Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

Zeng, Qingyu; He, Jun; Liu, Yanmin

doi:https://doi.org/10.1155/2020/3897981

Mathematical Problems in Engineering

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Abstract Introduction Notation and Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Advances in Numerical Optimisation: Theory, Models, and Applications

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Research Article | Open Access

Volume 2020 | Article ID 3897981 | https://doi.org/10.1155/2020/3897981

Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

Qingyu Zeng,¹Jun He,¹and Yanmin Liu¹

Guest Editor: Li-Tao Zhang

Received21 Jun 2020

Accepted08 Aug 2020

Published24 Aug 2020

Abstract

In this paper, some properties of structured rectangular tensors are presented, and the relationship among these structured rectangular tensors is also given. It is shown that all the V-singular values of rectangular P-tensors are positive. Some necessary and/or sufficient conditions for a rectangular tensor to be a rectangular P-tensor are also obtained. A new subclass of rectangular tensors, which is called rectangular S-tensors, is introduced and it is proved that rectangular S-tensors can be defined by the feasible vectors of the corresponding rectangular tensor complementarity problem.

1. Introduction

Consider the following degree homogeneous polynomial:where and is an th order -dimensional real square tensor. When is even, the positive definiteness of in (1) plays an important role in automatic control [1]. In order to verify the positive definiteness of in (1), Qi introduced the definitions of H-eigenvalue and Z-eigenvalue of and showed that when is even, is positive definite (i.e., in (1) is positive definite) if and only if all H-eigenvalues or Z-eigenvalues of are positive [2–4].

One important structured tensor is called copositive tensor, which can be viewed as a generalization of copositive matrices and plays an important role in tensor complementarity problem [5] and polynomial optimization problems [6]. In [7], Qi introduced the definition of copositive tensors and obtained some necessary and sufficient conditions for a real symmetric tensor to be a copositive tensor. In [6], a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors is presented. Che et al. [5] showed that the tensor complementarity problem with a strictly copositive tensor has a nonempty and compact solution set. Song and Qi [8] proved that a real symmetric tensor is semipositive if and only if it is copositive. A numerical algorithm for copositivity of square tensors is proposed in [9].

Another important structured tensor is called P-tensor. The P-tensors and -tensors are first introduced by Song and Qi [10], which can be viewed as generalizations of the P-matrices and -matrices [11]. The authors in [10] also showed that a symmetric tensor with even order is positive definite if and only if it is a P-tensor and a symmetric tensor with even order is positive semidefinite if and only if it is a -tensor. Another definition of P-tensors (-tensors) is presented, which includes many important structured tensors with odd order [12], and the authors also showed that the complementarity problem with a P-tensor has a nonempty compact solution set.

Consider the following degree homogeneous polynomial:where

is a th order -dimensional real rectangular tensor. is called a real partially symmetric rectangular tensor, if is invariant under any permutation of indices among , and any permutation of indices among , i.e.,where is the permutation group of indices. Let be a partially symmetric rectangular tensor, and and are even. Then, is positive definite if and only if all of its H-singular values (or V-singular values) are positive [13–19].

The definition of copositive rectangular tensors is introduced in [20], which can be viewed as a generalization of copositive square tensors, and some necessary and sufficient conditions for a real partially symmetric rectangular tensor to be a copositive rectangular tensor are also given in [20]. Based on the criteria for identifying copositive rectangular tensors, a numerical method for identifying the copositiveness of a partially symmetric rectangular tensor is obtained [21].

The rest of this first part is organized as follows. In Section 2, some preliminaries are given. In Section 3, we intend to introduce two new classes of rectangular tensors which are called rectangular P-tensors and rectangular -tensors. Moreover, we prove that all the V-singular values of rectangular P-tensors (rectangular -tensors) are positive (nonnegative). We also discuss some properties of quantities for rectangular P-tensors, and a necessary and sufficient condition for a rectangular tensor to be a rectangular P-tensor is also obtained. In Section 4, we extend the S-tensors to rectangular S-tensors, and some properties of rectangular S-tensors are also given. In Section 5, we introduce the rectangular tensor complementarity problem (RTCP), which can be used to define the rectangular S-tensors, and the relationship among positive definite rectangular tensors, strictly copositive rectangular tensors, rectangular P-tensors, and rectangular S-tensors is also presented.

2. Notation and Preliminaries

In this section, we list some definitions related to rectangular tensors, which are needed in the subsequent analysis.

Let and be the real field and complex field, . We use small letters , , … for scalars, small bold letters , … for vectors, capital letters , , … for matrices, and calligraphic letters , , … for tensors. The th entry of a vector is denoted by , the th entry of a matrix is denoted by , and the th entry of a rectangular tensor is denoted by . Let be the -dimensional real Euclidean space and the set of all nonnegative vectors in be denoted by .

Definition 1. A rectangular is said to be(a)A positive definite rectangular tensor [13, 14], iff for all and (b)A copositive rectangular tensor [21], iff for all and (c)A strictly copositive rectangular tensor [21], iff for all and In order to verify the positive definiteness of a th order -dimensional partially symmetric rectangular tensor, the definition of a singular value of rectangular tensors is introduced by Chang et al. [13].

Definition 2. (see [13]). Let , if there exist a number , vectors and such thatwhere , , andthen is called the H-singular value of and is the left and right H-eigenvectors pair of , associated with .
Some sufficient conditions for the positive definiteness of a th order -dimensional partially symmetric rectangular tensor are given in [14], based on the following definition of the V-singular value.

Definition 3. (see [14]). Let ; if there exist a number , vectors , , and such thatthen is called the V-singular value of and is the left and right V-eigenvectors pair of , associated with .
The definitions of P-tensors and P₀-tensors are listed as follows.

Definition 4. (see [10]). A tensor is called a P-tensor if for each nonzero , there exists some index such thatwhere . A tensor is called a P₀-tensor if for each nonzero , there exists some index such that

3. Rectangular P-Tensors and Rectangular P₀-Tensors

We now introduce the definitions of rectangular P-tensors and rectangular P₀-tensors.

Definition 5. A rectangular tensor is called a rectangular P-tensor if for each and , there exists some indices such thatA rectangular tensor is called a rectangular P₀-tensor if for each and , there exists some indices such thatThe following result is given to show the positivity (nonnegativity) of the V-singular values for a rectangular P-tensor (P₀-tensor).

Theorem 1. Let be a rectangular P-tensor (P₀-tensor); then, all the V-singular values of are positive (nonnegative).

Proof. If is a rectangular P-tensor, is a V-singular value of with eigenvectors pair ; then, we haveand then, there exists some indices such thatBy the definition of rectangular P-tensors, we have . The case for rectangular P₀-tensors can be obtained similarly.
A rectangular tensor is called a principal rectangular subtensor of a rectangular tensor iff the sets contain and elements such thatLet be a -dimensional subvector of a vector and be a -dimensional subvector of a vector . Note that, for , the principal rectangular subtensors are just the diagonal entries.

Lemma 1. Let with V-singular value . Then,

Proof. If is a V-singular value of with eigenvectors pair , then we havewhich implieswhere , and . Then, by , we have

Theorem 2. Let be a rectangular P-tensor (-tensor). Then, every principal rectangular subtensor of is a rectangular P-tensor (-tensor). In particular, all the diagonal entries of a rectangular P-tensor (-tensor) tensor are positive (nonnegative).

Proof. Let be a principal rectangular subtensor of : with if and if , and with if and if . If is a rectangular P-tensor, there exists some index such thatwhich implies that is a rectangular P-tensor. The case for rectangular -tensors can be obtained similarly.
A sufficient and necessary condition for a rectangular tensor to be a rectangular P-tensor is given as follows.

Theorem 3. Let . Then, is a rectangular P-tensor if and only if for each nonzero , there exists positive diagonal matrices such that

Proof. If is a rectangular P-tensor, then there exists some index such thatThen, for enough small , we haveTherefore, we havewhere with and for and with and for .
On the contrary, if there exists positive diagonal matricessuch thatSince for all , for all , then there exists and such thatLet , and a quantity of a P-matrix is introduced in [11]. In 2015, letwhereSong and Qi introduced the definitions of quantities and for a P-tensor and obtained monotonicity and boundedness of such two quantities, and they also showed that a tensor is a P-tensor if and only if is positive, and a tensor with even order is a P-tensor if and only if is positive [8]. We define the following two quantities for rectangular P-tensors:We present some properties of quantities for rectangular P-tensors.

Theorem 4. Let be a rectangular -tensor and be a principal rectangular subtensor of . Then,(i)(ii)

Proof. Let be a principal rectangular subtensor of : with if and if , and with if and if . Then, and . Hence,Let , where denotes the smallest of V-singular value (if any exists) of a rectangular P-tensor . Then, we have the following upper bounds for and . Let be the rectangular identity tensor with

Theorem 5. Let be a rectangular P-tensor. Then,(i)(ii)

Proof. Let be a rectangular P-tensor, by Theorem 2, and we haveSince is a principal rectangular subtensor of , by Lemma 1, is a V-singular value of ; therefore,Furthermore, is a V-singular value of ; then, is not a rectangular P-tensor. By Theorem 2, is not a rectangular P-tensor. Then, there exists vectors with and such thatThen,for all . Then,Similarly, we haveTherefore,Based on the quantities and , a necessary and sufficient conditions for a rectangular tensor to be a rectangular P-tensor is given as follows.

Theorem 6. Let . Then, is a rectangular P-tensor (-tensor) if and only if and are positive (nonnegative).

Proof. Let be a rectangular P-tensor. Then, for each and , there exists some index such thatThen,Conversely, if and , we havewhich implies is a rectangular P-tensor. The case for rectangular -tensors can be obtained similarly.

4. Rectangular S-Tensor

Definition 6. A rectangular tensor is called a rectangular S-tensor if and only if there exists and such thatA rectangular tensor is called a rectangular -tensor if and only if there exists and such thatThe conditions and in the definition of rectangular S-tensors can be relaxed to and .

Theorem 7. Let . Then, is a rectangular S-tensor if and only if there exists and such that

Proof. The necessity is obvious by the definition of rectangular S-tensors. We prove the sufficiency as follows.
If there exists and such thatLet and ; for some small enough , we havewhich means that is a rectangular S-tensor.
From Theorem 2, we know that every principal rectangular subtensor of a rectangular P-tensor is a rectangular P-tensor. However, such a property does not always hold for rectangular S-tensor by the following example, i.e., the principal rectangular subtensor of a rectangular S-tensor is not always a rectangular S-tensor.

Example 1. Let , whereand all other . Then, for any and , we obtainThen, for and , and , which implies, is a rectangular S-tensor.
Let be a principal rectangular subtensor of with ; then, for any and , we havewhich means that is not a rectangular S-tensor.
Some necessary and/or sufficient conditions for a rectangular tensor to be a rectangular S-tensor are presented as follows.

Theorem 8. Let be a rectangular S-tensor. Then, there exists and such that

Proof. Since is a rectangular S-tensor, then there exists and such thatLet and ; then, and , andwhich means

Theorem 9. Let . If there exists a principal rectangular subtensor of is a rectangular S-tensor, for all , , and for all and , ; then, is a rectangular S-tensor.

Proof. Since is a rectangular S-tensor, then there exists and such thatLet and withThen, for any and , we haveTherefore, if , we obtainif , we obtainwhich implies that . Similarly, we have . Then, is a rectangular S-tensor.
A sufficient and necessary condition for a rectangular tensor to be a rectangular S-tensor is given as follows.

Theorem 10. Let . Then, is a rectangular S-tensor if and only if there exists , for any nonzero nonnegative diagonal matrices such that

Proof. If is a rectangular S-tensor and , then for any such thatThen, for any , we haveTherefore, we havewhere with and for and with and for .
On the contrary, for any nonzero nonnegative diagonal matricessuch thatFor any , let with and for and with and for , and we obtainBy the conditions , we havewhich means that is a rectangular S-tensor.

5. Rectangular Tensor Complementarity Problems

Converting a bimatrix game to a linear complementarity problem, we have the following LCP [22]:

In this section, we study the rectangular tensor complementarity problem (RTCP), which can be viewed as the generalization of the linear complementarity problem (69) to the tensor case.

Let , and . The rectangular tensor complementarity problem, denoted by RTCP , is to find vectors and such that

Vectors and are said to be feasible iff and satisfy the following inequalities:

The following equivalent definition of rectangular S-tensor can be given by means of the solution of the RTCP .

Theorem 11. Let . Then, is a rectangular S-tensor if and only if the RTCP is feasible for all and .

Proof. If is a rectangular S-tensor and , thenThen, for each and , there exists some scalar such thatwhich means that and are the feasible vectors of the RTCP .
On the contrary, if the RTCP is feasible for all and , assume that , , and and are the feasible solution of the RTCP . Then,Therefore,Then, is a rectangular S-tensor by Theorem 7.
In the end of this section, we propose some relationships among these structured rectangular tensors as follows.

Theorem 12. (a)A positive definite rectangular tensor is a rectangular P-tensor and a rectangular P-tensor is a rectangular S-tensor. The inverse implications are not true.(b)A positive definite rectangular tensor is a strictly copositive rectangular tensor, and a rectangular S-tensor is a strictly copositive rectangular tensor. The inverse implications are not true.

Proof. If is a positive definite rectangular tensor, which means that for all , , and and are even, thenthen,therefore, there exists some indices such thatthen is a rectangular P-tensor.
If is a rectangular P-tensor, by definitions of rectangular P-tensors and rectangular S-tensors, is a rectangular S-tensor. The conclusion of (b) can be obtained similarly by the definitions of positive definite rectangular tensors, copositive rectangular tensors, and rectangular S-tensors.

6. Conclusions

In this paper, based on the definition of V-singular value for rectangular tensors, we extend the concept of P-tensors and -tensors to rectangular P-tensors and rectangular -tensors. It is shown that all the V-singular values of rectangular P-tensors are positive. Some properties of quantities for rectangular P-tensors are given, and a necessary and sufficient condition for a rectangular tensor to be a rectangular P-tensor is also obtained. The rectangular S-tensor can be viewed as a generalization of S-tensors, and an example is constructed to illustrate that the principal rectangular subtensor of a rectangular S-tensor is not always a rectangular S-tensor. Finally, we introduced the rectangular tensor complementarity problem, an equivalent definition of rectangular S-tensors is given by means of the solution of the rectangular tensor complementarity problem.

By the definition of H-singular value for rectangular tensors, another definitions of rectangular P-tensors and rectangular P₀-tensors can be given as follows.

Definition 7. A tensor is called a rectangular HP-tensor, if for each , and , there exists some indices such thatA tensor is called a rectangular -tensor, if for each and , there exists some indices such thatThen, all the results for rectangular P-tensors and rectangular P₀-tensors can be extended to rectangular HP-tensors and rectangular HP₀-tensors except Theorem 5, since the rectangular identity tensor for the definition of H-singular value is hard to define. Similarly, we can get that a positive definite rectangular tensor is a rectangular HP-tensor and a rectangular HP-tensor is a rectangular S-tensor.

Data Availability

No additional data are available for this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by NSF of China (no. 11661084), Innovative Talent Team in Guizhou Province (Qian Ke He Pingtai Rencai[2016]5619), Project of Teaching Quality and Teaching Reform of Higher Education in Guizhou Province (Qian Jiao gaofa[2015]337), and New Academic Talents and Innovative Exploration Fostering Project (Qian Ke He Pingtai Rencai[2017]5727-21), Zun Shi Ke He HZ Zhi[2020]27, [2020]30.

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Copyright © 2020 Qingyu Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mathematical Problems in Engineering

Advances in Numerical Optimisation: Theory, Models, and Applications

Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

Abstract

1. Introduction

2. Notation and Preliminaries

3. Rectangular P-Tensors and Rectangular P0-Tensors

4. Rectangular S-Tensor

5. Rectangular Tensor Complementarity Problems

6. Conclusions

Data Availability

Conflicts of Interest

Acknowledgments

References

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3. Rectangular P-Tensors and Rectangular P₀-Tensors