New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms

Deshu Sun; Dongjian Bai

doi:10.1515/math-2021-0042

Article Open Access

New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms

Deshu Sun and Dongjian Bai

Published/Copyright: July 7, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

Positive definite polynomials are important in the field of optimization. ℋ-tensors play an important role in identifing the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose an iterative scheme for identifying ℋ-tensor and prove that the algorithm can terminate within finite iterative steps. Some numerical examples are provided to illustrate the efficiency and validity of methods.

Keywords: positive definiteness; homogeneous multivariate form; ℋ-tensors; generalized diagonal dominance; iterative scheme

MSC 2010: 15A69; 15A18; 65F15; 65H17

1 Introduction

The m th degree homogeneous polynomial of n variables f ( x ) is denoted as

(1) f ( x ) = ∑ i 1 , i 2 , … , i m ∈ N a i 1 i 2 ⋯ i m x i 1 x i 2 ⋯ x i m ,

where x = ( x 1 , x 2 , … , x n ) ∈ R n . The form in (1) can be represented as

(2) f ( x ) ≡ A x m ,

where A = ( a i 1 i 2 ⋯ i m ) is an m th-order n -dimensional tensor [1,2] with entries

a i 1 i 2 ⋯ i m ∈ R , i j = 1 , 2 , … , n , j = 1 , 2 , … , m .

The homogeneous polynomial f ( x ) in (2) is said to be positive definitiveness if f ( x ) > 0 for any x ∈ R n ( x ≠ 0 ) . Positive-definite homogeneous polynomial has very important application in the field of dynamical and positive semi-definite polynomial as well as in the field of optimization. Since each homogeneous polynomial can be associated with a symmetric tensor, we can identify the positive (semi-)definiteness of homogeneous polynomial by identifying positive definiteness of the symmetric tensor associated with it [3,4, 5,6,7, 8,9].

For n ≤ 3 , the positive definiteness of the homogeneous polynomial form can be checked by Sturm theorem [10]. However, for n > 3 and m ≥ 4 , the problem of determining a given even-order multivariate polynomial f ( x ) is positive semi-definiteness or not is NP-hard [7]. For this case, by ℋ -tensors, Li et al. [11] proposed a method to identify the positive definiteness of homogeneous polynomial forms. It is well known that ℋ -tensor is a special kind of tensors and an even order symmetric ℋ -tensor with positive diagonal entries is positive definite [11,12]. Due to this, we may identify the positive definiteness of homogeneous polynomial forms via identifying ℋ -tensors. Recently, with the help of generalized diagonally dominant tensor, various criteria for ℋ -tensors are provided [13,14,15, 16,17,18]. In this paper, we continue finding new non-parameter-involved iterative criterion for identifying ℋ -tensors. Its validity will be theoretically guaranteed, and its performance will be illustrated in a set of numerical tests.

2 Preliminaries

In this section, some notations, definitions and lemmas are given. Let C [ m , n ] ( R [ m , n ] ) be the set of all complex(real) m th-order n -dimensional tensors and ℐ ∈ C [ m , n ] be the unit tensor, where

δ i 1 i 2 ⋯ i m = 1 , if i 1 = ⋯ = i m , 0 , otherwise .

Let Q be a nonempty subset of N and N ⧹ Q be the complement of Q in N . For a tensor A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] , we denote

R i ( A ) = ∑ i 2 , … , i m ∈ N δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ = ∑ i 2 , … , i m ∈ N ∣ a i i 2 ⋯ i m ∣ − ∣ a i i ⋯ i ∣ , N 1 = { i ∈ N : 0 < ∣ a i i ⋯ i ∣ = R i ( A ) } , N 2 = { i ∈ N : 0 < ∣ a i i ⋯ i ∣ < R i ( A ) } , N 3 = { i ∈ N : ∣ a i i ⋯ i ∣ > R i ( A ) } , Q m − 1 = { i 2 i 3 ⋯ i m : i j ∈ S , j = 2 , 3 , … , m } , N m − 1 ⧹ Q m − 1 = { i 2 i 3 ⋯ i m : i 2 i 3 ⋯ i m ∈ N m − 1 and i 2 i 3 ⋯ i m ∉ S m − 1 } , N 0 m − 1 = N m − 1 ⧹ ( N 2 m − 1 ∪ N 3 m − 1 ) , r = max i ∈ N 3 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ ∣ a i i ⋯ i ∣ − ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ , P i , r ( A ) = ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ + r ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 3 , r 1 = max i ∈ N 3 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ ∣ a i i ⋯ i ∣ − ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ ,

and

(3) ω i = R i ( A ) R i ( A ) + ∣ a i i ⋯ i ∣ , ∀ i ∈ N 2 ,

(4) δ 1 = max { ω i , r 1 } , ∀ i ∈ N 2 ,

(5) P i , r 1 ( A ) = ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ + r 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 3 ,

(6) h 1 = max i ∈ N 3 δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ P i , r 1 ( A ) − ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ .

In this paper, we always assume that N 1 ∪ N 2 ≠ ∅ and N 3 ≠ ∅ . In addition, we also assume that A satisfies: a i i ⋯ i ≠ 0 , R i ( A ) ≠ 0 , ∀ i ∈ N .

Definition 1

[12] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] . A is called a diagonally dominant tensor if

(7) ∣ a i i ⋯ i ∣ ≥ ∑ i 2 , … , i m ∈ N δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N .

A is a strictly diagonally dominant tensor if all inequalities in (7) hold.

Definition 2

[12] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] . A is called an ℋ -tensor if there exists an entrywise positive vector x = ( x 1 , x 2 , … , x n ) T ∈ R n such that

∣ a i i ⋯ i ∣ x i m − 1 > ∑ i 2 , … , i m ∈ N δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ x i 2 ⋯ x i m , ∀ i ∈ N .

Definition 3

[17] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] , X = diag ( x 1 , x 2 , … , x n ) . Denote

ℬ = ( b i 1 ⋯ i m ) = A X m − 1 , b i 1 i 2 ⋯ i m = a i 1 i 2 ⋯ i m x i 2 x i 3 ⋯ x i m , i j ∈ N , j ∈ N .

We call ℬ the product of the tensor A and the matrix X .

Definition 4

[11] Tensor A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] is called reducible if there exists a nonempty proper index subset I ⊂ N such that

a i 1 i 2 ⋯ i m = 0 , ∀ i 1 ∈ I , ∀ i 2 , … , i m ∉ I .

Otherwise, we say A is irreducible.

Definition 5

[18] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] , for i , j ∈ N ( i ≠ j ) , if there exist indices k 1 , k 2 , … , k r with

∑ i 2 , … , i m ∈ N δ k s i 2 ⋯ i m = 0 , k s + 1 ∈ { i 2 , … , i m } ∣ a k s i 2 ⋯ i m ∣ ≠ 0 , s = 0 , 1 , … , r ,

where k 0 = i , k r + 1 = j , we say there is a nonzero element chain from i to j .

Lemma 1

[18] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] . If

∣ a i i ⋯ i ∣ ≥ R i ( A ) , ∀ i ∈ N ,
J ( A ) = { i ∈ N : ∣ a i i ⋯ i ∣ > R i ( A ) } ≠ ∅ ,
for any i ∉ J ( A ) , there is a nonzero element chain from i to j such that j ∈ J ( A ) ,

then A is an ℋ -tensor.

Lemma 2

[17] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] . If there is a positive diagonal matrix X such that A X m − 1 is strictly diagonally dominant, then A is an ℋ -tensor.

Lemma 3

[11] Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] . If A is irreducible diagonally dominant tensor such that the inequality in (7) holds for at least one i , then A is an ℋ -tensor.

3 Criteria for identifying ℋ -tensors

In this section, we give some new criteria for ℋ -tensors.

Theorem 1

Let A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] and let ω i , δ 1 , P j , r 1 ( A ) , h 1 be defined in (3)–(6), respectively. If

(8) ∣ a i i ⋯ i ∣ ω i > δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + h 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 2 ,

and

(9) ∣ a i i ⋯ i ∣ ≠ ∑ i 2 ⋯ i m ∈ N 0 m − 1 δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 1 ≠ ∅ ( or N 1 = ∅ ) ,

then A is an ℋ -tensor.

Proof

By the definitions of r , P i , r ( A ) , r 1 and P i , r 1 ( A ) , we obtain

0 < r 1 ≤ r < 1 ,

and

r 1 ∣ a i i ⋯ i ∣ ≥ ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ + r 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 3 ,

that is, P i , r 1 ( A ) ≤ r 1 ∣ a i i ⋯ i ∣ . Then,

0 < P i , r 1 ( A ) ∣ a i i ⋯ i ∣ ≤ r 1 ≤ r < 1 , ∀ i ∈ N 3 .

By the definitions of ω i and δ 1 , we get that for each i ∈ N 3 ,

δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ P i , r 1 ( A ) − ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ ≤ δ 1 P i , r 1 ( A ) − r 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ P i , r 1 ( A ) − ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ < 1 .

By the definition of h 1 , for ∀ i ∈ N 3 , we obtain 0 < h 1 < 1 and

(10) h 1 P i , r 1 ( A ) > δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + h 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ .

Let

(11) M i = 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ ∣ a i i ⋯ i ∣ ω i − δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ − ∑ i 2 ⋯ i m ∈ N 2 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ − h 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 2 .

If ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ = 0 , we denote M i = + ∞ . From inequalities (8), (9) and (11), we obtain M i > 0 ( ∀ i ∈ N 2 ) . Hence, by 0 < h 1 P i , r 1 ( A ) ∣ a i i ⋯ i ∣ < 1 ( ∀ i ∈ N 3 ) , there exists a positive number ε such that

(12) 0 < ε < min i ∈ N 2 { M i } ≤ + ∞ , 0 < max i ∈ N 3 h 1 P i , r 1 ( A ) ∣ a i i ⋯ i ∣ + ε < δ 1 < 1 .

Set X = diag ( x 1 , x 2 , … , x n ) , where

x i = ( δ 1 ) 1 m − 1 , i ∈ N 1 , ( ω i ) 1 m − 1 , i ∈ N 2 , h 1 P i , r 1 ( A ) ∣ a i i ⋯ i ∣ + ε 1 m − 1 , i ∈ N 3 .

Let ℬ = ( b i 1 i 2 ⋯ i m ) = A X m − 1 . Now, we prove that ℬ is a diagonally dominant matrix.

For i ∈ N 1 , by inequalities (9) and (12), we obtain

(13) R i ( ℬ ) = ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ x i 2 ⋯ x i m + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ ( ω i 2 ) 1 m − 1 ⋯ ( ω i m ) 1 m − 1 + ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ ε + h 1 P i 2 , r 1 ( A ) ∣ a i 2 i 2 ⋯ i 2 ∣ 1 m − 1 ⋯ ε + h 1 P i m , r 1 ( A ) ∣ a i m i m ⋯ i m ∣ 1 m − 1 ≤ δ 1 ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 max j ∈ { i 2 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 ε + max j ∈ { i 2 , … , i m } h 1 P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ ≤ ∣ a i i ⋯ i ∣ δ 1 = ∣ b i i ⋯ i ∣ .

For i ∈ N 2 , by inequalities (11) and (12), we have

(14) R i ( ℬ ) = ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ x i 2 ⋯ x i m + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ ( ω i 2 ) 1 m − 1 ⋯ ( ω i m ) 1 m − 1 + ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ ε + h 1 P i 2 , r 1 ( A ) ∣ a i 2 i 2 ⋯ i 2 ∣ 1 m − 1 ⋯ ε + h 1 P i m , r 1 ( A ) ∣ a i m i m ⋯ i m ∣ 1 m − 1 ≤ δ 1 ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 δ i i 2 ⋯ i m = 0 max j ∈ { i 2 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 ε + max j ∈ { i 2 , … , i m } h 1 P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ < ∣ a i i ⋯ i ∣ ω i = ∣ b i i ⋯ i ∣ .

Finally, for i ∈ N 3 , combining (10) and (12), we have

(15) R i ( ℬ ) = ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ x i 2 ⋯ x i m + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ ( ω i 2 ) 1 m − 1 ⋯ ( ω i m ) 1 m − 1 + ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ ε + h 1 P i 2 , r 1 ( A ) ∣ a i 2 i 2 ⋯ i 2 ∣ 1 m − 1 ⋯ ε + h P i m , r 1 ( A ) ∣ a i m i m ⋯ i m ∣ 1 m − 1 ≤ δ 1 ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 max j ∈ { i 2 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 δ i i 2 ⋯ i m = 0 ε + max j ∈ { i 2 , … , i m } h 1 P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ < ε ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ + h 1 P i , r 1 ( A ) < ε ∣ a i i ⋯ i ∣ + h 1 P i , r 1 ( A ) = ∣ b i i ⋯ i ∣ .

Therefore, from inequalities (13)–(15), we obtain

∣ b i i ⋯ i ∣ ≥ R i ( ℬ ) ( ∀ i ∈ N 1 ) , ∣ b i i ⋯ i ∣ > R i ( ℬ ) ( ∀ i ∈ N 2 ∪ N 3 ) .

Combining (9), for all i ∈ N 1 , there exists i 2 ⋯ i m ∈ N 2 m − 1 ∪ N 3 m − 1 such that a i i 2 ⋯ i m ≠ 0 . Since ℬ = ( b i 1 i 2 ⋯ i m ) = A X m − 1 , then for ∀ i ∈ N 1 , there exists i 2 ⋯ i m ∈ N 2 m − 1 ∪ N 3 m − 1 such that b i i 2 ⋯ i m ≠ 0 . Hence, ℬ is a diagonally dominant tensor with nonzero elements chain, and by Lemma 1, ℬ is an ℋ -tensor. Furthermore, by Lemma 2, A is an ℋ -tensor.□

By a similar proof to that of Theorem 1 and Lemmas 1, 2 and 3, we can obtain the following theorem.

Theorem 2

Let A = ( a i 1 ⋯ i m ) ∈ C [ m , n ] with

(16) ∣ a i i ⋯ i ∣ ω i ≥ δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + h 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 2 .

If A satisfies one of the following conditions, then A is an ℋ -tensor,

A is irreducible and N 1 ≠ ∅ or a strict inequality holds for at least one i ∈ N 2 in inequality (12);
( N 1 ∪ N 2 ) ⧹ ( N ¯ 1 ∪ N ¯ 2 ) ≠ ∅ , and for all i ∈ ( N ¯ 1 ∪ N ¯ 2 ∪ N 3 ) , there exists a nonzero element chain from i to j such that j ∈ ( N 1 ∪ N 2 ) ⧹ ( N ¯ 1 ∪ N ¯ 2 ) , where
N ¯ 1 = i ∈ N 1 : ∣ a i i ⋯ i ∣ = ∑ i 2 ⋯ i m ∈ N 0 m − 1 δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ , N ¯ 2 = i ∈ N 2 : ∣ a i i ⋯ i ∣ ω i = δ 1 ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } { ω j } ∣ a i i 2 ⋯ i m ∣ + h 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ .

Example 1

Consider a tensor A = ( a i j k ) ∈ C [ 3 , 3 ] defined as follows:

A = [ A ( 1 , : , : ) , A ( 2 , : , : ) , A ( 3 , : , : ) ] , A ( 1 , : , : ) = 12 1 0 1 6 0 1 0 12 , A ( 2 , : , : ) = 1 0 0 0 10 2 0 2 2 , A ( 3 , : , : ) = 0 0 0 1 1 0 0 0 8 .

Obviously,

∣ a 111 ∣ = 12 , R 1 ( A ) = 21 , ∣ a 222 ∣ = 10 , R 2 ( A ) = 8 , ∣ a 333 ∣ = 8 , R 3 ( A ) = 2 .

So, N 1 = ∅ , N 2 = { 1 } , N 3 = { 2 , 3 } and

ω 1 = 7 11 , r = 1 4 , P 2 , r ( A ) = 5 2 , P 3 , r ( A ) = 5 4 , r 1 = 4 31 , δ 1 = 7 11 , P 2 , r 1 ( A ) = 145 124 , P 3 , r 1 ( A ) = 32 31 , h 1 ( A ) = 1736 2497 .

Since

δ 1 ∑ j k ∈ N 0 2 ∣ a 1 j k ∣ + ∑ j k ∈ N 2 2 δ 1 j k = 0 max l ∈ { j , k } { ω l } ∣ a 1 j k ∣ + h 1 ∑ j k ∈ N 3 2 max l ∈ { j , k } P l , r 1 ( A ) ∣ a l l l ∣ ∣ a 1 j k ∣ = 7 11 ( 1 + 1 + 1 ) + 1736 2497 × 6 × 145 124 × 1 10 + 12 × 32 31 × 1 8 = 8673 2497 < 84 11 = ∣ a 111 ∣ ω 1 ,

we know that A satisfies the conditions of Theorem 1, then A is an ℋ -tensor. But

∑ j k ∈ N 0 2 ⧹ N 3 2 δ 1 j k = 0 ∣ a 1 j k ∣ + ∑ j k ∈ N 3 2 max l ∈ { j , k } R l ( A ) ∣ a l l l ∣ ∣ a 1 j k ∣ = ( 1 + 1 + 1 ) + 7 10 × ( 6 + 12 ) = 78 5 > 12 = ∣ a 111 ∣ .

So, A does not satisfy the conditions of Theorem 3 in [13].

4 An iterative scheme for identifying ℋ -tensors

In this section, we provide an iterative algorithm for identifying ℋ -tensors on the basis of the results in Section 3. The numerical experiment will be performed via Matlab R2015 on a 2.2 GHz Intel computer with a 6-core i7-8750H processor.

Algorithm 4.1.

INPUT: A tensor A = ( a i 1 i 2 ⋯ i m ) ∈ C [ m , n ] with a i i ⋯ i ≠ 0 for all i ∈ N .

OUTPUT: A positive diagonal matrix X = X ( 0 ) X ( 1 ) ⋯ X ( p − 1 ) if A is an ℋ -tensor.

Step 1. If N 3 ( A ) = ∅ , then A is not an ℋ -tensor, stop; otherwise,

Step 2. Set A ( 0 ) = A , X ( 0 ) = ℐ , p = 1 .

Step 3. Compute A ( p ) = A ( p − 1 ) ( X ( p − 1 ) ) m − 1 = ( a i 1 i 2 ⋯ i m ( p ) ) .

Step 4. If N 2 ( A ( p ) ) = ∅ , then A is an ℋ -tensor, stop; otherwise,

Step 5. If N 3 ( A ( p ) ) = ∅ , then A is not an ℋ -tensor, stop; otherwise,

Step 6. Compute δ 1 ( p ) , ω i ( p ) , h 1 ( p ) , δ 1 ( p ) , P j , r 1 ( A ( p ) ) .

Step 7. For ∀ i ∈ N 2 ( A ( p ) ) , if A ( p ) satisfies

∣ a i i ⋯ i ( p ) ∣ ω i ( p ) > δ 1 ( p ) ∑ i 2 ⋯ i m ∈ N 0 m − 1 ( A ( p ) ) ∣ a i i 2 ⋯ i m ( p ) ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ( A ( p ) ) δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } { ω j ( p ) } ∣ a i i 2 ⋯ i m ( p ) ∣ + h 1 ( p ) ∑ i 2 ⋯ i m ∈ N 3 m − 1 ( A ( p ) ) max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ( p ) ) ∣ a j j ⋯ j ( p ) ∣ ∣ a i i 2 ⋯ i m ( p ) ∣ ,

and

∣ a i i ⋯ i ( p ) ∣ ≠ ∑ i 2 ⋯ i m ∈ N 0 m − 1 ( A ( p ) ) δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ( p ) ∣ , ∀ i ∈ N 1 ( A ( p ) ) ≠ ∅ ( or N 1 ( A ( p ) ) = ∅ ) ,

then A is an ℋ -tensor, stop; otherwise,

Step 8. Set X ( p ) = diag ( x 1 ( p ) , x 2 ( p ) , … , x n ( p ) ) , compute

x i ( p ) = ( δ 1 ( p ) ) 1 m − 1 , if i ∈ N 1 ( A ( p ) ) , ( ω i ( p ) ) 1 m − 1 , if i ∈ N 2 ( A ( p ) ) , h 1 ( p ) P i , r 1 ( A ( p ) ) ∣ a i i ⋯ i ( p ) ∣ 1 m − 1 , if i ∈ N 3 ( A ( p ) ) .

Step 9. Set X ( p ) = diag ( x 1 ( p ) , x 2 ( p ) , … , x n ( p ) ) , p = p + 1 ; go to step 3.

The theoretical basis for the functionality of Algorithm 4.1 as criteria for ℋ -tensor is provided by the following theorem.

Theorem 3

Let A = ( a i 1 ⋯ i m ) ∈ C [ m , n ] . If A is an ℋ -tensor, then Algorithm 4.1 terminates after a finite number of iterations by producing a strictly diagonally dominant tensor.

Proof

Without loss of generality, we assume that A is a non-negative tensor. Suppose, on the contrary, that Algorithm 4.1 produces the infinite sequence { A ( p ) } with

A ( p ) = A ( 0 ) ( X ( 1 ) ) m − 1 ( X ( 2 ) ) m − 1 ⋯ ( X ( p − 1 ) ) m − 1 .

From the fact that each diagonal entry of positive diagonal matrix X ( p ) is less than 1, we have

A = A ( 0 ) = A ( 1 ) ≥ ⋯ ≥ A ( p ) ≥ ⋯ ≥ 0 .

Hence, the sequence { A ( p ) } has a limitation,

ℬ = lim p → + ∞ A ( p ) ≥ 0 ,

where ℬ = A X m − 1 , X = X ( 1 ) X ( 2 ) ⋯ X ( p ) ⋯ is a positive diagonal matrix.

Next, we prove that

lim p → + ∞ N 3 ( A ( p ) ) = N 3 ( ℬ ) = ∅ .

In fact, suppose that lim p → + ∞ N 3 ( A ( p ) ) ≠ ∅ , then

1 − δ 1 ( p ) > 0 , 1 − ω i ( p ) > 0 , 1 − h 1 ( p ) P i , r 1 ( A ( p ) ) ∣ a i i ⋯ i ( p ) ∣ > 0 .

Therefore, there exists i ∈ N 3 ( A ( p ) ) and positive numbers ε 1 , ε 2 , ε 3 such that

a i i ⋯ i ( p ) ( 1 − δ 1 ( p ) ) > ε 1 , a i i ⋯ i ( p ) ( 1 − ω i ( p ) ) > ε 2 , a i i ⋯ i ( p ) 1 − h 1 ( p ) P i , r 1 ( A ( p ) ) ∣ a i i ⋯ i ( p ) ∣ > ε 3 .

Let ε 0 = min { ε 1 , ε 2 , ε 3 } . By Algorithm 4.1, for any i ∈ N , we obtain

0 < a i i ⋯ i ( p + 1 ) = a i i ⋯ i ( p ) δ 1 ( p ) < a i i ⋯ i ( p ) − ε 1 < a i i ⋯ i ( p ) − ε 0 , 0 < a i i ⋯ i ( p + 1 ) = a i i ⋯ i ( p ) ω i ( p ) < a i i ⋯ i ( p ) − ε 2 < a i i ⋯ i ( p ) − ε 0 , 0 < a i i ⋯ i ( p + 1 ) = a i i ⋯ i ( p ) h 1 ( p ) P i , r 1 ( A ( p ) ) ∣ a i i ⋯ i ( p ) ∣ < a i i ⋯ i ( p ) − ε 3 < a i i ⋯ i ( p ) − ε 0 .

Hence,

a i i ⋯ i ( 0 ) = a i i ⋯ i ( 1 ) > a i i ⋯ i ( 2 ) + ε 0 > ⋯ > a i i ⋯ i ( p ) + ( p − 1 ) ε 0 .

When p → + ∞ , one has a i i ⋯ i ( 0 ) → + ∞ . A contradiction arrives which means that

lim p → + ∞ N 3 ( A ( p ) ) = N 3 ( ℬ ) = ∅ .

That is,

∣ b i i ⋯ i ∣ ≤ R i ( ℬ ) , ∀ i ∈ N .

So, ℬ is not an ℋ -tensor [11]. On the other hand, since A is an ℋ -tensor, then there exists a positive diagonal matrix D such that A D m − 1 = ℬ ( X − 1 D ) m − 1 is strictly diagonally dominant. Then, ℬ is an ℋ -tensor. We also obtain a contradiction, then our hypothesis does not hold. The proof is completed.□

Example 2

[13] Randomly generate 100 m th -order n -dimensional tensors such that the elements of each tensor satisfying

a i 1 i 2 ⋯ i m ∈ ( − n m × 0.6 , n m × 0.6 ) , if i 1 = ⋯ = i m , ( − 1 , 1 ) , otherwise .

The numerical results of Algorithm 4.1 are shown in Table 1.

The numerical results of Algorithm 4.1 are shown in Table 1, where “ p 1 ” denotes the number of tensors that are ℋ -tensor, “ p 2 ” denotes the number of tensors that are not ℋ -tensor, “ p 3 ” denotes the number of tensors that whether they are ℋ -tensor that are not checkable by using Algorithm 4.1, “ R T ” denotes the running time when algorithm terminates.

Table 1

Numerical results of Example 2

m (order)	n (dimension)	p 1	p 2	R T
4	6	44	56	25.5520
4	7	52	48	32.2855
4	8	57	43	40.4317
4	9	61	39	42.3266
4	10	73	27	39.4375
5	6	66	34	55.07456
5	7	81	19	25.2355
5	8	86	14	98.4053
5	9	71	29	30.1723
5	10	71	29	58.3400
6	6	81	19	35.7658
6	7	83	17	105.6540
6	8	92	8	34.6045
6	9	91	9	30.4036
6	10	93	7	42.3576

5 An application

Based on the criteria for ℋ -tensors, we present some new conditions for the positive definiteness of even-order real symmetric tensors. First, we give the following lemma.

Lemma 4

[11] Let A = ( a i 1 i 2 ⋯ i m ) ∈ R [ m , n ] be an even-order real symmetric tensor with a i i ⋯ i > 0 ( ∀ i ∈ N ) . If A is an ℋ -tensor, then A is positive definite.

Combining Theorems 1, 2 and Lemma 4, we have the following result.

Theorem 4

Let A = ( a i 1 ⋯ i m ) ∈ R [ m , n ] be an even-order real symmetric tensor, and a i i ⋯ i > 0 ( ∀ i ∈ N ) . If A satisfies one of the following conditions:

all the conditions of Theorem 1,
all the conditions of Theorem 2,

then A is a positive definite tensor.

Example 3

Consider polynomial f ( x ) = A x 6 with A being a 6-order 6-dimensional real symmetric tensor with entries

a 111111 = 4 , a 222222 = 16 , a 333333 = 33 , a 444444 = 16 , a 555555 = 2 , a 666666 = 3 , a 122222 = a 212222 = a 221222 = a 222122 = a 222212 = a 222221 = − 1 , a 133333 = a 313333 = a 331333 = a 333133 = a 333313 = a 333331 = − 2 , a 144444 = a 414444 = a 441444 = a 444144 = a 444414 = a 444441 = − 1 , a 233333 = a 323333 = a 332333 = a 333233 = a 333323 = a 333332 = − 2 , a 244444 = a 424444 = a 442444 = a 444244 = a 444424 = a 444442 = − 1 , a 344444 = a 434444 = a 443444 = a 444344 = a 444434 = a 444443 = − 1 , a 222333 = a 223233 = a 223323 = a 223332 = a 232233 = a 232323 = a 232332 = − 1 , a 233223 = a 233232 = a 233322 = a 333222 = a 332322 = a 332232 = a 332223 = − 1 , a 323322 = a 323232 = a 323223 = a 322332 = a 322323 = a 322233 = − 1 ,

and other a i 1 i 2 i 3 i 4 i 5 i 6 = 0 . By calculations, one has

a 222222 = 16 < 18 = R 2 ( A ) ,

and

a 444444 ( a 222222 − R 2 ( A ) + ∣ a 244444 ∣ ) = − 16 < 15 = R 4 ( A ) ∣ a 244444 ∣ .

Then, A is not a strictly diagonally dominated tensor or a quasi-doubly strictly diagonally dominant tensor. Hence, we cannot identify the positive definiteness of A by Theorem 3 in [19] and Theorem 4 in [20]. However, by using Algorithm 4.1, we obtain that A is an ℋ -tensor. It follows from Theorem 4 that A is positive definite, that is, the f ( x ) is positive definite.

6 Conclusion

In this paper, we proposed some implementable criteria for identifying ℋ -tensors, which are used to identify the positive definiteness of an even degree homogeneous polynomial f ( x ) ≡ A x m . We also provided a non-parameter-involved iterative scheme for identifying ℋ -tensors, which can stop within finite steps.

Funding information: This work was supported by the National Natural Science Foundation of China (11861077), the Foundation of Science and Technology Department of Guizhou Province (20191161, 20181079) and the Research Foundation of Guizhou Minzu University (2019YB08).
Conflict of interest: Authors state no conflict of interest.

References

[1] L. Q. Qi , Eigenvalues of a real supersymetric tensor, J. Symbolic Comput. 40 (2005), 1302–1324. 10.1016/j.jsc.2005.05.007Search in Google Scholar

[2] L. H. Lim , Singular values and eigenvalues of tensors: a variational approach , in: CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing , 2005, pp. 129–132. Search in Google Scholar

[3] L. P. Zhang , L. Q. Qi , and G. L. Zhou , M -tensors and some applications, SIAM J. Matrix Anal. Appl. 32 (2014), 437–452. 10.1137/130915339Search in Google Scholar

[4] L. Q. Qi , G. H. Yu , and Y. Xu , Nonnegative diffusion orientation distribution function, J. Math. Imaging Vision 45 (2013), 103–113. 10.1007/s10851-012-0346-ySearch in Google Scholar

[5] S. L. Hu , L. Q. Qi , and J. Y. Shao , Cored hypergraphs, power hypergraphs and their Laplacian eigenvalues, Linear Algebra Appl. 439 (2013), 2980–2998. 10.1016/j.laa.2013.08.028Search in Google Scholar

[6] H. B. Chen and L. Q. Qi , Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim. 11 (2015), no. 4, 1263–1274. 10.3934/jimo.2015.11.1263Search in Google Scholar

[7] Q. Ni , L. Q. Qi , and F. Wang , An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control. 53 (2008), no. 3, 1096–1107. 10.1109/TAC.2008.923679Search in Google Scholar

[8] Y. S. Song and L. Q. Qi , Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra 63 (2015), no. 1, 120–131. 10.1080/03081087.2013.851198Search in Google Scholar

[9] K. L. Zhang and Y. J. Wang , An ℋ -tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math. 305 (2016), 1–10. 10.1016/j.cam.2016.03.025Search in Google Scholar

[10] N. K. Bose and A. R. Modaress , General procedure for multivariable polynomial positivity with control applications, IEEE Trans. Automat. Control. 21 (1976), 596–601. 10.1109/TAC.1976.1101356Search in Google Scholar

[11] C. Q. Li , F. Wang , J. X. Zhao , Y. Zhu , and Y. T. Li , Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math. 255 (2014), 1–14. 10.1016/j.cam.2013.04.022Search in Google Scholar

[12] W. Y. Ding , L. Q. Qi , and Y. M. Wei , M -tensors and nonsingular M -tensors, Linear Algebra Appl. 439 (2013), 3264–3278. 10.1016/j.laa.2013.08.038Search in Google Scholar

[13] Y. T. Li , Q. L. Liu , and L. Q. Qi , Programmable criteria for strong ℋ -tensors, Numer. Algorithms 74 (2017), 199–211. 10.1007/s11075-016-0145-4Search in Google Scholar

[14] Y. J. Wang , G. L. Zhou , and L. Caccetta , Nonsingular ℋ -tensor and its criteria, J. Ind. Manag. Optim. 12 (2016), 1173–1186. 10.3934/jimo.2016.12.1173Search in Google Scholar

[15] Y. J. Wang , K. L. Zhang , and H. Sun , Criteria for strong ℋ -tensors, Front. Math. China 11 (2016), no. 3, 577–592. 10.1007/s11464-016-0525-zSearch in Google Scholar

[16] R. J. Zhao , L. Gao , Q. L. Liu , and Y. T. Li , Criterions for identifying ℋ -tensors, Front. Math. China 11 (2016), no. 3, 661–678. 10.1007/s11464-016-0519-xSearch in Google Scholar

[17] M. R. Kannana , N. S. Mondererb , and A. Bermana , Some properties of strong ℋ -tensors and general ℋ -tensors, Linear Algebra Appl. 476 (2015), 42–55. 10.1016/j.laa.2015.02.034Search in Google Scholar

[18] F. Wang , D. S. Sun , J. X. Zhao , and C. Q. Li , New practical criteria for ℋ -tensors and its application, Linear Multi . Algebra 65 (2017), no. 2, 269–283. 10.1080/03081087.2016.1183558Search in Google Scholar

[19] L. Q. Qi and Y. S. Song , An even order symmetric B -tensor is positive definite, Linear Algebra Appl. 457 (2014), 303–312. 10.1016/j.laa.2014.05.026Search in Google Scholar

[20] C. Q. Li and Y. T. Li , Double B -tensors and quasi-double B -tensors, Linear Algebra Appl. 466 (2015), 343–356. 10.1016/j.laa.2014.10.027Search in Google Scholar

Received: 2020-06-22

Revised: 2021-03-06

Accepted: 2021-04-11

Published Online: 2021-07-07

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0042

Keywords for this article

positive definiteness; homogeneous multivariate form; ℋ-tensors; generalized diagonal dominance; iterative scheme

Creative Commons

BY 4.0