GLOBAL ERROR BOUNDS FOR THE TENSOR COMPLEMENTARITY PROBLEM WITH A P -TENSOR

. As a natural extension of the linear complementarity problem, the tensor complementarity problem has been studied recently; and many theoret- ical results have been obtained. In this paper, we investigate the global error bound for the tensor complementarity problem with a P -tensor. We give two global error bounds for this class of complementarity problems with the help of two positively homogeneous operators deﬁned by a P -tensor. When the order of the involved tensor reduces to 2, the results obtained in this paper coincide exactly with the one for the linear complementarity problem.

1. Introduction. Given M ∈ R n×n and q ∈ R n , the linear complementarity problem [7], denoted by the LCP(M, q), is to find a vector x ∈ R n such that x ≥ 0, M x + q ≥ 0, and x (M x + q) = 0.
The LCP (M, q) has been studied extensively due to its wide applications in bimatrix game, the contact problems, the free boundary problem for journal bearing, the network equilibrium problem, etc. [2,7,22].
For the LCP (M, q), one of important issues is to study the related error bound, which is an inequality that bounds the distance from vectors in a test set T ⊆ R n to the solution set of the LCP (M, q), denoted by S, in terms of some residual function. Recalled that a nonnegative valued function r : S T → R + is said to be a residual function for the LCP (M, q) if it satisfies the property that r(x) = 0 if and only if x ∈ S. An error bound for the LCP (M, q) in term of r is a pair of inequalities of the form for some positive constants c 1 , c 2 , γ 1 and γ 2 , where r 1 and r 2 are two residual functions for the LCP (M, q), and dist(x, S) is the distance from the vector x to the set S. If T = R n , then (1) is called a global error bound for the LCP (M, q).
The error bound has been studied extensively for the LCP (M, q). For example, in 1990, Mathias and Pang [21] established error bounds for the LCP(M, q) with a P -matrix; while Luo, Mangasarian, Ren, Solodov [19] discussed error bounds for the LCP(M, q) with a nondegenerate matrix. Then, perturbation bounds of the LCP(M, q) with a P -matrix and the computation of those error bounds were proposed by Chen and Xiang [5,6]. The error bounds for the LCP(M, q) with an Hmatrix were given by Li and Zheng [18]. Recently, a series of literatures studied the error bounds for the LCP(M, q) with B-type matrices, including the M B-matrix by Chen, Li, Wu, Vong [4], the DB-matrix by Dai [8], the SB-matrix by Dai, Li, Lu [9], and the B-matrix by García-Esnaola and Peña [12], Li, Gan and Yang [17], and Gao and Li [11].
Given a vector q ∈ R n and an mth-order n-dimensional tensor A = (a i1i2···im ) with a i1i2···im ∈ R for all i j ∈ {1, 2, · · · , n} and j ∈ {1, 2, · · · , m}, the tensor complementarity problem, denoted by the TCP(A , q), is to find a vector x ∈ R n such that This class of complementarity problems was used firstly by Song and Qi [24]. It is obvious that the TCP(A , q) is a nature extension of the LCP (M, q).
By using special properties of several classes of structured tensors [23], various properties of the solution set of the TCP(A , q) have been well studied, including the nonempty compactness of the solution set by Che, Qi and Wei [3], Song and Qi [25,26], Song and Yu [28], Gowda, Luo, Qi and Xiu [13], Luo, Qi and Xiu [20], and Wang, Huang and Bai [29]; the existence of solution by Huang, Suo and Wang [16], and Song and Qi [27]; the global uniqueness and solvability by Bai, Huang and Wang [1]; and the topological properties of the solution set and stability of the TCP(A , q) by Yu, Ling, He and Qi [30]. More recently, the strict feasibility of the TCP(A , q) was discussed with the help of S-tensor by Guo, Zheng and Huang [14]. In particular, Song and Qi [26] and Song and Yu [28] gave estimations of upper and lower bounds of the solution set of the TCP(A , q) with A being a P -tensor and a strictly semi-positive tensor, respectively. In addition, some applications of the TCP(A , q) were also given (see, for example, [15]).
Since the TCP(A , q) is a generalization of the LCP(M, q) and the error bound for the LCP(M, q) has been studied extensively, a nature question is whether can we extend some results of the error bound for the LCP(M, q) to the TCP(A , q) or not? We will give some answers to this question in this paper. We will give two results of the global error bound for the TCP(A , q) with A being a P -tensor. After recalling some basic concepts and related results in Section 2, we give our main results in Section 3. The final conclusions are given in Section 4.
Throughout this paper, we assume that m and n are two positive integers with m ≥ 3 and n ≥ 2 unless otherwise stated; and use T m,n to denote the set of all real mth-order n-dimensional tensors. For any positive integer n, we denote [n] := {1, 2, · · · , n}.

2.
Preliminaries. In this section, we recall some basic definitions and related facts, which are useful for our later discussions. Definition 2.1. [24] A tensor A ∈ T m,n is said to be a P -tensor if and only if for each x ∈ R n \ {0}, there exists an index i ∈ [n] such that Yuan and You [31] obtained the following property for P -tensor.
There does not exist an odd order P -tensor.
Bai, Huang and Wang showed the following result in [1].
For any given q ∈ R n and a P -tensor A ∈ T m,n , the solution set of the TCP(A , q) is nonempty and compact.
Definition 2.4. Let mapping F : K ⊆ R n → R n . Then, F is said to be a P -function if for all pairs of distinct vectors x and y in K, Clearly, A is a P -tensor if the mapping A x m−1 + q with any given q ∈ R n is a P -function.
Then, A is said to be a strong P -tensor if F (x) = A x m−1 + q for any given q ∈ R n is a P -function.
It is easy to check from Definitions 2.1 and 2.5 that each strong P -tensor is a P -tensor.
Lemma 2.6. [1] Suppose that A ∈ T m,n is a strong P -tensor, then for any q ∈ R n , the TCP(A ,q) has a unique solution. Similarly, for any A ∈ T m,n , Recall that an operator T : R n → R n is called positively homogeneous if and only if T (tx) = tT (x) for each t > 0 and all x ∈ R n . Song and Qi defined two positively homogeneous operators in [24]: • Let A ∈ T m,n . For any x ∈ R n , define an operator T A : R n → R n by • When m is even, for any x ∈ R n , define another operator F A : R n → R n by where

MENGMENG ZHENG, YING ZHANG AND ZHENG-HAI HUANG
In [21], Mathias and Pang defined an important quantity for a P matrix M : As its generalizations, Song and Qi [24] introduced two quantities: for any positive integer m; and when m is even. From Lemma 2.2, P -tensors are all even order, then α(T A ) and α(F A ) are both well defined for any P -tensor. Song and Qi obtained two necessary and sufficient conditions for a P -tensor in terms of α(F A ) and α(T A ).
Since every strong P -tensor is a P -tensor, the following corollary is obvious.
In [21], Mathias and Pang obtained the following global error bound for the LCP(M, q).
Theorem 2.8. Let M be an n × n P -matrix. Letx denote the unique solution of the LCP(M, q) and u be an arbitrary n-vector. Then, In the next section, we extend such a result to the TCP(A , q).
3. Error bounds for the TCP(A , q). In this section, we investigate the global error bound for the TCP(A , q). From Lemma 2.3, it follows that the solution set of the TCP(A , q), denoted by S, is nonempty and compact when A ∈ T m,n is a P -tensor. Therefore, for any u ∈ R n , there exists a vectorx ∈ S such that where dist(u, S) := min Then, we have the following result.

ERROR BOUNDS FOR TENSOR COMPLEMENTARITY PROBLEMS 937
Theorem 3.1. Given q ∈ R n and A ∈ T m,n with A being a P -tensor. For any u ∈ R n , letx be given by (7). Suppose that rx(·) is defined by (8). Then, Proof. For simplicity, we use v to replace vx(u) in the following. It is obvious from We first show sufficiency. Sincex is a solution to the TCP(A , q), it follows that min{x, Ax m−1 + q} = 0, which means that This is equivalent tô This, together with (9) and the condition that u =x, implies that v = 0. Next, we show necessity. Since v = 0, it follows from (9) that We divide the proof into the following two cases: Sincex is a solution of the TCP(A , q), then Sincex is a solution of the TCP(A , q), then In summary, we obtain that However, since A is a P -tensor, it follows that for any x ∈ R n \ {0}, there exists which contradicts (10).
Sincex is a solution to the TCP(A , q), Theorem 3.1 demonstrates that rx(·) is a residual function for the TCP (A , q). In the following, we give a global error bound for the TCP(A , q) in terms of the residual function rx(·) defined by (8) and the quantity α(F A ) defined by (5).
Theorem 3.2. Given q ∈ R n , A ∈ T m,n with A being a P -tensor and α(F A ) is defined by (5). For any u ∈ R n , letx be given by (7). Suppose that rx(·) is defined by (8). Then, for any u ∈ R n , Proof. For simplicity, we use v to denote vx(u) in the following. Then, we need to show that We divide the proof into the following two parts. Part 1. We show that the inequality on the right-hand side of (11) holds.
Sincex is a solution to the TCP(A , q), it is easy to see that ω ≥ 0,x ≥ 0,x ω = 0.
Thus, for any i ∈ [n], which implies that for any i ∈ [n], In addition, it follows from the definition of α(F A ) (i.e., (5)) that Then, Therefore, combining (13) and (14), we have that Furthermore, since A is a P -tensor, it follows from Lemma 2.7 that α(F A ) > 0. Thus, from (15) we obtain that Part 2. We show that the inequality on the left-hand side of (11) holds. We consider the following two cases: (C1) Suppose that v i > 0 for an arbitrarily given i ∈ [n]. Then, Sincex is a solution of the TCP(A , q), it follows thatx i = 0 or (Ax m−1 +q) i = 0.

MENGMENG ZHENG, YING ZHANG AND ZHENG-HAI HUANG
Thus, we conclude that (C2) Suppose that v i ≤ 0 for an arbitrarily given i ∈ [n]. Then, Sincex is a solution of the TCP(A , q), it follows thatx i ≥ 0 and ( Thus, it follows from (17) and (18) that Now, by Part 1 and Part 2 (i.e., combining (16) and (19)) we obtain that (11) holds, which completes the proof.
From Lemma 2.6 we know that the TCP(A , q) has a unique solution if the involved tensor A is a strong P tensor. Since every strong P tensor is a P -tensor, from Theorem 3.2 we can obtain the following result immediately. Corollary 2. Given q ∈ R n and A ∈ T m,n with A being a strong P -tensor. Let α(F A ) be defined by (5). For any u ∈ R n , letx be the unique solution of the T CP (A , q). Suppose that rx(·) is defined by (8). Then , for any u ∈ R n , Up to now, we have obtained a global error bound for the T CP (A , q) with the help of the quantity α(F A ). In the following, we investigate the global error bound for the T CP (A , q) with the help of the quantity α(T A ). For this purpose, we define the following functions: and the quantity α(T A ) is defined by (4). Similar to the proof of Theorem 3.1, we can obtain the following result.
Theorem 3.3. Given q ∈ R n and A ∈ T m,n with A being a P -tensor. For any u ∈ R n , letx be given by (7). Suppose that r 1 x (·) and r 2 x (·) are defined by (20). Then, r 1 x (u) = 0 ⇐⇒ u =x ⇐⇒ r 2 x (u) = 0. Theorem 3.3 demonstrates that both r 1 x (·) and r 2 x (·) are residual functions for the T CP (A , q). In the following, by using these two residual functions, we investigate the global error bound for the T CP (A , q).
Theorem 3.4. Given q ∈ R n and A ∈ T m,n with A being a P -tensor. For any u ∈ R n , letx be given by (7). Suppose that r 1 x (·) and r 2 x (·) are defined by (20). Then, for any u ∈ R n , x (u). Proof. For simplicity, we use v to replace v x (u) in the following. It is obvious that we only need to show that where v := min u, A (u −x) m−1 + Ax m−1 + q for any u ∈ R n . We first show that (22) holds.
From Theorem 3.4 and Lemma 2.6 we can obtain the following result immediately.
Corollary 3. Given q ∈ R n and A ∈ T m,n with A being a strong P -tensor. Letx be the unique solution to the TCP(A , q). Suppose that r 1 x (·) and r 2 x (·) are defined by (20). Then, for any given u ∈ R n , r 1 x (u) ≤ ||u −x|| ∞ ≤ r 2 x (u). We have obtained two global error bound results for the TCP(A , q). What is the relationship between the obtained results and some known results for the LCP(M, q)? Remark 1. When m = 2, the tensor A ∈ T m,n reduces to a matrix, denoted by M . Then, for any x ∈ R n , we have that Similarly, when m = 2, the result obtained in Theorem 3.4 also reduces to the one in Theorem 2.8.

4.
Conclusions. In this paper, we introduced three residual functions for the TCP(A , q), by which we obtained two results of the global error bound for the TCP(A , q) with a P -tensor in terms of two quantities α(F A ) and α(T A ). The results we obtained reduce to the one achieved by Mathias and Pang [21] when the TCP(A , q) reduces to the LCP(M, q).
It is well known that the error bound has important application in iterative methods for solving the related optimization problem. How to apply the obtained error bounds to the convergence analysis of iterative methods for solving the TCP(A , q)? This is a further issue to be studied.