? How about the global error bound estimation in R 2n for the ELCP? These constitute the main topics of this paper. In this paper, we shall deal with the two issues. In fact, based on some equivalent reformulation of the ELCP and using a new type residual function, we present a global error bound for the ELCP in R 2n under a milder condition, and the requirement of the column monotonicity, or R 0 -property, or non-degenerate solution, and so on is removed here. Furthermore, the global error bounds for the vertical linear complementarity problem (VLCP) and the mixed linear complementarity problem (MLCP) are also discussed in detail.

Methods and notations
The aim of this study is to design a new global error bound for the ELCP. More specifically, the ELCP is firstly converted into an equivalent extended complementarity problem, which eliminates the variable z in the ELCP. Then, we define a residual function of the transformed problem, based on which we derive some new error bounds for the transformed problem and the original ELCP. Furthermore, we deduce some global error bounds for the two special cases of the ELCP: VLCP and MLCP. Note that the obtained results can be viewed as some supplements to the results in [4].
We adopt the following notations throughout the paper. All vectors are column vectors and the superscript T denotes the transpose. The x + denotes the orthogonal projection of vector x ∈ R n onto R n + , that is, (x + ) i := max{x i , 0}, 1 ≤ i ≤ n; the norm · and · 1 denote the Euclidean 2-norm and 1-norm, respectively. For x, y ∈ R n , use (x; y) to denote the column vector (x , y ) , and min{x, y} means the componentwise minimum of x and y.
We use I m to denote an identity matrix of order m, use D + to denote the pseudo-inverse of matrix D, use diag(a 1 , a 2 , . . . , a n ) to denote the diagonal matrix with elements a 1 , a 2 , . . . , a n .
For any n × n real matrix A, we denote by A the transpose of A, by A the matrix norm of A, that is, A := max(λ(A A)) 1 2 , where λ(A A) is an eigenvalue of the matrix A A, denote a nonnegative vector x ∈ R n by x ≥ 0, denote an absolute value of the real number a by |a|, and use C k n to denote the combinatorial number, which is the number of combinations when k elements are arbitrarily taken from n elements. We denote the empty set by ∅.

List of abbreviations
In this section, we give the following tabular for abbreviations used in this paper (see Table 1).

Global error bound for ELCP
In this section, we first present an equivalent reformulation of the ELCP in which parameter z is not involved and then establish a global error bound for the ELCP under weaker conditions. From the definition of the ELCP, the following result is straightforward.
Let w = (x; y) and U = QQ + -I m . Using the fact that x = A + b is a solution to the linear equation Ax = b if it is consistent, we conclude that the last equation in Proposition 5.1 is equivalent to Define block matrices A = (I n , 0 n ), B = (0 n , I n ). Then the ELCP can be equivalently reformulated as the following extended complementarity problem w.r.t. w: We denote its solution set by W * , and let Then it holds that {w ∈ R 2n |f (w) = 0} = W * .
From the definition of f (w), a direct computation yields that Then, by the definition of σ i , we can get that the cardinality of the set H is Applying the related theory of linear algebra and (5.4), we give the following result for our analysis.
Proof Since the matrixQ is symmetric, there exists an orthogonal matrix C 1 ∈ R 2n×2n such that Then Then there exists vector p 2 such that v = ξp 2 . This and yields the desired result as follows.
In the following, we present our main error bound result for the ELCP.
Theorem 5.1 Suppose that W * = {w ∈ R 2n |f (w) = 0} is nonempty. Then there exists a constant ρ 0 > 0 such that, for any w ∈ R 2n , there existsw ∈ W * satisfying Proof Applying Lemma 5.1, there exists an affine transformation w = C 1 v + p 1 such that where a i > 0, c j , τ ∈ R, i ∈ I + ∪ I -, j ∈ J are defined in (5.9), I + , I -, and J are respectively non-overlapping subsets of {1, 2, . . . , 2n}, and C 1 is an orthogonal matrix. Therefore, given Now, we break the discussion into three cases.
Case 1. If J = ∅, we take an index j ∈ J such that |c j | = max{|c i ||i ∈ J} > 0, and letv with entries Then where the second equality follows from the fact that C 1 is an orthogonal matrix. Since the cardinality of the set H is 2 2n , then |c min | = min{|c j ||j = 1, 2, . . . , 2 2n }, so |c min | is independent of selection of the matrix M 2 , i.e., |c min | is independent of selection of vector w. Thus, the last inequality holds. The desired result is proved for this case.
In addition, based on the definition ofz and θ , we get where the first inequality comes from the fact that a 1 2 , ∀a, b ∈ R + , the second inequality uses assumption h(z) > 0 and the fact that h(z) ≤ i∈I + z 2 i + i∈Ĩ -z 2 i . Using (5.12), (5.16), and (5.17), one has where the fourth equality comes from the fact that C 1 is an orthogonal matrix, the last inequality uses the fact that C -1 2 ≤ max{1, 1 √σ }, andσ > 0 is a constant.
In fact, using the related theory of linear algebra, by the definition of the matrixQ, one has λ i = λ Thus, we deduce thatσ is independent of selection of the matrix M 1 . The desired result follows for this case. Case 3. If J = ∅ and τ > 0. Then it follows from and (5.10) that Letf (w) = -f (w),ĝ(v) = -g(v),Î + = I -,Î -= I + ,τ = -τ < 0. Then Considering this together with (5.19), using a similar argument to that of Case 2 above, there exist constantσ > 0 andw ∈ W * such that The desired result follows for this case. Now, we give another error bound for the ELCP.

Theorem 5.2
Suppose that X * is nonempty. Then, for any (x; y) ∈ R 2n , there exists a constant ρ 1 > 0 such that Proof By Theorem 5.1, one has where the first equality is by (5.3), the second and third inequalities follow from the fact that and the fourth inequality is obtained by the fact (-a) + ≤ | min{a, b}| for any a, b ∈ R. By the definitions of A, B, w, and setting ρ 1 = 2ρ 0 , we obtain the desired result.
Remark 5.1 Obviously, the condition needed in Theorem 5.2 in this paper is strictly weaker than that needed in [4]. The requirements of column monotonicity, R 0 -property, and rank Q = l are removed here. In addition, we establish this global error bound in R 2n rather than that in which is defined in [4]. On the other hand, we also present the following examples to compare.
On the other hand, from (5.20), we see that s(x k , y k ) = k -4 for the point (x k ; y k ). Then from (5.21), it follows that as k → ∞. Thus, the function s(x, y) + s 1 2 (x, y) and s(x, y) cannot provide an error bound for the point (x k ; y k ).

Global error bound for special cases of ELCP
In this section, we respectively establish the global error bound of the VLCP and the MLCP based on Theorem 5.2.

Global error bound for VLCP
Consider the VLCP of finding vector x ∈ R n such that Denote its solution set byX * . Certainly, the VLCP is a special case of the ELCP with where A, B ∈ R m×n , a, b ∈ R m , and A = 0 or B = 0. Applying Theorem 5.2 to the VLCP, we have the following conclusion.

Global error bound for MLCP
Consider the MLCP of finding vector (x, z) ∈ R n × R m such that where A ∈ R l×n , B ∈ R l×m , C ∈ R n×n , D ∈ R n×m , a ∈ R l , b ∈ R n . Denote the solution set byX * . Let y = Cx + Dz + b. Then system (6.5) can be rewritten as Certainly, the MLCP is a special case of the ELCP with From Theorem 5.2, we have the following conclusion.

Conclusions and remarks
In this paper, we established some new global error bounds for the VLCP and the MLCP based on the global error bound for the ELCP. These global error bounds extend some known results in the literature, which is verified by a numerical comparison.
As the error bound analysis has important applications in the sensitivity analysis and error bound estimation for optimization methods, it would be interesting to investigate whether our new error bound results will give effective global error estimates for some particular methods in solving a non-monotone ELCP (VLCP and MLCP) that does not require any non-degeneracy assumption, such as the Newton-type with quick convergence rate. These will be further considered in the future research.