# Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

Authors: Qingyu Zeng; Jun He; Yanmin Liu
Journal: Mathematical Problems in Engineering
 (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.


---

## Body

## 1. Introduction

Consider the followingm degree homogeneous polynomial:(1)fx=Axm,where Axm=∑i1,…,im=1nai1…imxi1…xim and A∈ℝm,n is an mth order n-dimensional real square tensor. When m is even, the positive definiteness of fx in (1) plays an important role in automatic control [1]. In order to verify the positive definiteness of fx 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., fx in (1) is positive definite) if and only if all H-eigenvalues or Z-eigenvalues of A 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 andP0-tensors are first introduced by Song and Qi [10], which can be viewed as generalizations of the P-matrices and P0-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 P0-tensor. Another definition of P-tensors (P0-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 followingp+q degree homogeneous polynomial:(2)fx,y=Axpyq,where(3)Axpyq=∑i2,…,ip=1m∑j1,…,jq=1nai1…ipj1…jqxi1…xipyj1…yjq.A=ai1…ipj1…jq∈ℝp;q;m;n is a p,qth order m×n-dimensional real rectangular tensor. A is called a real partially symmetric rectangular tensor, if ai1…ipj1…jq is invariant under any permutation of indices among i1…ip, and any permutation of indices among j1…jq, i.e.,(4)aπi1…ipσj1…jq=ai1…ipj1…jq,π∈Sp,σ∈Sq,where Sr is the permutation group of r indices. Let A∈ℝp;q;m;n be a partially symmetric rectangular tensor, and p and q are even. Then, A 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 Section2, 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 P0-tensors. Moreover, we prove that all the V-singular values of rectangular P-tensors (rectangular P0-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.Letℝ and ℂ be the real field and complex field, m=1,2,…,m. We use small letters a, b, … for scalars, small bold letters x,y, … for vectors, capital letters A, B, … for matrices, and calligraphic letters A, ℬ, … for tensors. The ith entry of a vector x is denoted by xi, the i,jth entry of a matrix A is denoted by aij, and the i1,…,ip,j1,…,jqth entry of a rectangular tensor A is denoted by ai1…ipj1…jq. Let ℝn be the n-dimensional real Euclidean space and the set of all nonnegative vectors in ℝn be denoted by ℝ+n.Definition 1.
A rectangularA∈ℝp;q;m;n is said to be(a)
A positive definite rectangular tensor [13, 14], iff Axpyq>0 for all x∈ℝm/0 and y∈ℝn/0(b)
A copositive rectangular tensor [21], iff Axpyq≥0 for all x∈ℝ+m and y∈ℝ+n(c)
A strictly copositive rectangular tensor [21], iff Axpyq>0 for all x∈ℝ+m/0 and y∈ℝ+n/0
In order to verify the positive definiteness of ap,qth 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∈ℝp;q;m;n, if there exist a number λ∈ℝ, vectors x∈ℝm/0 and y∈ℝn/0 such that(5)Axp−1yq=λxM−1,Axpyq−1=λyM−1,where xM−1=x1M−1,…,xmM−1T, M=p+q, and(6)Axp−1yqi=∑i2,…,ip=1m∑j1,…,jq=1naii2,…,ipj1,…,jqxi2…xipyj1…yjq,Axpyq−1j=∑i1,…,ip=1m∑j2,…,jq=1naii2,…,ipjj1,…,jqxi2…xipyj2…yjq,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 ap,qth 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∈ℝp;q;m;n; if there exist a number λ∈ℝ, vectors x∈ℝm/0, p,q≥2, and y∈ℝn/0 such that(7)Axp−1yq=λxp−1,Axpyq−1=λyq−1,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.
The definitions of P-tensors and P0-tensors are listed as follows.Definition 4.
(see [10]). A tensor A=ai1…im∈ℝm,n is called a P-tensor if for each nonzero x∈ℝn, there exists some index i such that(8)xim−1Axm−1i>0,where Axm−1=∑i2,…,im=1nai1…imxi2…xim. A tensor A=ai1,…,im∈ℝm,n is called a P0-tensor if for each nonzero x∈ℝn, there exists some index i such that(9)xim−1Axm−1i≥0.


## 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 tensorA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n is called a rectangular P-tensor if for each x∈ℝm/0 and y∈ℝn/0, there exists some indices i∈m,j∈n such that(10)xip−1Axp−1yqi>0,yjq−1Axpyq−1j>0.
A rectangular tensorA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n is called a rectangular P0-tensor if for each x∈ℝm/0 and y∈ℝn/0, there exists some indices i∈m,j∈n such that(11)xip−1Axp−1yqi≥0,yjq−1Axpyq−1j≥0.
The following result is given to show the positivity (nonnegativity) of the V-singular values for a rectangular P-tensor (P0-tensor).Theorem 1.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular P-tensor (P0-tensor); then, all the V-singular values of A are positive (nonnegative).Proof.
IfA is a rectangular P-tensor, λ is a V-singular value of A with eigenvectors pair x,y; then, we have(12)Axp−1yq=λxp−1,Axpyq−1=λyq−1,and then, there exists some indices i∈m,j∈n such that(13)xip−1Axp−1yqi=λxi2p−1>0,yjq−1Axpyq−1j=λyj2q−1>0.
By the definition of rectangular P-tensors, we haveλ>0. The case for rectangular P0-tensors can be obtained similarly.
A rectangular tensorC∈ℝp;q;rm;rn is called a principal rectangular subtensor of a rectangular tensor A=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n iff the sets I⊆m,J⊆n contain rm and rn elements such that(14)C=Arm,rnI,J=ci1i2…ipj1j2…jq,for alli1,i2,…,ip∈I,j1,j2,…,jq∈J.
LetxI be a rm-dimensional subvector of a vector x∈ℝm and yJ be a rn-dimensional subvector of a vector y∈ℝn. Note that, for rm=rn=1, the principal rectangular subtensors are just the diagonal entries.Lemma 1.
LetA=ai0,…,i0j0,…,j0∈ℝp;q;1;1 with V-singular value λ. Then,(15)λ=ai0,…,i0j0,…,j0.Proof.
Ifλ is a V-singular value of A with eigenvectors pair x,y, then we have(16)Axp−1yq=λxp−1,Axpyq−1=λyq−1,which implies(17)ai0,…,i0j0,…,j0xi0p−1yj0q=λxi0p−1,ai0,…,i0j0,…,j0xi0pyj0q−1=λyj0q−1,where x=xi0,y=yj0, and xi0≠0,yj0≠0. Then, by xi0p=1andyj0q=1, we have(18)λ=ai0,…,i0j0,…,j0.Theorem 2.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular P-tensor (P0-tensor). Then, every principal rectangular subtensor of A is a rectangular P-tensor (P0-tensor). In particular, all the diagonal entries of a rectangular P-tensor (P0-tensor) tensor are positive (nonnegative).Proof.
LetArm,rnI,J be a principal rectangular subtensor of A:(19)xI=xi1,…,xirmT∈ℝrm,yJ=yj1,…,yrnT∈ℝrn.
x∗=x1∗,…,xim∗T∈ℝm with xi∗=xi if i∈I and xi∗=0 if i∉I, and y∗=y1∗,…,yjm∗T∈ℝn with yj∗=yj if j∈J and yj∗=0 if j∉J. If A∈ℝp;q;m;n is a rectangular P-tensor, there exists some index i∈m,j∈n such that(20)xi∗p−1Ax∗p−1y∗qi=xip−1Arm,rnI,JxIp−1yJqi>0,yj∗p−1Ax∗py∗q−1j=yjq−1Arm,rnI,JxIpyJq−1j>0,which implies that Arm,rnI,J is a rectangular P-tensor. The case for rectangular P0-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.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n. Then, A is a rectangular P-tensor if and only if for each nonzero x∈ℝmandy∈ℝn, there exists positive diagonal matrices Dx,Dy such that(21)xp−1TDxAxp−1yq>0,yq−1TDyAxpyq−1>0.Proof.
IfA is a rectangular P-tensor, then there exists some index i∈m,j∈n such that(22)xkp−1Axp−1yqk>0,ylq−1Axpyq−1l>0.
Then, for enough smallμ,ν>0, we have(23)xkp−1Axp−1yqk+μ∑i=1,i≠kmxip−1Axp−1yqi>0,ylq−1Axpyq−1l+ν∑j=1,j≠lnyjq−1Axpyq−1j>0.
Therefore, we have(24)xp−1TDxAxp−1yq>0,yq−1TDyAxpyq−1>0,where Dx=diagd1,d2,…,dm with dk=1 and di=μ for i≠k and Dy=diage1,e2,…,en with el=1 and ej=ν for j≠l.
On the contrary, if there exists positive diagonal matrices(25)Dx=diagd1,d2,…,dm,Dy=diage1,e2,…,en,such that(26)xp−1TDxAxp−1yq>0,yq−1TDyAxpyq−1>0.
Sincedi>0 for all i∈m, ej>0 for all j∈n, then there exists k∈m and l∈n such that(27)xkp−1Axp−1yqk>0,ylq−1Axpyq−1l>0.
Letx∞:=maxxi,i∈n, and a quantity αA of a P-matrix A is introduced in [11]. In 2015, let(28)αTA=minx∞=1maxi∈nxiTAxi,αFA=minx∞=1maxi∈nxiFAxi,where(29)TAx=x22−mAxm−1,FAx=Axm−11/m−1.
Song and Qi introduced the definitions of quantitiesαTA and αFA 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 αTA is positive, and a tensor A with even order is a P-tensor if and only if αFA is positive [8]. We define the following two quantities for rectangular P-tensors:(30)αxA=minx∞=1maxi∈mxip−1Axp−1yqi,αyA=miny∞=1maxj∈nyjq−1Axpyq−1j.
We present some properties of quantities for rectangular P-tensors.Theorem 4.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular P0-tensor and Arm,rnI,J be a principal rectangular subtensor of A. Then,(i)
αxA≤αxArm,rnI,J(ii)
αyA≤αyArm,rnI,JProof.
LetArm,rnI,J be a principal rectangular subtensor of A:(31)xI=xi1,…,xirmT∈ℝrm/0,yJ=yj1,…,yrnT∈ℝrn/0.
x∗=x1∗,…,xim∗T∈ℝm with xi∗=xi if i∈I and xi∗=0 if i∉I, and y∗=y1∗,…,yjm∗T∈ℝn with yj∗=yj if j∈J and yj∗=0 if j∉J. Then, x∗∞=xI∞ and y∗∞=yJ∞. Hence,(32)αxA=minx∞=1maxi∈mxip−1Axp−1yqi≤minx∗∞=1maxi∈mxi∗p−1Ax∗p−1y∗qi=minxI∞=1maxi∈Ixip−1Arm,rnI,JxIp−1yJqi=αxArm,rnI,J,αyA=miny∞=1maxj∈nyjq−1Axpyq−1j≤miny∗∞=1maxi∈nyi∗q−1Ax∗p−1y∗q−1j=minyI∞=1maxj∈Jyiq−1Arm,rnI,JxIpyJq−1j=αyArm,rnI,J.
LetδA:=minλArm,rnI,J;I⊆m,J⊆n, where λA denotes the smallest of V-singular value (if any exists) of a rectangular P-tensor A. Then, we have the following upper bounds for αxA and αyA. Let ℐR=ϵi1,…,ipj1,…,jq∈ℝp;q;n;n be the rectangular identity tensor with(33)ϵi1,…,ipj1,…,jq=1,ifi=,…,=ip,j1=,…,=jq,0,otherwise.Theorem 5.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular P-tensor. Then,(i)
αxA≤δA≤mini∈m,j∈nai,…,ij,…,j(ii)
αyA≤δA≤mini∈m,j∈nai,…,ij,…,jProof.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular P-tensor, by Theorem 2, and we have(34)ai,…,ij,…,j>0,for alli∈m,j∈n.
SinceA1,1I,J is a principal rectangular subtensor of A, by Lemma 1, ai…ij…j is a V-singular value of A1,1I,J; therefore,(35)δA≤ai,…,ij,…,j.
Furthermore,δA is a V-singular value of A1,1I,J; then, A1,1I,J−δAℐRrm,rnI,J is not a rectangular P-tensor. By Theorem 2, A−δAℐR is not a rectangular P-tensor. Then, there exists vectors xandy with x∞=1 and y∞=1 such that(36)maxi∈mxip−1A−δAℐRxp−1yqi≤0,maxj∈nyjq−1A−δAℐRxpyq−1j≤0.
Then,(37)xkp−1Axp−1yqk−xkp−1δAℐRxp−1yqk≤0,ylq−1Axpyq−1l−ylq−1δAℐRxpyq−1l≤0,for all k∈m,l∈n. Then,(38)maxi∈mxip−1Axp−1yqi=xi0p−1Axp−1yqi0≤xi0p−1δAℐRxp−1yqi0≤δAx∞y∞=δA.
Similarly, we have(39)maxj∈nyjq−1Axpyq−1j≤δA.
Therefore,(40)αxA≤δA≤mini∈m,j∈nai,…,ij,…,j,αyA≤δA≤mini∈m,j∈nai,…,ij,…,j.
Based on the quantitiesαxA and αyA, a necessary and sufficient conditions for a rectangular tensor to be a rectangular P-tensor is given as follows.Theorem 6.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n. Then, A is a rectangular P-tensor (P0-tensor) if and only if αxA and αyA are positive (nonnegative).Proof.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular P-tensor. Then, for each x∈ℝm/0 and y∈ℝn/0, there exists some index i∈m,j∈n such that(41)xip−1Axp−1yqi>0,yjq−1Axpyq−1j>0.
Then,(42)αxA=minx∞=1maxi∈mxip−1Axp−1yqi>0,αyA=miny∞=1maxj∈nyjq−1Axpyq−1j>0.
Conversely, ifαxA>0 and αyA>0, we have(43)xip−1Axp−1yqi>0,yjq−1Axpyq−1j>0,which implies A is a rectangular P-tensor. The case for rectangular P0-tensors can be obtained similarly.


## 4. Rectangular S-Tensor

Definition 6.
A rectangular tensorA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n is called a rectangular S-tensor if and only if there exists 0<x∈ℝm and 0<y∈ℝn such that(44)Axp−1yq>0,Axpyq−1>0.
A rectangular tensorA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n is called a rectangular S0-tensor if and only if there exists 0≤x∈ℝm/0 and 0≤y∈ℝn/0 such that(45)Axp−1yq≥0,Axpyq−1≥0.
The conditions0<x∈ℝm and 0<y∈ℝn in the definition of rectangular S-tensors can be relaxed to 0≤x∈ℝm/0 and 0≤y∈ℝn/0.Theorem 7.
LetA=ai1i2…ipj1j2…jq∈ℝp;q;m;n. Then, A is a rectangular S-tensor if and only if there exists 0≤x∈ℝm/0 and 0≤y∈ℝn/0 such that(46)Axp−1yq>0,Axpyq−1>0.Proof.
The necessity is obvious by the definition of rectangular S-tensors. We prove the sufficiency as follows.
If there exists0≤x∈ℝm/0 and 0≤y∈ℝn/0 such that(47)Axp−1yq>0,Axpyq−1>0.
Letem=1,…,1T and en=1,…,1T; for some small enough t>0, we have(48)Ax+temp−1y+tenq>0,Ax+tempy+tenq−1>0,which means that A is a rectangular S-tensor.
From Theorem2, 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.
LetA=ai1i2j1j2∈ℝ2;2;2;2, where(49)a1111=1,a1112=−1,a2221=1,a2222=−1,and all other ai1i2j1j2=0. Then, for any 0≤x∈ℝm/0 and 0≤y∈ℝn/0, we obtain(50)Axy2=x1y12−x1y1y2x2y2y1−x2y22=x1y1y1−y2x2y2y1−y2,Ax2y=x12y1−x12y2x22y1−x22y2=x12y1−y2x22y1−y2.
Then, fory=2,1T>0 and x=1,1T>0, Axy2>0 and Ax2y>0, which implies, A is a rectangular S-tensor.
LetArm,rnI,J be a principal rectangular subtensor of A with I=J=2; then, for any x>0 and y>0, we have(51)Arm,rnI,Jxy2=−1xy2<0,Arm,rnI,Jx2y=−1x2y<0,which means that Arm,rnI,J 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.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n be a rectangular S-tensor. Then, there exists i∈m and j∈n such that(52)ai,…,ij,…,j+∑aii2,…,ipj1,…,jq>0,ii2,…,ipj1,…,jq≠i,…,ij,…,jaii2,…,ipj1,…,jq>0.Proof.
SinceA is a rectangular S-tensor, then there exists 0<x∈ℝm and 0<y∈ℝn such that(53)Axp−1yqi>0,i∈m.
Letxt=maxi∈mxi and ys=maxj∈nyj; then, xt>0 and ys>0, and(54)0<Axp−1yqt=∑i2,…,ip=1m∑j1,…,jq=1nati2,…,ipj1,…,jqxi2,…,xipyj1,…,yjq=at,…,ts,…,sxtp−1ysq+∑ati2,…,ipj1,…,jq>0,ti2,…,ipj1,…,jq≠t,…,ts,…,sati2,…,ipj1,…,jqxi2,…,xipyj1,…,yjq+∑ati2,…,ipj1,…,jq<0,ti2,…,ipj1,…,jq≠t,…,ts,…,sati2,…,ipj1,…,jqxi2,…,xipyj1,…,yjq≤at,…,ts,…,sxtp−1ysq+∑ati2,…,pj1,…,jq>0,ti2,…,ipj1,…,jq≠t,…,ts,…,sati2,…,ipj1,…,jqxi2,…,xipyj1,…,yjq≤at…ts…s+∑ati2,…,ipj1,…,jq>0,ti2,…,ipj1,…,jq≠t,…,ts,…,sati2,…,ipj1,…,jqxtp−1ysq,which means(55)at,…,ts,…,s+∑ati2,…,ipj1,…,jq>0,ti2,…,ipj1,…,jq≠t,…,ts,…,sati2,…,ipj1,…,jq>0.Theorem 9.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n. If there exists a principal rectangular subtensor Arm,rnI,J of A is a rectangular S-tensor, aii2,…,ipj1,…,jq>0 for all i2,…,ip∈I, j1,…,jq∈J, i∉I and ai1,…,ipjj2,…,jq>0 for all i1,…,ip∈I and j2,…,jq∈J, j∉J; then, A is a rectangular S-tensor.Proof.
SinceArm,rnI,J is a rectangular S-tensor, then there exists 0<x^∈ℝrm and 0<y^∈ℝrn such that(56)Ax^p−1y^qi>0,i∈I,Ax^py^q−1j>0,j∈J.
Let0≤x˜∈ℝm/0 and 0≤y˜∈ℝn/0 with(57)xi=x^i,ifi∈I,x˜i=0,ifi∉I,y˜j=y^j,ifj∈J,y˜j=0,ifj∉J.
Then, for anyi∈I and j∈J, we have(58)Ax˜p−1y˜qi=∑i2,…,ip=1m∑j1,…,jq=1naii2,…,ipj1,…,jqx˜i2,…,x˜ipy˜j1,…,y˜jq=∑it∈I,t∈2,…,m∑js∈J,s∈naii2,…,ipj1,…,jqx˜i2,…,x˜ipy˜j1,…,y˜jq,Axpyq−1j=∑i1,…,ip=1m∑j2,…,jq=1nai1,…,ipjj2,…,jqxi1,…,xipyj2,…,yjq=∑it∈I,t∈m∑js∈J,s∈2,…,nai2,…,ipjj2,…,jqx˜i1,…,x˜ipy˜j2,…,y˜jq.
Therefore, ifi∈I, we obtain(59)Ax˜p−1y˜qi=Arm,rnI,Jx˜p−1y˜qi>0,if i∉I, we obtain(60)Ax˜p−1y˜qi=∑it∈I,t∈2,…,m∑js∈J,s∈naii2,…,ipj1,…,jqx˜i2,…,x˜ipy˜j1,…,y˜jq>0,which implies that Ax˜p−1y˜qi>0. Similarly, we have Axpyq−1j>0. Then, 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.Theorem 10.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n. Then, A is a rectangular S-tensor if and only if there exists 0<x∈ℝmand0<y∈ℝn, for any nonzero nonnegative diagonal matrices DxandDy such that(61)xTDxAxp−1yq>0,yTDyAxpyq−1>0.Proof.
IfA is a rectangular S-tensor and 0<x∈ℝmand0<y∈ℝn, then for any k∈mandl∈n such that(62)xkAxp−1yqk>0,ylAxpyq−1l>0.
Then, for anyμ,ν≥0, we have(63)xkAxp−1yqk+μ∑i=1,i≠kmxiAxp−1yqi>0,ylAxpyq−1l+ν∑j=1,j≠lnyjAxpyq−1j>0.
Therefore, we have(64)xTDxAxp−1yq>0,yTDyAxpyq−1>0,where Dx=diagd1,d2,…,dm with dk=1 and di=μ for i≠k and Dy=diage1,e2,…,en with el=1 and ej=ν for j≠l.
On the contrary, for any nonzero nonnegative diagonal matrices(65)Dx=diagd1,d2,…,dm,Dy=diage1,e2,…,en,such that(66)xTDxAxp−1yq>0,yTDyAxpyq−1>0.
For anyi∈mandj∈n, let Dx=diagd1,d2,…,dm with di=1 and dk=0 for k≠i and Dy=diage1,e2,…,en with ej=1 and el=0 for l≠j, and we obtain(67)xTDxAxp−1yq=xiAxp−1yqi>0,yTDyAxpyq−1=yjAxpyq−1j>0.
By the conditionsx∈ℝm>0andy∈ℝn>0, we have(68)Axp−1yq>0,Axpyq−1>0,which means that A is a rectangular S-tensor.


## 5. Rectangular Tensor Complementarity Problems

Converting a bimatrix gameFA,A to a linear complementarity problem, we have the following LCP [22]:(69)u=qm+Ay≥0,x≥0,xTu=0,v=qn+ATx≥0,y≥0,yTv=0.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.LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n, qm∈ℝm and qn∈ℝn. The rectangular tensor complementarity problem, denoted by RTCP A,qm,qn, is to find vectors x∈ℝm and y∈ℝn such that(70)qm+Axp−1yq≥0,x≥0,xTqm+Axp−1yq=0,qn+Axpyq−1≥0,y≥0,yTqn+Axpyq−1=0.Vectorsx and y are said to be feasible iff x and y satisfy the following inequalities:(71)qm+Axp−1yq≥0,x≥0,qn+Axpyq−1≥0,y≥0.The following equivalent definition of rectangular S-tensor can be given by means of the solution of the RTCPA,qm,qn.Theorem 11.
LetA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n. Then, A is a rectangular S-tensor if and only if the RTCP A,qm,qn is feasible for all qm∈ℝm and qn∈ℝn.Proof.
IfA is a rectangular S-tensor and 0<x∈ℝmand0<y∈ℝn, then(72)Axp−1yq>0,Axpyq−1>0.
Then, for eachqm∈ℝm and qn∈ℝn, there exists some scalar t>0 such that(73)At1/p−1xp−1yq=tAxp−1yq≥−qm,At1/p−1xpyq−1=tp/p−1Axpyq−1≥−qn,which means that t1/p−1x and y are the feasible vectors of the RTCP A,qm,qn.
On the contrary, if the RTCPA,qm,qn is feasible for all qm∈ℝm and qn∈ℝn, assume that qm<0, qn<0, and x¯ and y¯ are the feasible solution of the RTCP A,qm,qn. Then,(74)qm+Ax¯p−1y¯q≥0,x¯≥0,qn+Ax¯py¯q−1≥0,y¯≥0.
Therefore,(75)Ax¯p−1y¯q≥−qm>0,Ax¯py¯q−1≥−qn>0.
Then,A 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.
IfA is a positive definite rectangular tensor, which means that Axpyq>0 for all x∈ℝm/0, y∈ℝn/0, and p and q are even, then(76)∑i=1mxiAxp−1yqi>0,∑j=1nyjAxpyq−1j>0,then,(77)∑i=1mxip−1Axp−1yqi>0,∑j=1nyjq−1Axpyq−1j>0,therefore, there exists some indices i0∈mandj0∈n such that(78)xi0p−1Axp−1yqi0>0,yj0q−1Axpyq−1j0>0,then A is a rectangular P-tensor.
IfA is a rectangular P-tensor, by definitions of rectangular P-tensors and rectangular S-tensors, A 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 andP0-tensors to rectangular P-tensors and rectangular P0-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 tensorA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n is called a rectangular HP-tensor, if for each x∈ℝm/0, and y∈ℝn/0, there exists some indices i∈m,j∈n such that(79)xiM−1Axp−1yqi>0,yjM−1Axpyq−1j>0.
A tensorA=ai1i2,…,ipj1j2,…,jq∈ℝp;q;m;n is called a rectangular HP0-tensor, if for each x∈ℝm/0 and y∈ℝn/0, there exists some indices i∈mandj∈n such that(80)xiM−1Axp−1yqi≥0,yjM−1Axpyq−1j≥0.
Then, 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.



---

*Source: 3897981-2020-08-24.xml*