Novel Pareto $Z$-eigenvalue inclusion intervals for tensor eigenvalue complementarity problems and its applications

1.   Introduction

Consider the tensor eigenvalue complementarity problems of finding $(\lambda, x)\in \mathbb{R}\times \mathbb{R}^{n}_{+}\backslash\{0\}$ such that

where $a \bot b$ means that vectors $a, b$ are perpendicular to each other, and $\mathcal{A} = (a_{{i_1}{i_2}\ldots{i_m}})\in \mathbb{R}^{[m, n]}$ is an $m$-th order $n$-dimensional real tensor, and $\mathcal{A} x^{m-1}$ is the vector in $\mathbb{R}^n$ with entries

If (1.1) holds, $(\lambda, x)\in \mathbb{R}\times \mathbb{R}^{n}_{+}\backslash\{0\}$ is called a Pareto $Z$-eigenpair of tensor $\mathcal{A}.$ Pareto $Z$-eigenvalue problems of tensors were introduced by Song ^[17], which can be seen generalizations of classical tensor (matrix) eigenvalue problems ^{[1,5,6,8,13,14,19,20,21,22]}, have broad applications in higher-order Markov chains ^[11] and magnetic resonance imaging ^[15,26,27]. Therefore, Pareto $Z$-eigenvalue problems of tensors garnered a lot of interest in the literature ^[4,10,30,32]. To achieve Pareto $Z$-eigenvalues of tensor eigenvalue complementarity problems, for instance, Zeng ^[32] suggested a semidefinite relaxation approach. Nonetheless, there exist a huge, potentially endless number of Pareto $Z$-eigenvalues of tensors ^[2,32]. Therefore, calculating all Pareto $Z$-eigenvalues is difficult. A few scholars have turned to investigating Pareto $Z$-eigenvalue intervals to describe the distribution of Pareto $Z$-eigenvalues ^[17,29]. Particularly, Yang et al. ^[29] proposed Pareto $Z$-eigenvalue intervals via key tensor elements, which have a significant impact on the Pareto $Z$-eigenvalue estimation. It is crucial to create new Pareto $Z$-eigenvalue intervals that are independent of certain tensor constituents. Note that matrices can be viewed as the large elements of tensors, and the spectral radius has relative stability. Can we use the spectral radius of the related symmetric matrices instead of tensor elements to accurately characterize the Pareto $Z$-eigenvalue? Different from the existing $Z$-eigenvalue inclusion sets ^{[16,24,25,31]}, we investigate the relations between the tensor and its induced matrix and establish Pareto $Z$-eigenvalue intervals from the spectral radius of the linked symmetric matrices.

As we know, tensor $\mathcal{A}$ is strictly copositive if $\mathcal{A}x^{m} > 0, \forall x\in\mathbb{R}^{n}_{+}\backslash\{0\},$ which has important applications in vacuum stability of a general scalar potential ^[9] and polynomial optimization ^[3,12]. Song et al. ^[17] pointed out that symmetric tensor $\mathcal{A}$ is strictly copositive if and only if its Pareto $Z$-eigenvalues are positive. Therefore, we can identify whether a tensor is copositive by the lower bounds of Pareto $Z$-eigenvalues. Inspired by the articles ^[17,29], we propose some criteria for judging strict copositivity via the spectral radius of the symmetric matrices extracted from the given tensor.

The remainder of this paper is organized as follows: In Section 2, crucial definitions and preliminary results are recalled. In Section 3, we establish two tight Pareto $Z$-eigenvalue intervals via the spectral radius of the symmetric matrices. In Section 4, sufficient conditions are proposed for identifying strict copositivity of symmetric tensors.

2.   Preliminaries

In this section, we first introduce important definitions and notations of tensors ^[2,13,29].

The set of all real numbers is denoted by $\mathbb{R},$ and the $n$-dimensional real Euclidean space is denoted by $\mathbb{R}^{n}$. For any $a\in \mathbb{R}$, we denote $[a]_{+}: = \max\{0, a\}$ and $[a]_{-}: = \max\{0, -a\}$. For any $\mathcal{A}\in \mathbb{R} ^{[m, n]},$ we define

Definition 2.1. Let $\mathcal{A} = (a_{{i_1}{i_2}\cdots{i_m}})\in\mathbb{R}^{[m, n]},$ and $\sigma_Z(\mathcal{A})$ be the set of all Pareto $Z$-eigenvalues of $\mathcal{A}.$

(i) The maximum Pareto $Z$-eigenvalue and the minimum Pareto $Z$-eigenvalue of $\mathcal{A}$ are denoted by

(ii) $\mathcal{A}$ is called symmetry if

where $\Gamma_m$ is the permutation group of $m$ indices.

(iii) $\delta_{{i_1}{i_2}\ldots{i_m}}$ is called the generalized Kronecker symbol:

We conclude this section with significant results of the symmetric matrices ^[7] and the bound of Pareto $Z$-eigenvalue.

Lemma 2.1. Let $P\in \mathbb{R}^{n\times n}$ be a symmetric matrix and $x \in \mathbb{R}^n$ be a unit vector, i.e., $x^{\top}x = 1.$ $\mu_{\min}(P)$ (or $\mu_{\max}(P))$ denotes the minimum (maximum) eigenvalue of a square matrix $P,$ and $\rho(P)$ is the spectral radius of $P.$ Then,

Lemma 2.2. (Theorem 2 of ^[29]) Let $\mathcal{A}\in \mathbb{R}^{[m, n]}$ and $\sigma(\mathcal{A})\neq \emptyset.$ Then,

where $R_{i}(\mathcal{A})_{+}: = \sum\limits_{i_{2}, \dots, i_{m} = 1}^{n}[a_{ii_{2}\cdots i_{m}}]_{+}, \quad R_{i}(\mathcal{A})_{-}: = \sum\limits_{i_{2}, \ldots, i_{m} = 1}^{n}[a_{ii_{2}\ldots i_{m}}]_{-}.$

3.   Pareto $Z$-eigenvalues inclusion intervals

We begin with the bounds of Pareto $Z$-eigenvalues for a third-order tensor based on the spectral radius of the symmetric matrix $V_{i}: = \frac{\mathcal{A}_{i::}+\mathcal{A}^{\top}_{i::}}{2}$.

Theorem 3.1. Let $\mathcal{A} \in \mathbb{R}^{[3, n]}$ with $\sigma_{Z}(\mathcal{A})\neq \emptyset.$ Then,

where $[V_{i}]_{+} = \frac{[\mathcal{A}_{i::}]_{+}+[\mathcal{A}_{i::}]^{\top}_{+}}{2}, [V_{i}]_{-} = \frac{[\mathcal{A}_{i::}]_{-}+[\mathcal{A}_{i::}]^{\top}_{-}}{2}$ and $\mathcal{A}_{i::}$ is the matrix by fixing $i$ indices of $\mathcal{A}$.

Proof. Suppose that $(\lambda, x)$ is a Pareto $Z$-eigenpair of $\mathcal{A}.$ On the one hand, since $x^{\top}x = 1$ and $x_i\geq 0$ hold for all $i\in N$, we obtain

where the second inequality holds from Cauchy–Schwarz inequality. It follows from the definition of $[V_{i}]_{+}$ and $x^{\top}[\mathcal{A}_{i::}]_{+}x = x^{\top}[\mathcal{A}_{i::}]^{\top}_{+}x$ that

Since $[V_{i}]_{+}$ is a real symmetric matrix, by (3.2), (3.3), and Lemma 2.1, we obtain

On the other hand, from $x^{\top}x = 1$ and $x_i\geq 0$ for all $i\in N$, one has

It follows from the definition of $[V_{i}]_{-}$ and $x^{\top}[\mathcal{A}_{i::}]_{-}x = x^{\top}[\mathcal{A}_{i::}]^{\top}_{-}x$ that

Taking into account that $[V_{i}]_{-}$ is a real symmetric matrix, by (3.5), (3.6), and Lemma 2.1, we deduce

Combining (3.4) with (3.7) yields

which implies $\lambda \in\Upsilon(\mathcal{A})$ and $\sigma_{Z}(\mathcal{A})\subseteq \Upsilon(\mathcal{A})$. □

The following example is proposed to test the efficiency of the obtained results.

Example 3.1. Consider a tensor $\mathcal{A} = (a_{ijk})\in \mathbb{R}^{[3,3]}$ defined by

By calculating, we have

Following the similar calculations to the above, one has

According to Theorem 3.1, we obtain

Recalling Theorem 1 of ^[29], we deduce

which implies that the bound of Theorem 3.1 is sharp.

The following example estimates the Pareto $Z$-eigenvalues to guarantee nonconstant trajectories of equilibrium systems.

Example 3.2. Consider the following differential equilibrium system:

Thus, $\sum$ can be written as $\dot{x}(t) = \mathcal{A}x^2,$ where $x = (x_1, x_2, x_3)^{\top}$ with $x^2_1+ x^2_2+x^3_3 = 1$ and $\mathcal{A} = (a_{ijk})\in {\mathbb{R}^{[3,3]}}$ with

In order to ensure nonconstant trajectories of the equilibrium system, we need to find $(\lambda, x)\in \mathbb{R}\times \mathbb{R}^{n}_{+}\backslash\{0\}$ such that

Using Algorithm 3.1 of ^[32], we obtain four Pareto $Z$-eigenvalues and the associated Pareto $Z$-eigenvectors about 3.25 seconds:

It follows from Theorem 3.1 that we estimate $0\leq \lambda\leq \sqrt{6}.$ We apply this estimation to Algorithm 3.1 of ^[32] and can calculate the above Pareto $Z$-eigenvalues in 2.65 seconds. Therefore, Algorithm 3.1 of ^[32] could be accelerated by establishing the bound of Pareto $Z$-eigenvalues.

Using the spectral radius of symmetric matrices extracted from the given tensor, we establish Pareto $Z$-eigenvalue intervals of an $m$-order tensor with $m\geq 4.$

Theorem 3.2. Let $\mathcal{A}\in \mathbb{R}^{[m, n]}$ with $m\geq4$, and $\sigma_{Z}(\mathcal{A})\neq \emptyset.$ Then,

where $[B_{i}]_{+} = \frac{[A_{i}]_{+}+[A_{i}]^{\top}_{+}}{2}, \quad[B_{i}]_{-} = \frac{[A_{i}]_{-}+[A_{i}]^{\top}_{-}}{2}$ and

Proof. Suppose that $(\lambda, x)$ is a Pareto $Z$-eigenpair of $\mathcal{A}$. Then,

Denote $x_{p} = \max\limits_{i\in N}\{x_i\}.$ Then, $0 < x_{p}\leq1$ as $x^{\top}x = 1$. Recalling the $p$-th equation of (3.8), we obtain

Taking modulus in the equation above, one has

where $[A_{p}]_{+}$ and $[A_{p}]_{-}$ are defined in Theorem 3.2. Certainly, $x^{\top}[A_{i}]_{+}x = x^{\top}[A_{i}]^{\top}_{+}x$ and $x^{\top}[A_{i}]_{-}x = x^{\top}[A_{i}]^{\top}_{-}x.$ It follows from the definitions of $[B_{i}]_{+}$ and $[B_{i}]_{-}$ that

Since $[B_{p}]_{+}$ and $[B_{p}]_{-}$ are real symmetric matrices, by (3.9), (3.10) and Lemma 2.1, we have

which implies $\lambda \in\Theta(\mathcal{A}),$ and hence $\sigma_{Z}(\mathcal{A})\subseteq \Theta(\mathcal{A})$. □

Now, we are in a position to establish tight Pareto $Z$-eigenvalues inclusion intervals by accurate classification of index sets.

Theorem 3.3. Let $\mathcal{A}\in \mathbb{R}^{[m, n]}$ with $m\geq 4,$ and $\sigma_{Z}(\mathcal{A})\neq \emptyset.$ Then,

where

$[B_{i}^{j}]_{+} = \frac{[A_{i}^{j}]_{+}+[A_{i}^{j}]^{\top}_{+}}{2}, \quad[B_{i}^{j}]_{-} = \frac{[A_{i}^{j}]_{-}+[A_{i}^{j}]^{\top}_{-}}{2}$ and

Proof. Let $(\lambda, x)$ be a Pareto $Z$-eigenpair of $\mathcal{A}.$ Setting $0 < x_{p} = \max\limits_{i\in N}\{x_i\}$ and referring to the $p$-th equation of (3.8), for any $q\in N, q\neq p$, we obtain

Clearly,

With the definitions of $[D_{ij}]_{+}, [D_{ij}]_{-}, [B_{i}^{j}]_{+}$ and $[B_{i}^{j}]_{-},$ it is easy to verify that

Since $[D_{pq}]_{+},$$[D_{pq}]_{-}, [B_{p}^{q}]_{+}$ and $[B_{p}^{q}]_{-}$ are real symmetric matrices, by (3.11), (3.12), and Lemma 2.1, we deduce

Recalling the $q$-th equation of (3.8), one has

where $[B_{q}]_{+}$ and $[B_{q}]_{-}$ are defined in Theorem 3.2. It follows from (3.14) and Lemma 2.1 that

We now break up the argument into two cases.

Case Ⅰ. $|\lambda| x_{p}^{2}\leq \rho([D_{pq}]_{+})x_{p}x_{q}+x_{p}^{2}\rho([B_{p}^{q}]_{+}).$ In this case, if $x_q > 0$, multiplying (3.13) with (3.15) and dividing $x_{p}^{2}x_{q}^{2}$ yield

which implies $\lambda\in \mathcal{M}_{p, q}(\mathcal{A}).$

Otherwise, $x_q = 0.$ From (3.13), it holds that

which shows that $\lambda\in\mathcal{M}_{p, q}(\mathcal{A}).$

Case Ⅱ. $|\lambda| x_{p}^{2}\leq \rho([D_{pq}]_{-})x_{p}x_{q}+x_{p}^{2}\rho([B_{p}^{q}]_{-}).$ Following the similar arguments to the proof of Case Ⅰ, we obtain $\lambda\in\mathcal{M}_{p, q}(\mathcal{A}).$ Combining Cases Ⅰ and Ⅱ, we obtain the desired results. □

In order to illustrate the validity of Theorems 3.2 and 3.3, we employ a running example.

Example 3.3. Consider a tensor $\mathcal{A} = (a_{ijkl})\in \mathbb{R}^{[4,3] }$ defined by

From Theorem 3.2, we compute

Recalling Theorem 3.3, one has

From Theorem 2 of ^[29], we obtain

By virtue of Theorem 3 of ^[29], one has

It follows from Theorem 4 of ^[29] that

Therefore, the bounds in Theorems 3.2 and 3.3 are sharper than those Theorems 2–4 in ^[29].

4.   Checking the strict copositivity of tensors

In this section, we focus on sufficient conditions for judging strict copositivity via the spectral radius of the symmetric matrices extracted from the given tensor. For this, we give a necessary condition for a strictly copositive tensor.

Lemma 4.1. (Proposition 2.1 of ^[18]) Let $\mathcal{A} = (a_{{i_1}{i_2}\ldots{i_m}})\in\mathbb{R}^{[m, n]}$. If $\mathcal{A}$ is strictly copositive, then $a_{i\ldots i} > 0, \forall i \in N$.

Theorem 4.1. Let $\mathcal{A} = (a_{{i_1}{i_2}{i_3}})\in \mathbb{R}^{[3, n]}$ be symmetric with $a_{iii} > 0$ for all $i\in N$. If

then $\mathcal{A}$ is strictly copositive.

Proof. Suppose that $(\lambda, x)$ is a Pareto $Z$-eigenpair of $\mathcal{A}.$ Setting $0 < x_{p} = \max\limits_{i\in N}\{x_i\}$ and referring to the $p$-th equation of (3.10), we obtain

Further,

Dividing both sides by $x_{p}^{2}$ on (4.2), we have

Since $x_{p} = \max\limits_{i\in N}\{x_i\}$ and $x^{\top}x = 1,$ we deduce $x_{p}\geq\frac{1}{\sqrt{n}}$. It follows from $a_{iii} > 0$ and (4.3) that

Combining (4.1) with (4.4), we have $\lambda > 0.$ Further, $\mathcal{A}$ is strictly copositive from Lemma 2.1. □

Theorem 4.2. Let $\mathcal{A} = (a_{{i_1}{i_2}\ldots{i_m}})\in \mathbb{R}^{[m, n]}$ be symmetric with $m\geq4$ and $a_{i\ldots i} > 0$ for all $i\in N$. If

then $\mathcal{A}$ is strictly copositive.

Proof. Let $(\lambda, x)$ be a Pareto $Z$-eigenpair of $\mathcal{A}.$ Setting $0 < x_{p} = \max\limits_{i\in N}\{x_i\}$ and referring to the $p$-th equation of (3.10), we obtain

Further,

where $[A_{p}]_{-}$ is defined in Theorem 3.2. Dividing both sides by $x_{p}^{2}$ on (4.6), we deduce

Since $x_{p} = \max\limits_{i\in N}\{x_i\}$ and $x^{\top}x = 1,$ we deduce $x_{p}\geq\frac{1}{\sqrt{n}}$. It follows from $a_{i\ldots i} > 0$ and (4.7) that

Combining (4.5) with (4.8), we obtain $\lambda > 0.$ Further, $\mathcal{A}$ is strictly copositive from Lemma 2.1. □

A nice consequence of our results is that Theorems 4.1 and 4.2 are better than that of Theorem 5 of ^[29].

Lemma 4.2. (Theorem 5 of ^[29]) Let $\mathcal{A} = (a_{{i_1}{i_2}\ldots{i_m}})\in \mathbb{R}^{[m, n]}$ be symmetric with $a_{i\ldots i} > 0$ for $i\in N$. Then, $\mathcal{A}$ is strictly copositive, provided that

where $R_{i}(\mathcal{A})_{-} = \sum\limits_{i_{2}, \ldots, i_{m}\in N}[a_{ii_{2}\ldots i_{m}}]_{-}.$

Corollary 4.1. Let $\mathcal{A} = (a_{{i_1}{i_2}\ldots{i_m}})\in \mathbb{R}^{[m, n]}$ be symmetric with $a_{i\ldots i} > 0$ for $i\in N$. Then,

Proof. It follows from $a_{i\ldots i} > 0$ that $R_{i}(\mathcal{A})_{-} = \sum\limits_{\delta _{ii_{2}\ldots i_{m} = 0}}[a_{ii_{2}\ldots i_{m}}]_{-}.$ We now break up the argument into two cases.

Case 1. $m = 3$. It is clear that

Further,

Case 2. $m\geq 4$. We obtain

Consequently,

Therefore, the desired results hold. □

Identifying the strict copositivity actually necessitates $\mathcal{A}$ being symmetric. Therefore, symmetry may be relatively strict for general tensors. We can solve this issue by symmetrizing the tensors $\mathcal{A} = (a_{{i_1}{i_2}\ldots{i_m}})\in\mathbb{R}^{[m, n]}$ as follows:

where $\widetilde{\mathcal{A}} = (\widetilde{a}_{{i_1}{i_2}\ldots{i_m}})\in\mathbb{R}^{[m, n]}$ is the symmetrization tensor under permutation group $\Gamma_{m}$.

The following example shows that Theorem 4.1 can verify the strict copositivity more accurately than that of Theorem 5 of ^[29] for $m = 3$ tensors.

Example 4.1. Consider a tensor $\mathcal{A} = (a_{ijk})\in \mathbb{R}^{[3,2]}$ defined by

It is easy to see that $\mathcal{A}$ is symmetric with

which means that $\mathcal{A}$ is strictly copositive.

Referring to Theorem 5 of ^[29], we deduce

Therefore, it is impossible to judge the strict copositivity of $\mathcal{A}$ with Theorem 5 of ^[29].

When $\mathcal{A}$ is asymmetric, we still identify the strict copositivity of tensors by Theorem 4.1.

Example 4.2. Consider a tensor $\mathcal{A} = (a_{ijk})\in \mathbb{R}^{[3,2]}$ defined by

Observe that $\mathcal{A}$ is asymmetric from $a_{112} = -\frac{1}{10}, a_{121} = -\frac{2}{5}$ and $a_{211} = -\frac{1}{10}.$ Therefore, we cannot directly use Theorem 4.1 to judge whether $\mathcal{A}$ is strictly copositive. Symmetrizing $\mathcal{A}$, we obtain $\widetilde{\mathcal{A}}$ with

It is easy to see that $\widetilde{\mathcal{A}}$ is symmetric with

which implies that $\widetilde{\mathcal{A}}$ is strictly copositive. Taking into account that $\mathcal{A}x^{3} = \widetilde{\mathcal{A}}x^{3} > 0,$ we deduce that $\mathcal{A}$ is strictly copositive.

In what follows, we reveal that the results of Theorem 4.2 are sharper than those of Theorem 5 of ^[29] for $m\geq 4$ tensors.

Example 4.3. Consider a tensor $\mathcal{A} = (a_{ijkl})\in \mathbb{R}^{[4,2]}$ defined by

First, we rewrite

and compute

which means that $\mathcal{A}$ is strictly copositive.

Recalling to Theorem 5 of ^[29], we obtain

Consequently, we cannot judge the strict copositivity of $\mathcal{A}$ from Theorem 5 of ^[29].

5.   Conclusions

In this paper, we proposed sharp Pareto $Z$-eigenvalue inclusion intervals for tensor eigenvalue complementarity problems via the spectral radius of symmetric matrices. Further, we proposed some criteria to confirm the strict copositivity of real tensors. It may be possible to conduct additional research to create some algorithms for tensor eigenvalue complementarity problems using Pareto $Z$-eigenvalue intervals, such as parametric algorithms and ADMM algorithms ^[5,23,28].

Author contributions

Xueyong Wang: Conceptualization, Methodology, Writing-original draft preparation; Gang Wang: Conceptualization, Writing-review, Project administration; Ping Yang: Software, Formal analysis. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (12071250) and the Natural Science Foundation of Shandong Province (ZR2021MA088).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.