# Complementarity Problem With Nekrasov $Z$ tensor

R. Deba,1 and A. K. Dasb,2
_a Jadavpur University, Kolkata, India._
_b Indian Statistical Institute, Kolkata, India._
_1 Email<EMAIL_ADDRESS>
_2 Email<EMAIL_ADDRESS>

###### Abstract

It is worth knowing that a particular tensor class belongs to $P$-tensor which
ensures the compactness to solve tensor complementarity problem (TCP). In this
study, we propose a new class of tensor, Nekrasov $Z$ tensor, in the context
of the tensor complementarity problem. We show that the class of $P$-tensor
contains the class of even ordered Nekrasov $Z$ tensors with positive diagonal
elements. In this context, we propose a procedure by which a Nekrasov $Z$
tensor can be transformed into a tensor which is diagonally dominant.

Keywords: Diagonally dominant tensor, Nekrasov tensors, Nonsingular $H$
tensor, Nekrasov $Z$ tensor, $P$-tensors, Tensor complementarity problem.

AMS subject classifications: 15A69, 99C33, 65F15

11footnotetext: Corresponding author

##  Introduction

$\mathscr{T}=(t_{j_{1}...j_{m}})$ has multidimensional array of elements
$t_{j_{1}...j_{m}}\in\mathbb{C}$ where $j_{i}\in[n]$ with $i\in[m],$
$[n]=\\{1,...,n\\}.$ The set of $m$th order $n$ dimensional complex (real)
tensors is denoted by $\mathbb{C}^{[m,n]}\;(\mathbb{R}^{[m,n]}).$ Several
classes of tensors have significant impact on different branches of science
and engineering. Applications of tensors can be found in physics, quantum
computing, diffusion tensor imaging, image authenticity verification problem,
spectral hypergraph theory, optimization theory and in several other areas. In
optimization theory, Song and Qi [47] introduced the tensor complementarity
problem (TCP) which is identified a subclass of the nonlinear complementarity
problem. Here functions are obtained with the help of a tensor.

Given $\mathscr{T}\in\mathbb{R}^{[m,n]}$ and $q\in\mathbb{R}^{n},$ TCP is to
obtain $x\in\mathbb{R}^{n}$ such that

$x\geq 0,\;\;\mathscr{T}x^{m-1}+q\geq 0,\;\;x^{T}(\mathscr{T}x^{m-1}+q)=0.$
(1.1)

This problem is denoted by TCP$(\mathscr{T},q)$ and the solution set of
TCP$(\mathscr{T},q)$ is denoted by SOL$(\mathscr{T},q).$ TCP are observed in
optimization theory, game theory and in other areas. Some mathematical problem
can be reformulated TCP, such as a class of multiperson noncooperative game
[15], hypergraph clustering problem and traffic equilibrium problem [16].

TCP can be viewed as one kind of extension of the linear complementarity
problem (LCP) where the involved functions are homogeneous polynomials
constructed with the help of a tensor. Given $M\in\mathbb{R}^{[2,n]}$ and
$q\in\mathbb{R}^{n}$, the LCP [3] is to obtain $u\in\mathbb{R}^{n}$ such that

$u\geq 0,\;\;Mu+q\geq 0,\;\;u^{T}(Mu+q)=0.$ (1.2)

The problem is denoted by LCP$(M,q)$ and the solution set of LCP$(M,q)$ is
denoted by SOL$(M,q).$ The stability of a large number of problems can be
explained with the formulation of a linear complementarity problem. For
details see [9], [33], [12] [13], [37], [20], [31], [36], [17], [21], [38],
[7], [8]. For the formulation of conflicting situations see [28], [34], [40],
[30], [29], [6], [35] and [27]. In connection with linear complementarity
problem computation methods are dependent on the matrix classes. For details
see [26], [25] [32], [5], [39], [17], [18], [21], [19].

Nekrasov $Z$-matrix was introduced by Orea and Pe$\tilde{n}$a [41] to study
the methods to compute its inverse with high relative accuracy.

Several matrix types play a crusial role in the theory of linear
complementarity problem, similarly, some structured tensors are important to
study tensor complementarity theory. In recent years some special types of
matrices are extended to tensors in connection with TCP. Among the several
tensor classes, the class of $P(P_{0})$-tensor, $Z$ tensor, $M$-tensor, $H$
tensor, and Nekrasov tensor are of special interest. $Z$ tensor and $M$-tensor
are introduced by Zhang et al. [51] in context of tensor complementarity
problem. They prove that for a $Z$ tensor the TCP has the least element
property. Ding et al. [11] introduce $H$ tensor and establish its importance
in the tensor complementarity problem. Zhang and Bu [50] introduce the
Nekrasov tensor and show that a Nekrasov tensor can be posed as a $H$ tensor.

Song and Qi [45] introduce $P$-tensor. Various properties of $P$-tensors are
considered in connection with TCP. For details see [1], [10], [49].

In this paper, we introduce Nekrasov $Z$ tensor. We study the connection of
Nekrasov $Z$ tensor and diagonally dominant tensor. We prove that the class of
Nekrasov $Z$ tensor of even order with positive diagonal elements is a
subclass of $P$-tensor.

The paper is organized as follows. Section 2 contains few essential and early
observations. In Section 3, we develop Nekrasov $Z$ tensor. We investigate its
connection with the $P$-tensor.

##  Preliminaries

Here we consider some basic notations and present few useful definitions and
theorems used in this paper. $x\in\mathbb{C}^{n}$ denotes a column vector and
row transpose of $x$ is denoted by $x^{T}.$ A diagonal matrix
$W=[w_{ij}]_{n\times n}=diag(w_{1},\;...,\;w_{n})$ is defined as
$w_{ij}=\left\\{\begin{array}[]{ll}w_{i}&;\;\forall\;i=j,\\\
0&;\;\forall\;i\neq j.\end{array}\right.$

######  definition 2.1:

[24] $A\in\mathbb{R}^{[2,n]},$ is a $Z$-matrix if all its off-diagonal
elements are nonpositive.

######  definition 2.2:

[4, 2, 22, 14] A matrix $M=(m_{ij})\in\mathbb{C}^{[2,n]},$ is a Nekrasov
matrix if $\lvert m_{ii}\rvert>\Lambda_{i}(M),\;\forall\;i\in[n],$ where

$\Lambda_{i}(M)=\left\\{\begin{array}[]{ll}\sum_{j=2}^{n}\lvert
m_{ij}\rvert&,i=1,\\\ \sum_{j=1}^{i-1}\lvert
m_{ij}\rvert\frac{\Lambda_{j}(M)}{\lvert m_{jj}\rvert}+\sum_{j=i+1}^{n}\lvert
m_{ij}\rvert&,i=2,3,...,n.\end{array}\right.$

######  definition 2.3:

[41] $A\in\mathbb{R}^{[2,n]}$ is Nekrasov $Z$-matrix, if $A$ is a Nekrasov
mtrix as well as a $Z$-matrix.

Let $\mathscr{O}$ be the zero tensor with zero at each entry of $\mathscr{O}.$
For $\mathscr{T}\in\mathbb{C}^{[m,n]}$ and
$x\in\mathbb{C}^{n},\;\mathscr{T}x^{m-1}\in\mathbb{C}^{n}$ is a vector defined
by

$(\mathscr{T}x^{m-1})_{i}=\sum_{i_{2},...,i_{m}=1}^{n}t_{i_{1}...i_{m}}x_{i_{2}}\cdots
x_{i_{m}},\mbox{ for all }i\in[n]$

$\mathscr{T}x^{m}\in\mathbb{C}$ is a scalar defined by

$x^{T}\mathscr{T}x^{m-1}=\mathscr{T}x^{m}=\sum_{i_{1},...,i_{m}=1}^{n}t_{i_{1}...i_{m}}x_{i_{1}}\cdots
x_{i_{m}}.$

Let $\mathscr{U}$ and $\mathscr{V}$ be two $n$ dimensional tensor of order
$p\geq 2$ and $r\geq 1,$ respectively. Shao [43] introduced the general
product of $\mathscr{U}$ and $\mathscr{V}.$ Let
$\mathscr{U}\cdot\mathscr{V}=\mathscr{T}.$ Then $\mathscr{T}$ is an $n$
dimensional tensor of order $((p-1)(r-1))+1$ where the elements of
$\mathscr{T}$ are given by

$t_{j\beta_{1}\cdots\beta_{p-1}}=\sum_{j_{2},\cdots,j_{p}\in[n]}u_{jj_{2}\cdots
j_{p}}v_{j_{2}\beta_{1}}\cdots v_{j_{p}\beta_{p-1}},$

where $j\in[n]$, $\beta_{1},\cdots,\beta_{p-1}\in[n]^{r-1}$

######  definition 2.4:

[48] Given $\mathscr{T}\in\mathbb{R}^{[m,n]}$ and $q\in\mathbb{R}^{n}$, $x$
solves TCP$(\mathscr{T},q)$ if there exists a $x$ satisfying the equation
(1.1).

######  definition 2.5:

[46] $\mathscr{T}\in\mathbb{R}^{[m,n]}$ is a $P$-tensor, if for any
$x\in\mathbb{R}^{n}\backslash\\{0\\}$, $\exists$ a $j\in[n]$ such that
$x_{j}\neq 0$ and $x_{j}(\mathscr{T}x^{m-1})_{j}>0.$

The row subtensors are defined in [44]. Here for the sake of convenience we
denote $i$th rowsubtensor of $\mathscr{T}$ by $\mathscr{R}_{i}(\mathscr{T}).$

######  definition 2.6:

[44] For each $i$ the $i$th row subtensor of
$\mathscr{T}\in\mathbb{C}^{[m,n]}$ is denoted by
$\mathscr{R}_{i}(\mathscr{T})$ and its elements are given as
$(\mathscr{R}_{i}(\mathscr{T}))_{i_{2}...i_{m}}=(t_{ii_{2}...i_{m}})$, where
$i_{l}\in[n]$ and $2\leq l\leq m.$

######  definition 2.7:

[50] Let $\mathscr{T}\in\mathbb{C}^{[m,n]}$ such that
$\mathscr{T}=(t_{i_{1}i_{2}...i_{m}})$ and

$R_{i}(\mathscr{T})=\sum_{(i_{2}...i_{m})\neq(i...i)}\lvert
t_{ii_{2}...i_{m}}\rvert,\;\forall\;i.$

The tensor $\mathscr{T}$ is diagonally dominant (strict diagonally dominant)
if $\lvert t_{i...i}\rvert\geq(>)R_{i}(\mathscr{T})$ for all $i\in[n].$

######  definition 2.8:

[50] $\mathscr{T}\in\mathbb{C}^{[m,n]}$ is quasidiagonally dominant tensor if
there exists a diagonal matrix $W=diag(w_{1},w_{2},\cdots,w_{n})$ such that
$\mathscr{T}W$ is strictly diagonally dominant tensor, i.e.,

$\lvert t_{i...i}\rvert w_{i}^{m-1}>\lvert t_{ii_{2}\cdots i_{m}}\rvert
w_{i_{2}}\cdots w_{i_{m}},\;\;\;\forall\;i\in[n].$

Qi [42] and Lim [23] proposed the idea of eigenvalues and eigenvectors in case
of tensors. If a pair
$(\lambda,x)\in\mathbb{C}\times(\mathbb{C}^{n}\backslash\ \\{0\\})$ satisfies
the equation $\mathscr{T}x^{m-1}=\lambda x^{m-1},$ then $\lambda$ is an
eigenvalue of $\mathscr{T}$, and $x$ is an eigenvector with respect to
$\lambda$ of $\mathscr{T}$, for which
$x^{[m−1]}=(x_{1}^{m−1},...,x_{n}^{m−1})^{T}.$ The spectral radius of
$\mathscr{T}$ is defined as
$\rho(\mathscr{T})=\max\\{|\lambda|\;:\;\lambda\text{ is an eigen value of
}\mathscr{T}\\}.$

######  definition 2.9:

[11] $\mathscr{T}$ is a $Z$ tensor if all its off diagonal elements are
nonpositive.

######  definition 2.10:

[11] A $Z$ tensor $\mathscr{T}$ is an $M$ tensor if $\exists$
$\mathscr{B}\;\geq\mathscr{O}$ and $s>0$ such that
$\mathscr{T}=s\mathscr{I}-\mathscr{B}$, where $s\geq\rho(\mathscr{B})$.
$\mathscr{T}$ is a nonsingular $M$ tensor if $s>\rho(\mathscr{B}).$

######  definition 2.11:

[11] For a tensor $\mathscr{T}=t_{i_{1}i_{2}...i_{m}}\in\mathbb{C}^{[m,n]},$
$\mathscr{C}(\mathscr{T})=c_{i_{1}i_{2}...i_{m}}$ is a comparison tensor of
$\mathscr{T}$ if

$c_{i_{1}i_{2}...i_{m}}=\left\\{\begin{array}[]{ll}|t_{i_{1}i_{2}...i_{m}}|&,\text{
if }(i_{1},i_{2},...,i_{m})=(i,i,...,i),\\\ -|t_{i_{1}i_{2}...i_{m}}|&,\text{
if }(i_{1},i_{2},...,i_{m})\neq(i,i,...,i).\end{array}\right.$

######  definition 2.12:

[11] $\mathscr{T}$ is an $H$ tensor, if its comparison tensor is an
$M$-tensor, and it is called a nonsingular $H$ tensor if its comparison tensor
is a nonsingular $M$-tensor.

######  definition 2.13:

[50] For
$\mathscr{T}=(t_{i_{1}i_{2}...i_{m}})\in\mathbb{C}^{[m,n]},\;t_{ii...i}\neq
0,$ $\forall\;i$ and

$\displaystyle\Lambda_{1}(\mathscr{T})$ $\displaystyle=R_{1}(\mathscr{T}),$
$\displaystyle\Lambda_{i}(\mathscr{T})$
$\displaystyle=\sum_{i_{2}...i_{m}\in[i-1]^{m-1}}\lvert
t_{ii_{2}...i_{m}}\rvert\left(\frac{\Lambda_{i_{2}}(\mathscr{T})}{\lvert
t_{i_{2}...i_{2}}\rvert}\right)^{\frac{1}{m-1}}\cdots\left(\frac{\Lambda_{i_{m}}(\mathscr{T})}{\lvert
t_{i_{m}...i_{m}}\rvert}\right)^{\frac{1}{m-1}}$
$\displaystyle\qquad+\sum_{i_{2}...i_{m}\notin[i-1]^{m-1},(i_{2}...i_{m})\neq(i...i)}\lvert
t_{ii_{2}...i_{m}}\rvert,\;\;\;\;i=2,3,...,n.$

$\mathscr{T}=(t_{i_{1}i_{2}...i_{m}})\in\mathbb{R}^{[m,n]}$ is a Nekrasov
tensor if

$\lvert t_{i...i}\rvert>\Lambda_{i}(\mathscr{T}),\;\forall\;i\in[n].$ (2.1)

######  theorem 2.1:

[11] $\mathscr{T}$ is a nonsingular $H$ tensor iff it is a quasi-strictly
diagonally dominant tensor.

######  theorem 2.2:

[50] If a tensor $\mathscr{T}=(t_{i_{1}i_{2}...i_{m}})\in\mathbb{C}^{[m,n]}$
is a Nekrasov tensor, then $\mathscr{T}$ is a nonsingular $H$ tensor.

######  theorem 2.3:

[10] A nonsingular $H$ tensor of even order with all positive diagonal
elements is a $P$-tensor.

######  theorem 2.4:

[1] For $q\in\mathbb{R}^{n}$ and a $P$-tensor
$\mathscr{T}\in\mathbb{R}^{[m,n]}$, the solution set of TCP$(\mathscr{T},q)$
is nonempty and compact.

##  Main results

We begin by introducing Nekrasov $Z$ tensor.

######  definition 3.14:

$\mathscr{T}\in\mathbb{R}^{[m,n]}$ is a Nekrasov $Z$ tensor if $\mathscr{T}$
is a Nekrasov tensor as well as a $Z$ tensor.

The following is an example of a Nekrasov $Z$ tensor.

######  example 3.1:

We consider the Example 3.3 of [50]. Let $\mathscr{B}\in\mathbb{R}^{[4,4]}$ be
such that
$b_{1111}=8,\;b_{2222}=3.8,\;b_{3333}=3,\;b_{4444}=10,\;b_{1112}=b_{2111}=b_{1211}=b_{1121}=-1,\;$
$b_{3222}=b_{2322}=b_{2232}=b_{2223}=-1,$
$b_{4441}=b_{4414}=b_{4144}=b_{1444}=-3$ and all other elements of
$\mathscr{B}$ are zeros. Then
$R_{1}(\mathscr{B})=6,\;R_{2}(\mathscr{B})=4,\;R_{3}(\mathscr{B})=1,\;R_{4}(\mathscr{B})=9.$
Since $|b_{2222}=|<R_{2}(\mathscr{B}),$ so $\mathscr{B}$ is not diagonally
dominant tensor. However
$\lambda_{1}(\mathscr{B})=6,\;\lambda_{2}(\mathscr{B})=3.75,\;\lambda_{3}(\mathscr{B})=0.98,\;\lambda_{4}(\mathscr{B})=9.$
Also $|b_{ii...i}|>\lambda_{i}(\mathscr{B}),\;\forall\;i\in[4].$ Hence
$\mathscr{B}$ is Nekrasov tensor. Also all the off-diagonal elements of
$\mathscr{B}$ are nonpositive. Therefore $\mathscr{B}$ is a $Z$ tensor. Hence
$\mathscr{B}$ is a Nekrasov $Z$ tensor.

Now we show that $P$-tensor contains even order Nekrasov $Z$ tensor with
positive diagonal elements.

######  theorem 3.1:

A Nekrasov $Z$ tensor of even order with positive diagonal elements is a
$P$-tensor.

###### Proof.

Let $\mathscr{T}$ be a Nekrasov $Z$ tensor which has positive diagonal
elements. By Theorem 2.2, a Nekrasv tensor is a $H$ tensor which is also
nonsingular. Therefore $\mathscr{T}$ is a nonsingular $H$ tensor which has
positive diagonal elements. Therefore by Theorem 2.3 $\mathscr{T}$ is a
$P$-tensor. ∎

Since a Nekrasov tensor is a nonsingular $H$ tensor, then there exists a
positive diagonal matrix $W$ such that $\mathscr{T}$ is a quasi-diagonally
dominant tensor. Here we are developing a diagonal matrix $W$ for which
$\mathscr{T}W$ is a diagonally dominant tensor.

######  theorem 3.2:

Suppose $\mathscr{T}\in\mathbb{R}^{[m,n]},$ $m$ is even. Let $\mathscr{T}$ be
a Nekrasov $Z$ tensor whose diagonal elements are positive and let $W$ be a
diagonal matrix such that,

$W=diag\left(\left(\frac{\Lambda_{1}(\mathscr{T})}{t_{11\cdots
1}}\right)^{\frac{1}{m-1}},\;\left(\frac{\Lambda_{2}(\mathscr{T})}{t_{22\cdots
2}}\right)^{\frac{1}{m-1}},\;\cdots\;,\left(\frac{\Lambda_{n}(\mathscr{T})}{t_{nn\cdots
n}}\right)^{\frac{1}{m-1}}\right)$ (3.1)

Then $\mathscr{T}W$ is a diagonally dominant $Z$ tensor.

###### Proof.

Firstly note that $\frac{\Lambda_{i}(\mathscr{T})}{t_{ii\cdots i}}\geq 0$ and
so $\left(\frac{\Lambda_{i}(\mathscr{T})}{t_{ii\cdots
i}}\right)^{\frac{1}{m-1}}\geq 0,\;\forall\;i\in[n],$ as $\mathscr{T}$ is a
Nekrasov $Z$ tensor with positive diagonal elements and $m$ is even. So $W$ is
a nonnegative diagonal matrix. Let
$\mathscr{B}=(b_{i_{1}i_{2}...i_{m}})\in\mathbb{R}^{[m,n]}$ be such that
$\mathscr{B}=\mathscr{T}W.$ Then

$b_{i_{1}i_{2}...i_{m}}=t_{i_{1}i_{2}...i_{m}}\left(\frac{\Lambda_{i_{2}}(\mathscr{T})}{t_{{i_{2}}{i_{2}}\cdots{i_{2}}}}\right)^{\frac{1}{m-1}}\cdots\left(\frac{\Lambda_{i_{m}}(\mathscr{T})}{t_{{i_{m}}{i_{m}}\cdots{i_{m}}}}\right)^{\frac{1}{m-1}}$
(3.2)

and

$b_{ii...i}=\Lambda_{i}(\mathscr{T}).$ (3.3)

It is easy to observe that the signs of $\mathscr{B}$ are same as the signs of
$\mathscr{T}.$ This infers that $\mathscr{B}$ is a $Z$ tensor, since
$\mathscr{T}$ is a $Z$ tensor. Now we show that $\mathscr{B}$ is a diagonally
dominant tensor. We write

$t_{k\cdots k}=\lvert t_{k\cdots
k}\rvert>\Lambda_{k}(\mathscr{T}),\;\;\forall\;k\in[n],$ (3.4)

since $\mathscr{T}$ is a Nekrasov tensor. Now for each $i\in[n]$ we obtain,

$\displaystyle|b_{i\cdots i}|$ $\displaystyle=b_{i\cdots i}$
$\displaystyle=\Lambda_{i}(\mathscr{T})$
$\displaystyle=\sum_{i_{2}...i_{m}\in[i-1]^{m-1}}\lvert
t_{ii_{2}...i_{m}}\rvert\left(\frac{\Lambda_{i_{2}}(\mathscr{T})}{\lvert
t_{i_{2}...i_{2}}\rvert}\right)^{\frac{1}{m-1}}\cdots\left(\frac{\Lambda_{i_{m}}(\mathscr{T})}{\lvert
t_{i_{m}...i_{m}}\rvert}\right)^{\frac{1}{m-1}}$
$\displaystyle\qquad+\sum_{i_{2}...i_{m}\notin[i-1]^{m-1},(i_{2}...i_{m})\neq(i...i)}\lvert
t_{ii_{2}...i_{m}}\rvert$
$\displaystyle\geq\sum_{i_{2}...i_{m}\in[i-1]^{m-1}}\lvert
t_{ii_{2}...i_{m}}\rvert\left(\frac{\Lambda_{i_{2}}(\mathscr{T})}{\lvert
t_{i_{2}...i_{2}}\rvert}\right)^{\frac{1}{m-1}}\cdots\left(\frac{\Lambda_{i_{m}}(\mathscr{T})}{\lvert
t_{i_{m}...i_{m}}\rvert}\right)^{\frac{1}{m-1}}$
$\displaystyle\qquad+\sum_{i_{2}...i_{m}\notin[i-1]^{m-1},(i_{2}...i_{m})\neq(i...i)}\lvert
t_{ii_{2}...i_{m}}\rvert\left(\frac{\Lambda_{i_{2}}(\mathscr{T})}{\lvert
t_{i_{2}...i_{2}}\rvert}\right)^{\frac{1}{m-1}}\cdots\left(\frac{\Lambda_{i_{m}}(\mathscr{T})}{\lvert
t_{i_{m}...i_{m}}\rvert}\right)^{\frac{1}{m-1}}\text{ by (\ref{for the
conclusion of Nekrasov Z})}$
$\displaystyle=\sum_{i_{2}...i_{m}\in[n]^{m-1},(i_{2}...i_{m})\neq(i...i)}\lvert
t_{ii_{2}...i_{m}}\rvert\left(\frac{\Lambda_{i_{2}}(\mathscr{T})}{\lvert
t_{i_{2}...i_{2}}\rvert}\right)^{\frac{1}{m-1}}\cdots\left(\frac{\Lambda_{i_{m}}(\mathscr{T})}{\lvert
t_{i_{m}...i_{m}}\rvert}\right)^{\frac{1}{m-1}}$
$\displaystyle=\sum_{(i_{2}...i_{m})\neq(i...i)}|b_{ii_{2}...i_{m}}|.$

This implies that $\mathscr{B}$ is a diagonally dominant tensor. ∎

Now we want to find a class of tensors that will contain Nekrasov $Z$ tensors
with positive diagonal elements. To do so we use a decomposition of tensors
which will be useful in our discussion.

Given a tensor $\mathscr{T}=(t_{i_{1}i_{2}...i_{m}})\in\mathbb{R}^{[m,n]},$ we
can write $\mathscr{T}$ as

$\mathscr{T}=\mathscr{B}^{+}+\mathscr{C},$ (3.5)

where $\forall\;i\in[n]$ the elements of rowsubtensors of $\mathscr{B}$ and
$\mathscr{C}$ are given by,

$(R_{i}(\mathscr{B}^{+}))_{i_{2}\cdots
i_{m}}=t_{ii_{2}...i_{m}}-r_{i}^{+},\;\forall\;i_{l}\in[n],\;l\in[m]$ (3.6)

and

$(R_{i}(\mathscr{C}))_{i_{2}\cdots
i_{m}}=r_{i}^{+},\;\forall\;i_{l}\in[n],\;l\in[m]..$ (3.7)

with

$r_{i}^{+}=\max_{i_{2},...,i_{m}\in[n],\;(i_{2}\cdots i_{m})\neq(i\cdots
i)}\\{0,t_{ii_{2}...i_{m}}\\}$ (3.8)

######  proposition 3.1:

Let $\mathscr{T}=(t_{i_{1}i_{2}...i_{m}})\in\mathbb{R}^{[m,n]}.$ If
$\mathscr{T}$ is decomposed as given in the equation (3.5), then
$\mathscr{B}^{+}$ is a $Z$ tensor and $\mathscr{C}$ is nonnegative tensor.

###### Proof.

Firstly, by the definition of the tensor $\mathscr{B}^{+}$ we have
$b_{i_{1}i_{2}...i_{m}}=t_{i_{1}i_{2}...i_{m}}-r_{i_{1}}^{+},$ for all
$i_{1},i_{2},...,i_{m}\in[n].$ Now from the definition of $r_{i_{1}}^{+}$
given by equation (3.8) for all $i_{1},i_{2},...,i_{m}\in[n],$ we write

$\displaystyle b_{i_{1}i_{2}...i_{m}}$
$\displaystyle=t_{i_{1}i_{2}...i_{m}}-r_{i_{1}}^{+}$
$\displaystyle=t_{i_{1}i_{2}...i_{m}}-\max_{i_{2},...,i_{m}\in[n],\;(i_{2}\cdots
i_{m})\neq(i\cdots i)}\\{0,t_{ii_{2}...i_{m}}\\}$ $\displaystyle\leq 0,\text{
when }(i_{2}\cdots i_{m})\neq(i\cdots i).$

Therefore all off diagonal elements of the tensor $\mathscr{B}^{+}$ are
nonpositive. Hence the tensor $\mathscr{B}^{+}$ is a $Z$ tensor.

Secondly, observe that the elements of $\mathscr{C}$ are $r_{i}^{+},$
$\forall\;i\in[n].$ Now by equation (3.8) we write

$r_{i}^{+}=\max_{i_{2},...,i_{m}\in[n],\;(i_{2}\cdots i_{m})\neq(i\cdots
i)}\\{0,t_{ii_{2}...i_{m}}\\}\geq 0.$

This implies $\mathscr{C}$ is a nonnegative tensor. ∎

######  remark 3.1:

If the tensor $\mathscr{B}^{+}$ is a Nekrasov tensor then $\mathscr{B}^{+}$
becomes a Nekrasov $Z$ tensor.

######  remark 3.2:

The class of tensors $\mathscr{T}$ that can be expressed as
$\mathscr{T}=\mathscr{B}^{+}+\mathscr{C},$ where $\mathscr{B}^{+}$ and
$\mathscr{C}$ are constructed by (3.6), (3.7) and (3.8) with $\mathscr{B}^{+}$
as a Nekrasov tensor contains the class of Nekrasov $Z$ tensors.

######  remark 3.3:

The $P$ tensors play a crucial role in tensor complementarity theory. We know
that the SOL$(\mathscr{T},q)$ is nonempty and compact if $\mathscr{T}$ is a
$P$ tensor. Here we show that $\mathscr{T},$ a Nekrasov $Z$ tensor of even
order which has positive diagonal elements, is a $P$-tensor. Hence we conclude
that the solution set of TCP$(\mathscr{T},q)$ is nonempty and compact.

##  Conclusion

In this article, we introduce Nekrasov $Z$ tensor. For a Nekrasov $Z$ tensor
$\mathscr{T}$ we find $W$ such that $\mathscr{T}W$ is a diagonally dominant
tensor. We prove that the Nekrasov $Z$ tensors of even order with positive
diagonal elements are contained in the class of $P$-tensor.

##  Acknowledgment

The author R. Deb acknowledges CSIR, India, JRF scheme for financial support.

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  * [17] R Jana, AK Das, and A Dutta.  On hidden ${Z}$-matrix and interior point algorithm.  Opsearch, 56(4):1108–1116, 2019.
  * [18] R Jana, AK Das, and VN Mishra.  Iterative descent method for generalized leontief model.  Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 91(2):237–244, 2021.
  * [19] R Jana, AK Das, and S Sinha.  On processability of Lemke’s algorithm.  Applications & Applied Mathematics, 13(2), 2018.
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  * [30] SK Neogy and AK Das.  Linear complementarity and two classes of structured stochastic games.  Operations Research with Economic and Industrial Applications: Emerging Trends, eds: SR Mohan and SK Neogy, Anamaya Publishers, New Delhi, India, pages 156–180, 2005.
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  * [37] SK Neogy and AK Das.  On weak generalized positive subdefinite matrices and the linear complementarity problem.  Linear and Multilinear Algebra, 61(7):945–953, 2013.
  * [38] SK Neogy, AK Das, and RB Bapat.  Modeling, computation and optimization, volume 6.  World Scientific, 2009.
  * [39] SK Neogy, AK Das, and A Gupta.  Generalized principal pivot transforms, complementarity theory and their applications in stochastic games.  Optimization Letters, 6(2):339–356, 2012.
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  * [49] Pingzhi Yuan and Lihua You.  Some remarks on p, p0, b and b0 tensors.  Linear Algebra and its Applications, 459:511–521, 2014.
  * [50] Junli Zhang and Changjiang Bu.  Nekrasov tensors and nonsingular $\mathscr{H}$\- tensors.  Computational and Applied Mathematics, 37(4):4917–4930, 2018.
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