Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

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.


Introduction
Consider the following m degree homogeneous polynomial: where Ax m � n i 1 ,...,i m �1 a i 1 ...i m x i 1 . . . x i m and A ∈ R [m,n] is an mth order n-dimensional real square tensor. When m is even, the positive definiteness of f(x) in (1) plays an important role in automatic control [1]. In order to verify the positive definiteness of f(x) in (1), Qi introduced the definitions of H-eigenvalue and Z-eigenvalue of A and showed that when m is even, A is positive definite (i.e., f(x) in (1) is positive definite) if and only if all H-eigenvalues or Z-eigenvalues of A are positive [2][3][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. e P-tensors and P 0 -tensors are first introduced by Song and Qi [10], which can be viewed as generalizations of the P-matrices and P 0 -matrices [11]. e 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 P 0 -tensor. Another definition of P-tensors (P 0 -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 p + q degree homogeneous polynomial: f(x, y) � Ax p y q , (2) where Ax p y q � m i 2 ,...,i p �1 n j 1 ,...,j q �1 a i 1 ...i p j 1 ...j q x i 1 . . . x i p y j 1 . . . y j q . (3) A � (a i 1 ...i p j 1 ...j q ) ∈ R [p;q;m;n] is a (p, q)th order (m × n)-dimensional real rectangular tensor. A is called a real partially symmetric rectangular tensor, if a i 1 ...i p j 1 ...j q is invariant under any permutation of indices among i 1 . . . i p , and any permutation of indices among j 1 . . . j q , i.e., where S r is the permutation group of r indices. Let A ∈ R [p;q;m;n] be a partially symmetric rectangular tensor, and p and q are even. en, A is positive definite if and only if all of its H-singular values (or V-singular values) are positive [13][14][15][16][17][18][19]. e 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]. e 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 P 0 -tensors. Moreover, we prove that all the V-singular values of rectangular P-tensors (rectangular P 0 -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.

Notation and Preliminaries
In this section, we list some definitions related to rectangular tensors, which are needed in the subsequent analysis.
Let R and C be the real field and complex field, e ith entry of a vector x is denoted by x i , the (i, j)th entry of a matrix A is denoted by a ij , and the (i 1 , . . . , i p , j 1 , . . . , j q )th entry of a rectangular tensor A is denoted by a i 1 ...i p j 1 ...j q . Let R n be the n-dimensional real Euclidean space and the set of all nonnegative vectors in R n be denoted by R n + .

Definition 1.
A rectangular A ∈ R [p;q;m;n] is said to be (a) A positive definite rectangular tensor [13,14], iff Ax p y q > 0 for all x ∈ R m / 0 { } and y ∈ R n / 0 { } (b) A copositive rectangular tensor [21], iff Ax p y q ≥ 0 for all x ∈ R m + and y ∈ R n + (c) A strictly copositive rectangular tensor [21], iff Ax p y q > 0 for all In order to verify the positive definiteness of a (p, q)th order (m × n)-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 A ∈ R [p;q;m;n] , if there exist a number λ ∈ R, vectors x ∈ R m / 0 { } and y ∈ R n / 0 { } such that where a ii 2 ,...,i p jj 1, ...,j q x i 2 . . . x i p y j 2 . . . y j q , then λ is called the H-singular value of A and (x, y) is the left and right H-eigenvectors pair of A, associated with λ. Some sufficient conditions for the positive definiteness of a (p, q)th order (m × n)-dimensional partially symmetric rectangular tensor are given in [14], based on the following definition of the V-singular value.
Definition 3 (see [14]). Let A ∈ R [p;q;m;n] ; if there exist a number λ ∈ R, vectors x ∈ R m / 0 { }, p, q ≥ 2, and y ∈ R n / 0 { } such that then λ is called the V-singular value of A and (x, y) is the left and right V-eigenvectors pair of A, associated with λ. e definitions of P-tensors and P 0 -tensors are listed as follows.
Definition 4 (see [10]). A tensor A � (a i 1 ...i m ) ∈ R [m,n] is called a P-tensor if for each nonzero x ∈ R n , there exists some index i such that where is called a P 0 -tensor if for each nonzero x ∈ R n , there exists some index i such that

Rectangular P-Tensors and Rectangular P 0 -Tensors
We now introduce the definitions of rectangular P-tensors and rectangular P 0 -tensors.
e following result is given to show the positivity (nonnegativity) of the V-singular values for a rectangular P-tensor (P 0 -tensor).
Proof. If A is a rectangular P-tensor, λ is a V-singular value of A with eigenvectors pair (x, y); then, we have and then, there exists some indices i ∈ [m], j ∈ [n] such that By the definition of rectangular P-tensors, we have λ > 0. e case for rectangular P 0 -tensors can be obtained similarly.
A rectangular tensor C ∈ R [p;q;r m ;r n ] is called a principal rectangular subtensor of a rectangular tensor A � (a i 1 i 2 ,...,i p j 1 j 2 ,...,j q ) ∈ R [p;q;m;n] iff the sets I ⊆ [m], J ⊆ [n] contain r m and r n elements such that Let x I be a r m -dimensional subvector of a vector x ∈ R m and y J be a r n -dimensional subvector of a vector y ∈ R n . Note that, for r m � r n � 1, the principal rectangular subtensors are just the diagonal entries.
Proof. If λ is a V-singular value of A with eigenvectors pair (x, y), then we have which implies ..,i p j 1 j 2 ,...,j q ) ∈ R [p;q;m;n] be a rectangular P-tensor (P 0 -tensor). en, every principal rectangular subtensor of A is a rectangular P-tensor (P 0 -tensor).
In particular, all the diagonal entries of a rectangular P-tensor (P 0 -tensor) tensor are positive (nonnegative).
Proof. Let A I,J r m ,r n be a principal rectangular subtensor of A: which implies that A I,J r m ,r n is a rectangular P-tensor. e case for rectangular P 0 -tensors can be obtained similarly.

Mathematical Problems in Engineering
A sufficient and necessary condition for a rectangular tensor to be a rectangular P-tensor is given as follows. □

en, A is a rectangular P-tensor if and only if for each nonzero
x ∈ R m and y ∈ R n , there exists positive diagonal matrices D x , D y such that Proof. If A is a rectangular P-tensor, then there exists some en, for enough small μ, ] > 0, we have erefore, we have where . . , e n ) with e l � 1 and e j � ] for j ≠ l.
On the contrary, if there exists positive diagonal matrices such that , and a quantity α(A) of a P-matrix A is introduced in [11]. In 2015, let where Song and Qi introduced the definitions of quantities α(T A ) and α(F A ) for a P-tensor A and obtained monotonicity and boundedness of such two quantities, and they also showed that a tensor A is a P-tensor if and only if α(T A ) is positive, and a tensor A with even order is a P-tensor if and only if α(F A ) is positive [8]. We define the following two quantities for rectangular P-tensors: We present some properties of quantities for rectangular P-tensors. Proof. Let A I,J r m ,r n be a principal rectangular subtensor of A: Mathematical Problems in Engineering en, Based on the quantities α x (A) and α y (A), a necessary and sufficient conditions for a rectangular tensor to be a rectangular P-tensor is given as follows. en, Conversely, if α x (A) > 0 and α y (A) > 0, we have which implies A is a rectangular P-tensor. e case for rectangular P 0 -tensors can be obtained similarly.

Rectangular S-Tensor
Definition 6. A rectangular tensor A � (a i 1 i 2 ,...,i p j 1 j 2 ,...,j q ) ∈ R [p;q;m;n] is called a rectangular S-tensor if and only if there exists 0 < x ∈ R m and 0 < y ∈ R n such that

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A rectangular tensor A � (a i 1 i 2 ,...,i p j 1 j 2 ,...,j q ) ∈ R [p;q;m;n] is called a rectangular S 0 -tensor if and only if there exists e conditions 0 < x ∈ R m and 0 < y ∈ R n in the definition of rectangular S-tensors can be relaxed to

en, A is a rectangular S-tensor if and only if there exists
Proof. e necessity is obvious by the definition of rectangular S-tensors. We prove the sufficiency as follows. If Ax p y q−1 > 0.

(47)
Let e m � (1, . . . , 1) T and e n � (1, . . . , 1) T ; for some small enough t > 0, we have which means that A is a rectangular S-tensor. From eorem 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.
Let A I,J r m ,r n be a principal rectangular subtensor of A with I � J � 2 { }; then, for any x > 0 and y > 0, we have which means that A I,J r m ,r n 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.
Proof. Since A is a rectangular S-tensor, then there exists 0 < x ∈ R m and 0 < y ∈ R n such that Let x t � max i∈ [m] x i and y s � max j∈[n] y j ; then, x t > 0 and y s > 0, and 6 Mathematical Problems in Engineering  . If there exists a principal rectangular subtensor A I,J r m ,r n of A is a rectangular S-tensor, a ii 2 ,...,i p j 1 ,...,j q > 0 for all i 2 , . . . , i p ∈ I, j 1 , . . . , j q ∈ J, i ∉ I and a i 1 ,...,i p jj 2 ,...,j q > 0 for all i 1 , . . . , i p ∈ I and j 2 , . . . , j q ∈ J, j ∉ J; then, A is a rectangular S-tensor.
Proof. Since A I,J r m ,r n is a rectangular S-tensor, then there exists 0 < x ∈ R r m and 0 < y ∈ R r n such that (56) en, for any i ∈ I and j ∈ J, we have a ii 2 ,...,i p j 1 ,...,j q x i 2 , . . . , x i p y j 1 , . . . , y j q , (58) erefore, if i ∈ I, we obtain if i ∉ I, we obtain a ii 2 ,...,i p j 1 ,...,j q x i 2 , . . . , x i p y j 1 , . . . , y j q > 0, which implies that (Ax p−1 y q ) i > 0. Similarly, we have (Ax p y q− 1 ) j > 0. en, A is a rectangular S-tensor. A sufficient and necessary condition for a rectangular tensor to be a rectangular S-tensor is given as follows. □

en, A is a rectangular S-tensor if and only if there exists
Mathematical Problems in Engineering 0 < x ∈ R m and 0 < y ∈ R n , for any nonzero nonnegative diagonal matrices D x and D y such that Proof. If A is a rectangular S-tensor and 0 < x ∈ R m and 0 < y ∈ R n , then for any k ∈ [m] and l ∈ [n] such that en, for any μ, ] ≥ 0, we have erefore, we have where such that For any i ∈ [m] and j ∈ [n], let D x � diag(d 1 , d 2 , . . . , d m ) with d i � 1 and d k � 0 for k ≠ i and D y � diag(e 1 , e 2 , . . . , e n ) with e j � 1 and e l � 0 for l ≠ j, and we obtain By the conditions x ∈ R m > 0 and y ∈ R n > 0, we have which means that A is a rectangular S-tensor.

Rectangular Tensor Complementarity Problems
Converting a bimatrix game F(A, A) 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 A � (a i 1 i 2 ,...,i p j 1 j 2 ,...,j q ) ∈ R [p;q;m;n] , q m ∈ R m and q n ∈ R n . e rectangular tensor complementarity problem, denoted by RTCP (A, q m , q n ), is to find vectors x ∈ R m and y ∈ R n such that q m + Ax p−1 y q ≥ 0, x ≥ 0, x T q m + Ax p− 1 y q � 0, q n + Ax p y q− 1 ≥ 0, y ≥ 0, y T q n + Ax p y q− 1 � 0.

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

Proof. If
A is a rectangular S-tensor and 0 < x ∈ R m and 0 < y ∈ R n , then Ax p−1 y q > 0, Ax p y q−1 > 0.
(72) en, for each q m ∈ R m and q n ∈ R n , there exists some scalar t > 0 such that which means that t (1/p− 1) x and y are the feasible vectors of the RTCP (A, q m , q n ).
On the contrary, if the RTCP (A, q m , q n ) is feasible for all q m ∈ R m and q n ∈ R n , assume that q m < 0, q n < 0, and x and y are the feasible solution of the RTCP (A, q m , q n ).
(74) erefore, en, A is a rectangular S-tensor by eorem 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. e 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. e inverse implications are not true.
Proof. If A is a positive definite rectangular tensor, which means that Ax p y q > 0 for all x ∈ R m / 0 { }, y ∈ R n / 0 { }, and p and q are even, then then, therefore, there exists some indices i 0 ∈ [m] and j 0 ∈ [n] such that then A is a rectangular P-tensor. If A is a rectangular P-tensor, by definitions of rectangular P-tensors and rectangular S-tensors, A is a rectangular S-tensor. e conclusion of (b) can be obtained similarly by the definitions of positive definite rectangular tensors, copositive rectangular tensors, and rectangular S-tensors.

Conclusions
In this paper, based on the definition of V-singular value for rectangular tensors, we extend the concept of P-tensors and P 0 -tensors to rectangular P-tensors and rectangular P 0 -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. e 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 P 0 -tensors can be given as follows.
en, all the results for rectangular P-tensors and rectangular P 0 -tensors can be extended to rectangular HPtensors and rectangular HP 0 -tensors except eorem 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 HPtensor and a rectangular HP-tensor is a rectangular S-tensor.

Data Availability
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Conflicts of Interest
e authors declare that they have no conflicts of interest.