Abstract

In this paper, we consider the problem of detecting the copositivity of partially symmetric rectangular tensors. We first propose a semidefinite relaxation algorithm for detecting the copositivity of partially symmetric rectangular tensors. Then, the convergence of the proposed algorithm is given, and it shows that we can always catch the copositivity of given partially symmetric tensors. Several preliminary numerical results confirm our theoretical findings.

1. Introduction

Let be a real -th order -dimensional rectangular tensor, where for , , , and . If the entries of the tensor are invariant under any permutation of and , is called a partially symmetric tensor. For the sake of simplicity, let be the set of all partially symmetric rectangular tensors with order and dimension . By the relationship between partially symmetric tensors and homogeneous polynomials, we always use the following notation:

By this notation, we know that is strictly copositive if and only if

Particularly, if and , then it reduces to the copositivity of symmetric tensors [110].

The copositive tensor has attracted many researches’ attention since it plays an important role in polynomial optimization [11], hypergraph theory [1], vacuum stability of a general scalar potential [12], tensor complementarity problem [13, 14], tensor eigenvalue complementarity problem [15, 16], and so on [1737]. Kannike proved the vacuum stability conditions for more complicated potentials with the help of the copositive tensor [12]. Ling et al. [16] proposed that the tensor generalized eigenvalue complementarity problem is solvable and has one solution at least under assumptions that the related square tensor is strictly copositive. During the process of application, a challenging problem is how to detect the copositivity of tensors numerically.

Recently, several numerical algorithms are proposed to check the copositivity of symmetric tensors. To the best of our knowledge, the first numerical algorithm was proposed by Chen et al. in [2], where the algorithm is based on the representation of the multivariate form in barycentric coordinates with respect to the standard simplex. Then, by a suitable convex subcone of a copositive tensor cone, an updated numerical algorithm for copositivity of tensors was proposed in [1]. It must be pointed out that the methods of [1, 2] can only capture strictly copositive tensors and noncopositive tensors. To overcome this drawback, in [38], Li et al. proposed an SDP relaxation algorithm to test the copositivity of higher-order tensors. Very recently, Nie et al. gave a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor [39]. If the potential tensor is copositive, the algorithm can get a certificate for the copositivity. Otherwise, the algorithm can get a point that refutes the copositivity. Furthermore, it is showed that the detection can be done by solving a finite number of semidefinite relaxations for all matrices and tensors.

For the copositivity of partially symmetric tensors, Gu et al. gave the first two spectral properties in [40], and some necessary or sufficient conditions for a real partially symmetric rectangular tensor to be copositive are further established. Moreover, an equivalent notion of strictly copositive rectangular tensors is presented [40]. In [41], Wang et al. extended the simplicial partition method for symmetric tensors to check the copositivity of partially symmetric tensors. However, as we discussed above, it can only capture all strictly copositive rectangular tensors or noncopositive rectangular tensors. When the input tensor is copositive but not strictly copositive, the algorithm may not stop in general. To solve this, motivated by the algorithm of [38, 39], we propose a new algorithm to check the copositivity of partially symmetric tensors in this paper.

The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries on polynomials. In Section 3, we formulate the potential problem as a proper polynomial optimization problem which can be efficiently solved by Lasserre-type semidefinite relaxations. Then, a numerical method is proposed to check whether a given partially symmetric tensor is copositive or not, and the convergence of this algorithm is established. Several numerical experiments are listed in Section 4, and final remarks are given in Section 5.

2. Preliminaries

Let be the ring of the polynomial with variables . Let denote the vector space of polynomials with degree at most . The degree of a polynomial is denoted as deg (). Denote as the vector of all entries which equals one. A polynomial is called SOS if there exist such that . Denote by the set of all SOS polynomials. For , let . Then, for any polynomial , it can be denoted by , and denotes its sequence of coefficients in the monomial basis of . For matrix , its transpose is denoted by . For a symmetric matrix means is positive semidefinite. More details about polynomial optimization can be seen in [4245].

For , , and denote . For , denotes the smallest integer that is not smaller than . If the subset satisfies that and , then is called ideal. For a polynomial tuple , the ideal generated by is defined such that

The -th truncation ideal generated by is

For complex and real algebraic varieties of polynomial tuple , define

The quadratic module generated by is (denote )

For , where is the space of real vectors indexed by , define

For a polynomial , the -th localizing matrix of is the symmetric matrix satisfyingfor all with , , where denotes the coefficient vector of the polynomial . When is the moment matrix Let be a polynomial tuple; its localizing matrix is defined such that

3. The SDP Algorithm for Copositivity of Partially Symmetric Tensors

In this section, we establish an equivalent condition for the copositivity of partially symmetric tensors. Then, the concerned problem can be reformulated as a polynomial optimization problem. To continue, recall that a partially symmetric tensor is strictly copositive if and only ifwhich is equivalent with the following optimization problem:

Clearly, tensor is strictly copositive if and only if . Problem (11) can be solved by classical Lasserre relaxations [46]. Since the objection function is continuous and the feasible region is compact, problem (11) always has a solution. Without loss of generality, assume is one of the solutions of (11); then, it satisfies the following KKT-conditions with :

By (12), we obtain that , , and

Combining this with the fact that , we consider the following optimization problem:

It is clear to see that problems (11) and (14) are equivalent in the sense that they have the same optimal solution. To solve (14), we introduce the following notations:

So, the problem of (14) can be rewritten such that

By the Lasserre-type semidefinite relaxations of (16), consider the semidefinite programwhere , with . It is obvious that the feasible set is compact, and the Archimedean condition holds. Thus, the asymptotic convergence of (17) is always guaranteed. Moreover, is copositive if for some , and is a monotonically decreasing sequence, with the decreasing of order , i.e.,

Now, we present an algorithm to check the copositivity of a given partially symmetric rectangular tensor (Algorithm 1).

Step 0: given an arbitrary vector . Let .
 Step 1: solve the semidefinite relaxation (17). If , then stop, and is copositive. If , go to Step 2.
 Step 2: solve the following semidefinite program:
 for an optimizer if it is feasible. If it is infeasible, let and go to Step 1.
 Step 3: let . If , then is not copositive and stop. Otherwise, let and go to Step 1.

The following theorem shows the convergence of Algorithm 1 for any partially symmetric tensor.

Theorem 1. Suppose is a partially symmetric tensor. Then, the following properties hold:(i)For all , problem (16) is feasible and achieves its optimal value for all sufficiently large(ii)For all , problem (19) has an optimizer if it is feasible(iii)If is copositive, Algorithm 1 must stop with when is sufficiently large(iv)If is not copositive, Algorithm 1 must stop with for almost all when is sufficiently large

Proof. (i)Since the feasible set of (11) is compact, it must have a minimizer . On the contrary, is a feasible point for (16), which implies that the semidefinite relaxation (17) is always feasible. Since , let ; then, it holds that . We now show that the feasible set of (17) is compact as follows. First of all, we haveThen, ; since , . Furthermore, for all , the -th diagonal entry of is nonnegative, which implies thatTake in the following analysis. By the same argument as (21) and repeating times, we can show that for all . By the definition of , we know that the diagonal entries are precisely , . Since , all the entries of must be between −1 and 1. So, is bounded, and the feasible set of (17) is compact. Hence, the optimal value can always be achieved. In the following, we will show that for all sufficiently large.By direct computation, the optimization (16) is equivalent with the following problem:For simplicity, denoteCorresponding Lasserre’s relaxations for (22) areFor , where , any feasible solution of (17) is also a feasible solution of (24), soNext, we show that the set of polynomialsis Archimedean, i.e., there exists such that the inequality defines a compact set in . Let and ; we haveSo, is Archimedean by Theorem 3.3 of [47]; we know that when is sufficiently large. Hence, when all values are sufficiently large.(ii)The proof is the same with (i).(iii)Clearly, is copositive if and only if . By item (i), for all big enough. Therefore, if is copositive, we must have for all large enough.(iv)If is not copositive, then . By (i), there exists such that for all . Hence, for all , problem (19) is equivalent with the following problem:It is -th Lasserre’s relaxation for the polynomial optimizationThe feasible region of (29) is clearly compact. When is arbitrary, (29) has a unique optimizer . Hence, for almost all , is the unique optimizer. For notation convenience, denote by the optimizer of (19) with the relaxation order . Let . By Corollary 3.5 of [48] or Theorem 3.3 of [49], the sequence must converge to . Since , we must have when is sufficiently large.

4. Numerical Examples

In this section, we give several numerical examples to show the efficiency of Algorithm 1. Let denote the set of all permutations of , and let when . All experiments are done in Matlab2014b on a desktop computer with Intel (R) Core (TM)i7-6500 CPU @ 2.50 GHz 2.60 GHz and 16 GB of RAM.

Example 1. Suppose that is given byThe corresponding polynomial for tensor isBy Algorithm 1, we know that with , , which implies that rectangular tensor is copositive.

Example 2. Suppose that with entries such thatThe corresponding polynomial of isBy Algorithm 1, we obtain that with optimal solution , which implies that is copositive but not strictly copositive.

Example 3. Suppose that is given bySo, the corresponding polynomial of is thatwhere . By Algorithm 1, we have with , , which implies that the rectangular tensor is copositive.

Example 4. Suppose that is given byThe corresponding polynomial of the partially symmetric rectangular tensor iswhere . By Algorithm 1, we know that with , , which implies that the rectangular tensor is not copositive.

Example 5. Suppose is a tensor with entries such thatThe corresponding polynomial of the partially symmetric rectangular tensor iswhere . By Algorithm 1, we know that with , , which implies that the rectangular tensor is strictly copositive.

5. Conclusions

In this paper, based on Lasserre’s hierarchy of semidefinite relaxations, we propose a new criterion to judge whether a given partially symmetric rectangular tensor is copositive or not. The convergence for the proposed algorithm is established. Furthermore, numerical examples demonstrate that the proposed algorithm is effective when the input rectangular tensor has lower dimension and orders, and it is difficult for the case with higher order or higher dimension. We will continue to study this problem in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Each author contributed equally to this paper and read and approved the final manuscript.

Acknowledgments

This project was supported by the Natural Science Foundation of China (11601261), the Shandong Provincial Natural Science Foundation (ZR2019MA022), and Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J14LI52).