New error bounds for the tensor complementarity problem

1.   Introduction

The set of all real $r$th-order $n$-dimensional tensors is denoted by $\mathbb{T}_{r, n}$, where $r\geq{3}$ and $n\geq{2}$ are positive integers. A tensor $\mathscr{A}\in\mathbb{T}_{r, n}$ is called a $P$-tensor ^[1], if for each vector $y \in \mathbb{R}^{n}\setminus\{0\}$, there exists an index $j\in J_{n}$ such that

where $J_{n}$ defines the index set $\{1, 2, ..., n\}$. For a given vector $q\in\mathbb{R}^{n}$ and a tensor $\mathscr{A} = (a_{j_{1}...j_{r}})\in\mathbb{T}_{r, n}$ with a multi-array of real entries $a_{j_{1}...j_{r}}$, where $j_{i}\in J_{n}$ for $i\in\{1, ..., r\}$, the tensor complementarity problem denoted by the TCP$(\mathscr{A}, q)$, is to find a real vector $y \in \mathbb{R}^{n}$ such that

where $\mathscr{A}y^{r-1}\in\mathbb{R}^{n}$ and the $j$th element of $\mathscr{A}y^{r-1}$ is determined by

for $j\in J_{n}$. When the tensor $\mathscr{A}$ is a matrix, the TCP$(\mathscr{A}, q)$ reduces to the linear complementarity problem ^[2], denoted by the LCP$(M, q)$, which is to find a real vector $y\in\mathbb{R}^{n}$ such that

where $M\in\mathbb{R}^{n\times n}$ and $q\in\mathbb{R}^{n}$. Error bounds of the LCP$(M, q)$ have been extensively studied in recent years. For example, Mathias and Pang in ^[3] introduced fundamental quantities associated with an arbitrary $P$-matrix and derived global upper and lower error bounds for the approximate solution of the LCP$(M, q)$ with a $P$-matrix. Global upper and lower error bounds to the LCP$(M, q)$ with a $P$-matrix were given in ^[2] by Cottle et al. We refer to ^[4,5,6,7] for more new results on the estimation of error bounds to the LCP$(M, q)$ with a $P$-matrix. We denote the LCP$(M, q)$ with a $P$-matrix by the LCP$(M, q; p)$ and the TCP$(\mathscr{A}, q)$ with a $P$-tensor by the TCP$(\mathscr{A}, q; p)$.

In recent years, the properties of several types of structured tensors in ^[8,9] were used to investigate some properties about the TCP$(\mathscr{A}, q)$ solution set. The nonempty of the solution set of the TCP$(\mathscr{A}, q)$ and the compactness of the solution set of the TCP$(\mathscr{A}, q)$ were explored in ^[10,11] by Che et al. Huang et al. ^[12,13] proved the existence of the solution of the TCP$(\mathscr{A}, q)$. Yu et al. ^[14] evaluated the stability features of the TCP$(\mathscr{A}, q)$ solution set. Zheng et al. ^[15] extended the results in ^[3] to the tensors and presented two TCP$(\mathscr{A}, q; p)$ global error bounds. In addition, some applications of the TCP$(\mathscr{A}, q)$ were given in ^[16]. In this paper, we mainly expand the error bounds of the approximate solution of the LCP$(M, q; p)$ in ^[2,7] to the cases of the TCP$(\mathscr{A}, q; p)$. We also extend the bound of the norm of the exact solution of the LCP$(M, q; p)$ in ^[7] to the case of the TCP$(\mathscr{A}, q; p)$.

The rest of this paper is arranged as follows. Section 2 presents some fundamental concepts and summarizes some related results that will be used later. Some new error bounds of the TCP$(\mathscr{A}, q; p)$ are given in Section 3. Section 4 drawn some conclusions.

2.   Preliminaries

In this section, we summarize some results and introduce some denotations that will be used later. The following results are true for a $P$-tensor.

Lemma 2.1. (^[17]) For any $P$-tensor $\mathscr{A}\in\mathbb{T}_{r, n}$ and any $q\in \mathbb{R}^{n}$, the solution set of the TCP$(\mathscr{A}, q)$ is nonempty and compact.

Lemma 2.2. (^[15,18]) There does not exist an odd order $P$-tensor.

For any tensor $\mathscr{A}\in\mathbb{T}_{r, n}$, the infinite norm of $\mathscr{A}$ is defined as follows:

Let $T: \mathbb{R}^{n}\to \mathbb{R}^{n}$ be an operator, $T$ is said to be positively homogeneous iff $T(ty) = tT(y)$ holds, for any positive $t$, $y$ $\in$ $\mathbb{R}^{n}$. Song and Qi in ^[1] introduced two quantities of the form

for an arbitrary positive even integer $r$, and

Here $T_{\mathscr{A}}: \mathbb{R}^{n}\to \mathbb{R}^{n}$ and $F_{\mathscr{A}}: \mathbb{R}^{n}\to \mathbb{R}^{n}$ are two operators that are positively homogenous and defined by

and

respectively. Here $y^{[\frac{1}{r-1}]} = (y_{1}^{\frac{1}{r-1}}, y_{2}^{\frac{1}{r-1}}, ..., y_{n}^{\frac{1}{r-1}})^{T}$ for $y = (y_{1}, y_{2}, ..., y_{n})^{T}$. It is true that

The following result in ^[1] gives two criterions of a $P$-tensor based on $\alpha{(F_{\mathscr{A}})}$ and $\alpha{(T_{\mathscr{A}})}$.

Lemma 2.3. (^[1]) Let $\mathscr{A} \in\mathbb{T}_{r, n}$, $\mathscr{A}$ is a $P$-tensor iff $\alpha{(T_{\mathscr{A}})} > 0$. When $r$ is even, it is especially true that $\mathscr{A}$ is a $P$-tensor iff $\alpha{(F_{\mathscr{A}})} > 0$.

3.   New error bounds to the TCP$(\mathscr{A}, q)$

In this section, new error bounds of the approximate solution of (1.1) are proposed. Denote the solution set of (1.1) by the set of the form

then according to Lemma 2.1, $S$ is nonempty and compact when $\mathscr{A} \in\mathbb{T}_{r, n}$ is a $P$-tensor. Thus for any $v\in\mathbb{R}^n$, there is a vector $\tilde{y}\in S$ such that

We also need the results as follows.

Lemma 3.1. For an arbitrary positive integer $r$, it is true that

Proof. In fact

This completes the proof.

Theorem 3.1. Suppose $q \in \mathbb{R}^{n}$, $\mathscr{A}\in \mathbb{T}_{r, n}$ is a $P$-tensor. Let $\tilde{y}$ be the unique solution of the TCP$(\mathscr{A}, q)$. We have

where $\tilde{v}_{\tilde{y}}$ is defined by

Proof. First show that for any $v, w\in\mathbb{R}^{n}$, the following inequality holds,

where $\tilde{v}_{\tilde{y}}$ is defined in (3.5) and $\tilde{w}_{\tilde{y}}$ is determined by (3.5) with $v$ being replaced by $w$. Denote $y = \tilde{v}_{\tilde{y}}$ and $x = \tilde{w}_{\tilde{y}}$, suppose that $\mid{y_{j}-x_{j}}\mid = y_{j}-x_{j}, {\forall}j\in J_{n}$, then we have

Thus,

In the following, we prove this result in two cases. If $x_{j} = w_{j}$, then

Therefore

If $x_{j} = (\mathscr{A}(w-\tilde{y})^{r-1})_{j}^{\frac{1}{r-1}}+(q+\mathscr{A}\tilde{y}^{r-1})_{j}^{\frac{1}{r-1}}$, then

It follows from Lemma 3.1, that

Thus it holds that

Combining (3.7) and (3.8) results in (3.6). Let $w = \tilde{y}$ in (3.6), then

Given that $\tilde{y}$ solves the TCP$(\mathscr{A}, q)$, then we have

It follows from (3.9) that

which implies (3.4). This completes the proof.

We remark that a lower error bound (Theorem 3.1) for the TCP$(\mathscr{A}, q)$ with a $P$-tensor in this paper is sharper than the result (Theorem $3.2$) in ^[15]. Because

The following result gives a global error bound on the approximate solution of (1.1).

Theorem 3.2. Suppose $\mathscr{A}\in \mathbb{T}_{r, n}$ is a $P$-tensor. Let $\tilde{y}$ be the unique solution of the TCP$(\mathscr{A}, q)$. We have

where

with $k_{1}$ satisfying $(\tilde{v}_{\tilde{y}})_{k_{1}}\neq{0}$,

and $\alpha{(F_{\mathscr{A}})}$ is defined in (2.2).

Proof. We first show that the following inequality holds

Suppose $w = (q+\mathscr{A}\tilde{y}^{r-1})^{[\frac{1}{r-1}]}$. Since $\tilde{y}$ solves (1.1), we can prove that

Let $x = v-\tilde{v}_{\tilde{y}}$ and $z = (\mathscr{A}(v-\tilde{y})^{r-1})^{[\frac{1}{r-1}]}+(q+\mathscr{A}\tilde{y}^{r-1})^{[\frac{1}{r-1}]}-\tilde{v}_{\tilde{y}}$, where $\tilde{v}_{\tilde{y}}$ is defined as that in (3.5), then we have that

Therefore, for any $j\in J_{n}$,

Thus

Let

then

From Lemma 3.1, we have that

Furthermore, we can deduce from (2.5) that

Therefore, combining (3.12)–(3.14) results in

If $(\tilde{v}_{\tilde{y}})_{k_{1}} = 0$, then $v = \tilde{y}$; otherwise, (3.11) holds. According to Lemma 2.3, we have $\alpha{(F_{\mathscr{A}})} > 0$, here $\mathscr{A}$ is a $P$-tensor. We can derive (3.10) by solving (3.11). The proof is completed.

If we set $v = 0$ in Theorem 3.2, we obtain a new bound for the norm of the exact solution of (1.1).

Corollary 1. Let $v = 0$, then under the conditions of Theorem 3.2, we have that

where

with $k_{2}$ satisfying

and $\alpha{(F_{\mathscr{A}})}$ is defined in (2.2).

Proof. The proof is similar to that of Theorem 3.2 and is omitted.

We have obtained a new lower error bound in Theorem 3.1 and a global error bound in Theorem 3.2 to the TCP$(\mathscr{A}, q; p)$.

We remark that when $r = 2$, Theorem 3.2 and Corollary 1 reduce to Theorem $3.1$ in ^[7] and Corollary $3.1$ in ^[7], respectively.

When $r = 2$, the tensor $\mathscr{A}\in \mathbb{T}_{r, n}$ reduces to a matrix $M$. Thus, $F_{\mathscr{A}}y: = (\mathscr{A}y^{r-1})^{[\frac{1}{r-1}]} = My$, we have

Furthermore, we can deduce from (3.5) that

which implies that

with $k_{1}$ satisfying $u_{k_{1}}\neq{0}$ and

and that

with $k_{2}$ satisfying $((-q)_{+})_{k_{2}}\neq{0}$ and

Therefore, Theorem 3.1 reduces to Theorem $5.10.6$ in ^[2].

Similarly, when $r = 2$, Theorem 3.2 is an extension of Theorem $3.1$ in ^[7], and Corollary 1 is an extension of Corollary $3.1$ in ^[7].

4.   Conclusions

In this paper, we present a new lower error bound and a result of the global error bound on the approximate solution to the TCP$(\mathscr{A}, q)$ with a $P$-tensor. The new bound on the norm of the exact solution to the TCP$(\mathscr{A}, q)$ with a $P$-tensor is derived.

As we all know, error bounds have important applications in iterative methods for solving related optimization problems. For example, we use the obtained error bounds to analyze the convergence of the iterative method for solving the TCP$(\mathscr{A}, q)$. We can futher study the numerical algorithm to compute the error bounds of the TCP$(\mathscr{A}, q)$ with a $P$-tensor (see ^[19]).

Acknowledgments

Research by G.X. Huang was supported in part by Key Laboratory of bridge nondestructive testing and engineering calculation Open fund projects (grant 2020QZJ03).

Conflict of interest

The authors declare there is no conflicts of interest.