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.
National Natural Science Foundation of China11601261Natural Science Foundation of Shandong ProvinceZR2019MA022Project of Shandong Province Higher Educational Science and Technology ProgramJ14LI521. Introduction
Let A=ai1i2,…,ipj1j2,…,jq be a real p,q-th order m×n-dimensional rectangular tensor, where ai1i2,…,ipj1j2,…,jq∈ℝ for ik∈m, k∈p, jl∈n, and l∈q. If the entries of the tensor are invariant under any permutation of i1,i2,…,ip and j1,j2,…,jq, A is called a partially symmetric tensor. For the sake of simplicity, let ℙSp,qm×n be the set of all partially symmetric rectangular tensors with order p,q and dimension m×n. By the relationship between partially symmetric tensors and homogeneous polynomials, we always use the following notation:(1)fx,y=Axpyq=∑i1,…,ip∈mj1,…,jq∈nai1i2,…,ipj1j2,…,jqxi1xi2,…,xipyj1yj2,…,yjq.
By this notation, we know that A=ai1,…,ipj1,…,jq∈ℙSp,qm×n is strictly copositive if and only if(2)Axpyq≥>0,for all x∈ℝ+m,y∈ℝ+n with x=1,y=1.
Particularly, if m=n and x=y, then it reduces to the copositivity of symmetric tensors [1–10].
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 [17–37]. 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 ℝx be the ring of the polynomial with variables x=x1,x2,…,xn∈ℝn. Let ℝxd⊆ℝx denote the vector space of polynomials with degree at most d. The degree of a polynomial f is denoted as deg (f). Denote e as the vector of all entries which equals one. A polynomial p is called SOS if there exist p1,p2,…,pr∈ℝx such that p=p12+p22+,…,+pr2. Denote by ∑x the set of all SOS polynomials. For x∈ℝn and α∈Nn, let xα=x1α1x2α2,…,xnαn. Then, for any polynomial f∈ℝx, it can be denoted by fx=∑α∈Nnfαxα, and vecf:=fαα∈ℕn denotes its sequence of coefficients in the monomial basis of ℝx. For matrix A, its transpose is denoted by A⊤. For a symmetric matrix X,X≽0 means X is positive semidefinite. More details about polynomial optimization can be seen in [42–45].
For α=α1,α2,…,αn∈Nn, α=α1+,⋯,+αn, and denote Ndn=α∈Nnα≤d. For t∈ℝ, ⌈t⌉ denotes the smallest integer that is not smaller than t. If the subset I⊆ℝx satisfies that I+I⊆I and I⋅ℝx⊆ℝx, then I is called ideal. For a polynomial tuple h=h1,h2,…,hs, the ideal generated by h is defined such that(3)ℐh=h1ℝx+h2ℝx+,…,+hsℝx.
The k-th truncation ideal generated by h is(4)ℐhk=h1ℝxk−degh1+h2ℝxk−degh2+,⋯,+hsℝxk−deghs.
For complex and real algebraic varieties of polynomial tuple h, define(5)VCh=x∈Cnhx=0,VRh=VCh∩ℝn.
The quadratic module generated by g=g1,g2,…,gt is (denote g0=1)(6)Qgk=∑x+g1∑x+,…,+gt∑x.
For y=yα∈ℝNdn,α∈Ndn, where ℝNdn is the space of real vectors indexed by α∈Ndn, define(7)∑α∈Ndnpαx1α1x2α2,…,xnαn,y=∑α∈Ndnpαyα.
For a polynomial q∈ℝx2k, the k-th localizing matrix of q is the symmetric matrix Lkqy satisfying(8)vecp1⊤Lqkyvecp2=qp1p2,y,for all p1,p2∈ℝx with degp1, degp2≤k−⌈degq/2⌉, where vecpi denotes the coefficient vector of the polynomial pi. When q=1,Lqky is the moment matrix Mky=L1ky. Let f=f1,f2,…,fr be a polynomial tuple; its localizing matrix is defined such that(9)Lfky=Lf1ky,Lf2ky,…,Lfrky.
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 A∈ℙSp,qm×n is strictly copositive if and only if(10)Axpzq≥0>0,for all x∈ℝ+m,z∈ℝ+n with x=1,z=1,which is equivalent with the following optimization problem:(11)f∗=minAxpzqs.t.e1⊤x=1,e2⊤z=1x∈ℝ+m,z∈ℝ+n.
Clearly, tensor A is strictly copositive if and only if f∗≥0>0. 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 x∗,z∗ is one of the solutions of (11); then, it satisfies the following KKT-conditions with λ,μ∈ℝ,v∈ℝm, and w∈ℝn:(12)pAx∗p−1z∗q−λe1−v=0,qAx∗pz∗q−1−μe2−w=0,e1⊤x∗=1,e2⊤z∗=1,x∗≥0,z∗≥0,v≥0,w≥0,x∗⊤v=0,z∗⊤w=0.
By (12), we obtain that λ=pAx∗pz∗q, μ=qAx∗pz∗q, and(13)pAx∗p−1zq−λe1≥0,qAx∗pz∗q−1−μe2≥0,x∗⊤pAx∗p−1z∗q−λx∗⊤e1=0,z∗⊤qAx∗pz∗q−1−μz∗⊤e2=0.
Combining this with the fact that x∗≤1,z∗≤1, we consider the following optimization problem:(14)minAxpzqs.t.x⊤Axp−1zq−Axpzqx⊤e1=0,z⊤Axpzq−1−Axpzqz⊤=0,Axp−1zq−Axpzqe1≥0,Axpzq−1−Axpzqe2≥0,e1⊤x=1,e2⊤z=1,1−x2≥0,1−z2≥0x∈ℝ+m,z∈ℝ+n.
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:(15)fx,z=Axpzq,gx,z=Axp−1zq−Axpzqe1,Axpzq−1−Axpzqe2,1−x2,1−z2,xi,zj,hx,z=e⊤x,0n+m=1,e⊤0,zn+m=1,xiAxp−1zqi−Axpzqxi,zjAxpzq−1j−Axpzqzj.
So, the problem of (14) can be rewritten such that(16)f∗=minfx,zs.t.gx,z≥0,hx,z=0.
By the Lasserre-type semidefinite relaxations of (16), consider the semidefinite program(17)ρk=min∑α∈Nn+mfαyαs.t.Lgky≽0,Lhky=0,y0=1,Mky≽0,y∈ℝN2kn+m,where k=k0,k0+1,…, with k0=max⌈p/2⌉,⌈q/2⌉. It is obvious that the feasible set is compact, and the Archimedean condition holds. Thus, the asymptotic convergence of (17) is always guaranteed. Moreover, A is copositive if ρk≥0 for some k, and ρk is a monotonically decreasing sequence, with the decreasing of order k, i.e.,(18)ρk0≤ρk0+1≤,…,≤ρk≤,…,≤f∗.
Now, we present an algorithm to check the copositivity of a given partially symmetric rectangular tensor (Algorithm 1).
Algorithm 1: An SDP method for copositivity of a partially symmetric tensor A∈ℙSp,qm×n.
Step 0: given an arbitrary vector ξ∈ℝℕp+qn+m. Let k=max⌈p/2⌉,⌈q/2⌉.
Step 1: solve the semidefinite relaxation (17). If ρk≥0, then stop, and A is copositive. If ρk<0, go to Step 2.
Step 2: solve the following semidefinite program:
for an optimizer y∗ if it is feasible. If it is infeasible, let k=k+1 and go to Step 1.
Step 3: let x∗,z∗=y∗e1,…,y∗em,y∗em+1,…,y∗em+n. If Ax∗pz∗q<0, then A is not copositive and stop. Otherwise, let k=k+1 and go to Step 1.
The following theorem shows the convergence of Algorithm 1 for any partially symmetric tensor.
Theorem 1.
Suppose A∈ℙSp,qm×n is a partially symmetric tensor. Then, the following properties hold:
For all k≥0, problem (16) is feasible and achieves its optimal value ρk=f∗ for all k sufficiently large
For all k≥0, problem (19) has an optimizer if it is feasible
If A is copositive, Algorithm 1 must stop with ρk≥0 when k is sufficiently large
If A is not copositive, Algorithm 1 must stop with fx,z<0 for almost all ξ∈ℝℕp+qn+m when k is sufficiently large
Proof.
Since the feasible set of (11) is compact, it must have a minimizer x∗,z∗. On the contrary, x∗,z∗ is a feasible point for (16), which implies that the semidefinite relaxation (17) is always feasible. Since L1−x2k≽0, let X=x,0,0,zx∈ℝm,z∈ℝn⊆ℝm+n; then, it holds that L1−x2k≽0. We now show that the feasible set of (17) is compact as follows. First of all, we have
(20)1≥y2e1+y2e2+,…,+y2en+m.
Then, 0≤y2ei≤1; since Mky≽0, i∈m+n. Furthermore, for all 0<α≤k−1, the α,α-th diagonal entry of L1−x2k is nonnegative, which implies that
(21)y2α≥y2e1+2α+y2e2+2α+,…,+y2en+m+2α.
Take α=e1,e2,…,em+n in the following analysis. By the same argument as (21) and repeating k−1 times, we can show that 0≤y2β≤1 for all β≤k. By the definition of Mky, we know that the diagonal entries Mky are precisely y2β, β≤k. Since Mky≽0, all the entries of Mky must be between −1 and 1. So, y 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 ρk=f∗ for all k sufficiently large.
By direct computation, the optimization (16) is equivalent with the following problem:
So, F is Archimedean by Theorem 3.3 of [47]; we know that ρk′=f∗ when k is sufficiently large. Hence, ρk=f∗ when all k values are sufficiently large.
The proof is the same with (i).
Clearly, A is copositive if and only if f∗≥0. By item (i), ρk=f∗ for all k big enough. Therefore, if A is copositive, we must have ρk≥0 for all k large enough.
If A is not copositive, then f∗<0. By (i), there exists k1∈N such that ρk=f∗ for all k≥k1. Hence, for all k≥k1, problem (19) is equivalent with the following problem:
It is k-th Lasserre’s relaxation for the polynomial optimization(29)minξ⊤x,zm+ns.t.1−e⊤x,0m+n≥0,1−e⊤0,zm+n≥0,x≥0,z≥0,f∗−fX≥0.
The feasible region of (29) is clearly compact. When ξ∈ℝℕp+qn+m is arbitrary, (29) has a unique optimizer X∗=x∗,z∗. Hence, for almost all ξ∈ℝℕp+qn+m, X∗ is the unique optimizer. For notation convenience, denote by y^k the optimizer of (19) with the relaxation order k. Let Xk=y^ke1,…,y^ken+m. By Corollary 3.5 of [48] or Theorem 3.3 of [49], the sequence Xkk=k0∞ must converge to X∗. Since f∗≤ρk∗<0, we must have fXk<0 when k is sufficiently large.
4. Numerical Examples
In this section, we give several numerical examples to show the efficiency of Algorithm 1. Let Sπi1i2,…,im denote the set of all permutations of i1i2,…,im, and let ρk∗=0 when ρk∗<1e−5. 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 A∈ℙS2,22×4 is given by(30)a1111=1,a1122=1,a1133=1,a1144=1,a2211=1,a2222=1,a2233=1,a2244=1,∑i1i2j1j2∈Sπ1112ai1i2j1j2=2,∑i1i2j1j2∈Sπ1134ai1i2j1j2=2,∑i1i2j1j2∈Sπ1113ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1114ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1123ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1124ai1i2j1j2=−2,∑i1i2j1j2∈Sπ2212ai1i2j1j2=2,∑i1i2j1j2∈Sπ2234ai1i2j1j2=2,∑i1i2j1j2∈Sπ2213ai1i2j1j2=−2,∑i1i2j1j2∈Sπ2214ai1i2j1j2=−2,∑i1i2j1j2∈Sπ2223ai1i2j1j2=−2,∑i1i2j1j2∈Sπ2224ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1211ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1222ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1212ai1i2j1j2=−4,∑i1i2j1j2∈Sπ1244ai1i2j1j2=−2,∑i1i2j1j2∈Sπ1234ai1i2j1j2=−4,∑i1i2j1j2∈Sπ1213ai1i2j1j2=4,∑i1i2j1j2∈Sπ1214ai1i2j1j2=4,∑i1i2j1j2∈Sπ1223ai1i2j1j2=4,∑i1i2j1j2∈Sπ1224ai1i2j1j2=4,∑i1i2j1j2∈Sπ1233ai1i2j1j2=−2.
The corresponding polynomial for tensor A is(31)fx,y=x1−x22y1+y2−y3−y42,x=x1,x2,y=y1,y2,y3,y4.
By Algorithm 1, we know that f∗=0 with x=0.5000,0.5000, y=0.2500,0.2500,0.2500,0.2500, which implies that rectangular tensor A is copositive.
Example 2.
Suppose that A∈ℙS2,21×2 with entries such that(32)a1111=1,a1122=1,∑i1i2j1j2∈Sπ11,12ai1i2j1j2=−2.
The corresponding polynomial of A is(33)fx,y=x12y12−2x12y1y2+x12y22,x=x1,y=y1,y2.
By Algorithm 1, we obtain that f∗=0 with optimal solution x,y=1.0000,0.7071,0.7071, which implies that A is copositive but not strictly copositive.
Example 3.
Suppose that A∈ℙS2,22×2 is given by(34)a1111=1,a1122=1,a2211=1,a2222=1,∑i1i2j1j2∈Sπ11,12ai1i2j1j2=−2,∑i1i2j1j2∈Sπ22,12ai1i2j1j2=−2,∑i1i2j1j2∈Sπ12,11ai1i2j1j2=2,∑i1i2j1j2∈Sπ12,22ai1i2j1j2=2,∑i1i2j1j2∈Sπ12,12ai1i2j1j2=−4.
So, the corresponding polynomial of A is that(35)fx,y=x12y12+x12y22−2x12y1y2+x22y12+x22y22−2x22y1y2+2x1x2y12+2x1x2y22−4x1x2y1y2,where x=x1,x2,y=y1,y2. By Algorithm 1, we have f∗=0 with x∗=0.5126,0.4874, y∗=0.5000,.5000, which implies that the rectangular tensor is copositive.
Example 4.
Suppose that A∈ℙS3,23×2 is given by(36)a11122=1,a22222=1,a33311=1,∑i1i2i3j1j2∈Sπ123,12ai1i2i3j1j2=−3.
The corresponding polynomial of the partially symmetric rectangular tensor A is(37)fx,y=x13y22+x23y22+x33y12−4x1x2x3y1y2,where x=x1,x2,x3,y=y1,y2. By Algorithm 1, we know that f∗=−0.0639 with x∗=0.7652,0.4702,0.7652, y∗=0.3572,0.8724, which implies that the rectangular tensor is not copositive.
Example 5.
Suppose A∈ℙS2,22×2 is a tensor with entries such that(38)a1111=1,a1122=−1,a2211=1,a2222=1,∑i1i2j1j2∈Sπ11,12ai1i2j1j2=2,∑i1i2j1j2∈Sπ22,12ai1i2j1j2=2,∑i1i2j1j2∈Sπ12,11ai1i2j1j2=2,∑i1i2j1j2∈Sπ12,22ai1i2j1j2=2,∑i1i2j1j2∈Sπ12,12ai1i2j1j2=−4.
The corresponding polynomial of the partially symmetric rectangular tensor A is(39)fx,y=x12y12−x12y22+2x12y1y2+x22y12+x22y22+2x22y1y2+2x1x2y12+2x1x2y22−4x1x2y1y2,where x=x1,x2,y=y1,y2. By Algorithm 1, we know that f∗=0.3333 with x∗=0.6666,0.3334, y∗=0.5000,.5000, 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).
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