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Pareto Z-eigenvalue inclusion theorems for tensor eigenvalue complementarity problems
Journal of Inequalities and Applications volume 2022, Article number: 77 (2022)
Abstract
This paper presents some sharp Pareto Z-eigenvalue inclusion intervals and discusses the relationships among different Pareto Z-eigenvalue inclusion intervals for tensor eigenvalue complementarity problems. As an application, we propose a sufficient condition for identifying the strict copositivity of tensors. Some examples are provided to illustrate the obtained results.
1 Introduction
Let \(\mathcal{A}=(a_{{i_{1}}{i_{2}}\cdots{i_{m}}})\in \mathbb{R}^{[m,n]}\) be an mth-order n-dimensional real tensor, x be a real n-vector and \(N=\{1,2,\dots,n\}\). Denote by \(\mathcal{A} x^{m-1}\) the vector in \(\mathbb{R}^{n}\) with entries
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 and b are perpendicular to each other. For the problem, its solution \((\lambda,x)\in \mathbb{R}\times \mathbb{R}^{n}_{+}\backslash \{0\}\) is called a Pareto Z-eigenpair of tensor \(\mathcal{A}\).
The Pareto Z-eigenpair of a tensor was introduced by Song [1], which is a natural generalization of that of a matrix [2–5]. It is worth noting that Pareto Z-eigenvalues of \(\mathcal{A}\) are closely related to Z (\(Z^{+}\))-eigenvalues of \(\mathcal{A}\) introduced by Lim [6] and Qi [7, 8], respectively.
Definition 1
For a tensor \(\mathcal{A}=(a_{{i_{1}}{i_{2}}\cdots{i_{m}}})\in \mathbb{R}^{[m,n]}\), if there exist \((\lambda,x)\in \mathbb{R}\times \mathbb{R}^{n}\backslash \{0\}\) such that
then \((\lambda,x)\) is called a Z-eigenpair of tensor \(\mathcal{A}\). Further, Z-eigenvalue λ of \(\mathcal{A}\) is said to be a \(Z^{+}\)-eigenvalue, if its eigenvector \(x\in \mathbb{R}^{n}_{+}\backslash \{0\}\).
Obviously, \(Z^{+}\)-eigenvalues of \(\mathcal{A}\) are Pareto Z-eigenvalues. However, the converse may not hold as pointed by Zeng [9]. Therefore, the tensor Pareto Z-eigenvalue received much attentions of researchers [9–12]. For instance, Zeng [9] proposed a semidefinite relaxation algorithm to obtain Pareto Z-eigenvalues of tensor eigenvalue complementarity problems. Since it is not easy to find all Pareto Z-eigenvalues in practice [1, 9, 13], it is significant to make some characterizations to the distribution of Pareto Z-eigenvalues. Inspired by the results obtained in [14–18], we establish some Pareto Z-eigenvalues inclusion intervals, give comparisons among these Pareto Z-eigenvalue inclusion intervals, and propose a sufficient condition to identify the strict copositivity of real tensors in this paper.
The remainder of this paper is organized as follows. In Sect. 2, we recall some preliminary results and establish Pareto Z-eigenvalue inclusion intervals. Further, we give comparisons among these Pareto Z-eigenvalue inclusion intervals. In Sect. 3, we propose a sufficient condition to identify the strict copositivity of tensors.
To end this section, we give some notations needed. 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 \(x\in \mathbb{R} ^{n}\), \(x^{{\otimes}m}\) denotes a tensor whose entries are defined by \((x^{{\otimes}m})_{i_{1}i_{2}\cdots i_{m}}=x_{i_{1}}x_{i_{2}} \cdots x_{i_{m}}\) for all \(i_{1},i_{2},\dots,i_{m}\in N\). For any \(\mathcal{A}\in \mathbb{R} ^{[m,n]}\) and \(x\in \mathbb{R} ^{n}\), we define
For any \(i,j\in N\), set
2 Pareto Z-eigenvalues inclusion intervals
First, we recall some results of strictly copositive tensors [19, 20], and then establish Pareto Z-eigenvalue inclusion theorems of tensor \(\mathcal{A}\). Some comparisons among different Pareto Z-eigenvalue inclusion intervals are also made in this section.
Definition 2
Tensor \(\mathcal{A} \in \mathbb{R}^{[m,n]}\) is said to be:
(i) strictly copositive if \(\mathcal{A}x^{m}>0\) for any \(x\in \mathbb{R}^{n}_{+}\backslash \{0\}\);
(ii) symmetric if \(a_{{i_{1}}{i_{2}}\cdots{i_{m}}}= a_{i_{\pi (1)}\cdots{i}_{\pi (m)}}, \forall \pi \in \Gamma _{m}\), where \(\Gamma _{m}\) is the permutation group of m indices.
Lemma 1
([1, Corollary 3.5])
Let \(\mathcal{A}=(a_{{i_{1}}{i_{2}}\cdots{i_{m}}})\in \mathbb{R}^{[m,n]}\) be symmetric. Then \(\mathcal{A}\) always has Pareto Z-eigenvalues; \(\mathcal{A}\) is strictly copositive if and only if all of its Pareto Z-eigenvalues are positive.
Lemma 2
([20, Proposition 2.1])
Let \(\mathcal{A}=(a_{{i_{1}}{i_{2}}\cdots{i_{m}}})\in \mathbb{R}^{[m,n]}\). If \(\mathcal{A}\) is strictly copositive, then \(a_{i\cdots i}>0,\forall i \in N\).
Based on the above lemmas, we have the following conclusion.
Theorem 1
Let \(\mathcal{A} \in \mathbb{R}^{[m,n]}\). Denote the set of Pareto Z-eigenvalues by \(\sigma (\mathcal{A})\) and assume \(\sigma (\mathcal{A})\neq \emptyset \). Then,
where \(\bar{a}=\max_{i_{1},\dots,i_{m}\in N}|a_{i_{1}i_{2}\cdots i_{m}}|\).
Proof
Suppose that \((\lambda,x)\) is a Pareto Z-eigenpair of \(\mathcal{A}\). Then
and
Meanwhile, from the definition of Pareto Z-eigenpair, we obtain
where the second inequality holds via Cauchy–Schwartz inequality. The desired result follows by combining (4) and (5). □
In the following, we will use some important elements of tensor to describe Pareto Z-eigenvalues inclusion intervals.
Theorem 2
Let \(\mathcal{A}\in \mathbb{R}^{[m,n]}\) and \(\sigma (\mathcal{A})\neq \emptyset \). Then,
Proof
Suppose that \((\lambda,x)\) is a Pareto Z-eigenpair of \(\mathcal{A}\). Then
Denote \(x_{p}=\max_{i\in N}\{x_{i}\}\). Then, \(0< x_{p}\leq 1\) as \(x^{\top}x=1\). Recalling the pth equation of (6), we get
Taking the absolute value of the equation above, one has
Dividing both sides by \(x_{p}^{2}\), one has
which implies \(\lambda \in \Omega _{p}(\mathcal{A})\), and hence \(\sigma (\mathcal{A})\subseteq \Omega (\mathcal{A})\). □
Theorem 3
Let \(\mathcal{A}\in \mathbb{R}^{[m,n]}\) and \(\sigma (\mathcal{A})\neq \emptyset \). Then,
where
Proof
Suppose that \((\lambda,x)\) is a Pareto Z-eigenpair of \(\mathcal{A}\). Setting \(0< x_{p}=\max_{i\in N}\{x_{i}\}\) and referring to the pth equation of (6), for any \(q\in N, q\neq p\), we obtain
which implies
Recalling the qth equation of (6), one has
which shows
We now break up the argument into two cases for (8).
Case I. \(|\lambda | x_{p}^{2}\leq R_{p}^{q}(\mathcal{A})_{+}x_{p}^{2}+[a_{pq \cdots q}]_{+}x_{p}x_{q}\). In this case, if \(x_{q}>0\), multiplying (8) with (9) and dividing \(x_{p}^{2}x_{q}^{2}\) yields
which implies \(\lambda \in \Phi _{p,q}(\mathcal{A})\).
Otherwise, \(x_{q}=0\). From (8), it holds that
which shows that \(\lambda \in \Phi _{p,q}(\mathcal{A})\).
Case II. \(|\lambda | x_{p}^{2}\leq R_{p}^{q}(\mathcal{A})_{-}x_{p}^{2}+[a_{pq \cdots q}]_{-}x_{p}x_{q}\). Following similar arguments as in the proof of Case I, we obtain \(\lambda \in \Phi _{p,q}(\mathcal{A})\).
Combining Cases I and II, we obtain the desired results. □
Compared with Theorem 2, the result of Theorem 3 requires relatively many calculations but has accurate results. Detailed investigation is given in Corollary 1.
Corollary 1
For a tensor \(\mathcal{A}\in \mathbb{R}^{[m,n]}\), it holds that
where \(\Phi (\mathcal{A})\) and \(\Omega (\mathcal{A})\) are defined in Theorems 2and 3.
Proof
For any \(\lambda \in \Phi (\mathcal{A})\), there exist \(p,q\in N\) with \(p\neq q\) such that
or
We now break up the argument into two cases.
Case I. \((|\lambda |-R_{p}^{q}(\mathcal{A})_{+})|\lambda |\leq [a_{pq\cdots q}]_{+} \max \{R_{q}(\mathcal{A})_{+},R_{q}(\mathcal{A})_{-}\}\).
If \([a_{pq\cdots q}]_{+}\max \{R_{q}(\mathcal{A})_{+},R_{q}(\mathcal{A})_{-} \}=0\), it holds that
or
which indicates that
Otherwise, \([a_{pq\cdots q}]_{+}\max \{R_{q}(\mathcal{A})_{+},R_{q}(\mathcal{A})_{-} \}> 0\). Then,
which implies
Consequently, (10) holds.
Case II. \((|\lambda |-R_{p}^{q}(\mathcal{A})_{-})|\lambda |\leq [a_{pq\cdots q}]_{-} \max \{R_{q}(\mathcal{A})_{+},R_{q}(\mathcal{A})_{-}\}\). Following similar arguments as in the proof of Case I, we can prove that \(\lambda \in \Omega (\mathcal{A})\).
Combining Case I with Case II, we conclude that \(\Phi (\mathcal{A})\subseteq \Omega (\mathcal{A})\). □
To get accurate results, we divide precisely the index set of \(\mathcal{A}\) and establish Theorem 4.
Theorem 4
Let \(\mathcal{A}\in \mathbb{R}^{[m,n]}\) and \(\sigma (\mathcal{A})\neq \emptyset \). Then,
where \(\mathcal{N}_{i,j}(\mathcal{A})=\{\lambda \in \mathbb{R}:(|\lambda |- \max \{P_{i}^{j}(\mathcal{A})_{+},P_{i}^{j}(\mathcal{A})_{-}\})| \lambda |\leq \max \{R_{i}(\mathcal{A})_{+}-P_{i}^{j}(\mathcal{A})_{+},R_{i}( \mathcal{A})_{-}-P_{i}^{j}(\mathcal{A})_{-}\}\cdot \max \{R_{j}( \mathcal{A})_{+},R_{j}(\mathcal{A})_{-}\}\}\).
Proof
Suppose that \((\lambda,x)\) is a Pareto Z-eigenpair of \(\mathcal{A}\). Setting \(0< x_{p}=\max_{i\in N}\{x_{i}\}\) and referring to the pth equation of (6), for any \(q\in N, q\neq p\), one has
Taking the absolute value of the equation above, we obtain
where the third inequality holds from \(0< x_{p}^{m-1}\leq x_{p}\leq 1\) and \(0\leq x_{q}<1\). Further,
In view of the qth equation of (6), we deduce
We now break up the argument into two cases.
Case I: \(x_{q} >0\). Multiplying (11) with (12) and dividing \(x_{p}^{2}x_{q}^{2}\), we obtain
which implies \(\lambda \in \mathcal{N}_{p,q}(\mathcal{A})\subseteq \mathcal{N}( \mathcal{A})\).
Case II: \(x_{q}=0\). It follows from (11) that
that is,
which implies \(\lambda \in \mathcal{N}_{p,q}(\mathcal{A})\subseteq \mathcal{N}( \mathcal{A})\). □
In what follows, we now test the efficiency of the obtained results.
Example 1
Consider a 3rd order 3-dimensional tensor \(\mathcal{A}=(a_{ijk})\) defined by
By calculating, we have
According to Theorem 1, we obtain
Referring to Theorem 2, we deduce
Recalling Theorem 3, one has
where
\(\Phi _{1,2}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 3\}\) | \(\Phi _{1,3}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 1+\sqrt{6}\}\) |
\(\Phi _{2,1}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 3\}\) | \(\Phi _{2,3}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 3\}\) |
\(\Phi _{3,1}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 2+\sqrt{7}\}\) | \(\Phi _{3,2}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 1+\sqrt{10} \}\). |
It follows from Theorem 4 that
where
\(\mathcal{N}_{1,2}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq \frac{3+\sqrt{21}}{2}\}\) | \(\mathcal{N}_{1,3}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq \frac{1+\sqrt{41}}{2} \) |
\(\mathcal{N}_{2,1}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq \frac{3+\sqrt{21}}{2}\}\) | \(\mathcal{N}_{2,3}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 1+\sqrt{6} \) |
\(\mathcal{N}_{3,1}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 2+\sqrt{7}\}\) | \(\mathcal{N}_{3,2}(\mathcal{A})=\{\lambda \in \mathbb{R}: |\lambda |\leq 3\}\). |
3 Judging strict copositivity of tensors
In this section, we mainly propose a sufficient condition for judging strict copositivity of \(\mathcal{A}\).
Theorem 5
Let \(\mathcal{A}=(a_{{i_{1}}{i_{2}}\cdots{i_{m}}})\in \mathbb{R}^{[m,n]} \) be symmetric with \(a_{i\cdots i}>0\) for \(i\in N\). Then \(\mathcal{A}\) is strictly copositive provided that
Proof
Suppose that \((\lambda,x)\) is a Pareto Z-eigenpair of \(\mathcal{A}\). Setting \(0< x_{p}=\max_{i\in N}\{x_{i}\}\) and referring to the pth equation of (6), we obtain
Further,
Dividing both sides by \(x_{p}^{2}\), we have
Since \(x_{p}=\max_{i\in N}\{x_{i}\}\) and \(x^{\top}x=1\), we deduce \(x_{p}\geq \frac{1}{\sqrt{n}}\). It follows from \(a_{i\cdots i}>0\) and (14) that
Combining (13) with (15), we have \(\lambda >0\) and \(\mathcal{A}\) is strictly copositive. □
From the conclusion, identifying the strict copositivity of tensor \(\mathcal{A}\) requires that it is symmetric. For general tensors, symmetry is a relatively strict condition. To tackle this problem, we may symmetrize the tensors \(\mathcal{A}=(a_{{i_{1}}{i_{2}}\cdots{i_{m}}})\in \mathbb{R}^{[m,n]}\) as follows:
where \(\widetilde{\mathcal{A}}=(\widetilde{a}_{{i_{1}}{i_{2}}\cdots{i_{m}}}) \in \mathbb{R}^{[m,n]}\) is the symmetrization tensor under permutation group \(\Gamma _{m}\).
The following example shows that the result given in Theorem 5 can verify the strict copositivity of tensors.
Example 2
Consider a 3rd order 2-dimensional tensor \(\mathcal{A}=(a_{ijk})\) defined by
It is easy to see that \(\mathcal{A}\) is symmetric with
According to Theorem 5, we have
which means that \(\mathcal{A}\) is strictly copositive.
When \(\mathcal{A}\) is asymmetric, we still identify the strict copositivity by Theorem 5.
Example 3
Consider a 3rd order 2-dimensional tensor \(\mathcal{A}=(a_{ijk})\) defined by
Since \(a_{112}=-1, a_{121}=-2\), and \(a_{211}=-1\), we know that \(\mathcal{A}\) is asymmetric. Therefore, we cannot directly use Theorem 5 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
According to Theorem 5, we have
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.
4 Conclusion
In this paper, we proposed sharp Pareto Z-eigenvalue inclusion intervals and established comparisons among different Pareto Z-eigenvalue inclusion intervals for tensor eigenvalue complementarity problems. Meanwhile, we gave a sufficient condition to check strict copositivity of real tensors. Further studies can be considered to develop some algorithms by Pareto Z-eigenvalue inclusion intervals for tensor eigenvalue complementarity problems, as done in [5] for solving the matrix eigenvalue complementarity problems.
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Acknowledgements
The authors would like to thank anonymous referees and editors for their helpful comments and suggestions, which greatly improved the quality of this paper.
Funding
This research is supported by the Natural Science Foundation of Shandong Province (ZR2020MA025, ZR2019PA016) and the National Natural Science Foundation of China (12071250, 11901343).
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PY: original draft writing, review writing, and editing. YJW: conceptualization, supervision, and funding acquisition. GW: computation and review writing. QLH: computation and review writing. All authors read and approved the final manuscript.
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Yang, P., Wang, Y., Wang, G. et al. Pareto Z-eigenvalue inclusion theorems for tensor eigenvalue complementarity problems. J Inequal Appl 2022, 77 (2022). https://doi.org/10.1186/s13660-022-02816-x
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DOI: https://doi.org/10.1186/s13660-022-02816-x