SDP relaxation method for detecting P -tensors

P -tensor and P 0 -tensor are introduced in tensor complementarity problem, which have wide applications in many ﬁelds such as game theory, tensor complementarity problem. In this paper, we discuss how to check whether a given symmetric tensor is P ( P 0 ) -tensor or not. For a symmetric tensor, it is a P ( P 0 ) -tensor is equivalent to the positivity(nonnegativity) of a polynomial optimization problem. For such polynomial optimization problem, a SDP relaxation method is proposed. By the proposed method, the P ( P 0 ) -tensor can be detected by solving a ﬁnite number of SDP relaxations. Furthermore, numerical examples are reported to show the efﬁciency of the proposed algorithm.


Introduction
For positive integers m and n 1 , n 2 , • • • , n m , an m-order and (n 1 , n 2 , • • • , n m )-dimensional real tensor is an array in the space R n1×n2×•••×nm .Every tensor A from this space can be indexed as ( When A is called an m-order n-dimensional square tensor.In such case, the tensor space A tensor in T m (R n ) is said to be symmetric if its entries are invariant under permutations of indices (i 1 , i 2 , . . ., i m ).The subspace of symmetric tensors in T m (R n ) is denoted as S m (R n ).
Using the notation as in Qi [16], for A ∈ T m (R n ) and x := (x 1 , . . ., x n ) T ∈ R n , we denote Note that Ax m−1 ∈ R n .Denote [p] = {1, 2, . . ., p} for every positive integer p in this paper.With these notations, the definition of P(P 0 )-tensor is presented, introduced in [18].

Definition 1
For A ∈ T m (R n ), A is said to be a 296 SDP RELAXATION METHOD FOR DETECTING P-TENSORS (i) P 0 -tensor if and only if for any nonzero vector x ∈ R n , there exists i ∈ [n] such that x i ̸ = 0 and (ii) P-tensor if and only if for any nonzero vector This definition can be regarded as an extension of P(P 0 )-matrix, which plays important roles in linear complementarity problems and variational inequalities, see [4,6,14].
For symmetric P(P 0 )-tensors, some properties of tensor are presented in [18].It is shown that a given symmetric tensor is P(P 0 )-tensor if and only if its smallest Z(H)-eigenvalue is positive(nonnegative).Furthermore, there does not exist odd order P-tensor and nonzero P 0 -tensor.For even order symmetric tensor, it is a P(P 0 )-tensor if and only if it is positive (semi-)definite.For computing the smallest Z(H)-eigenvalue, Qi et al. [17] discuss the case of (m, n) = (3, 2).Later shifted power methods are proposed in [8,19].Recently, SDP relaxation method is applied to compute Z(H)-eigenvalues in [1].
Motivated by these fact, in this paper, we propose a numerical method by solving SDP relaxations to check whether the given symmetric tensor is P(P 0 )-tensor or not.Furthermore, it is a P-tensor if the minimum of SDP relaxation is positive; it is a P 0 -tensor but not P-tensor if the optimal value of the polynomial optimization problem is zero; it is not a P 0 -tensor if the optimal value of the polynomial optimization problem is negative.
This paper is organized as follows.Section 2 gives preliminaries on polynomial optimization.Section 3 presents SDP relaxation method of polynomial optimization problem to check symmetric P(P 0 )-tensor.Numerical examples are presented in Section 4.

Preliminaries
In this section, we review some basics in polynomial optimization.We refer to [10,11] for surveys in polynomial optimization.
In the space R n , the symbol ∥ • ∥ denotes the standard Euclidean norm.Let R[x] be the ring of polynomials with real coefficients and in variables x := (x 1 , . . ., x n ), and let R[x] d be the set of real polynomials in x whose degrees are at most d.
For a polynomial tuple h = (h 1 , h 2 , • • • , h s ), the ideal generated by h is the set The k-th truncation of I(h) is the set The complex and real algebraic varieties of h are respectively defined as A polynomial p is said to be sum of squares (SOS) if there exist The set of all SOS polynomials is denoted as Σ[x].For a given degree m, denote The quadratic module generated by a polynomial tupe g = (g The k-th truncation of the quadratic module Q(g) is the set Note that if g = ∅ is an empty tuple, then Let N be the set of nonnegative integers.For x := (x 1 , . . ., x n ), α := (α 1 , . . ., α n ) and a degree d, denote Denote by R N n d the space of all real vectors y that are indexed by α ∈ N n d .For y ∈ R N n d , we can write it as For an integer t ≤ d and y ∈ R N n d , denote the t-th truncation of y as q (y) be the symmetric matrix such that The matrix q (y) is called the k-th localizing matrix of q generated by y.It is linear in y.For instance, when If q = (q 1 , . . ., q r ) is a tuple of polynomials, we then define When q = 1 (the constant 1 polynomial), 1 (y) is called the k-th moment matrix generated by y, and we denote For instance, when n = 2 and k = 2, For a degree d, denote the monomial vector 3. Checking symmetric P(P 0 )-tensor In this section, we propose a numerical method to check whether a given symmetric tensor is P(P 0 )-tensor or not.
From [18], for a symmetric tensor, it is a P(P 0 )-tensor if and only if it is positive (semi-)definite.Furthermore, the positive (semi-)definiteness is equivalent to positivity(nonnegativity) of the smallest Z(H)-eigenvalue.Motivated by this, we propose a numerical method to check the P(P 0 )-tensor.Before proceeding, we recall the definition of Z(H)-eigenvalues.

Definition 2
(The superscript T denotes the transpose.)Such u is called a Z-eigenvector associated with λ, and (λ, u) is called a Such v is called an Heigenvector associated with α, and (α, v) is called an H-eigenpair.
From [13], the smallest Z(H)-eigenvalue is the optimal value of the following optimization problem where The computation of ( 13) for m ′ = 1 and m ′ = m − 1 are similar, hence we only adopt m ′ = 1 for cleanness in the following.
The first order optimality condition of ( 13) with m ′ = 1 can be written as following for some λ ∈ R. It is clear to see that λ = mAx m .Based on this observation, we consider the following optimization problem min Ax m s.t.
It is clear to see that problems ( 13) and ( 14) are equivalent, that is, they have the same optimal solution.Hence, it suffices to consider problem (14).For convenience, we introduce the following notations Now problem ( 14) can be rewritten as Lasserre's hierarchy ( [9]) of semidefinite relaxations for problem (16) is Stat., Optim.Inf.Comput.Vol. 5, December 2017 L. LI AND X. WANG

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where k = k 0 , k 0 + 1, . . .and k 0 = ⌈ m+1 2 ⌉.Here ⌈t⌉ is the smallest integer that is larger than or equal to t. Matrix X ≽ 0 means that X is positive semidefinite, and matricesL (k) h (y), M k (y) are defined in (8) and (9).The dual optimization problem of ( 17) is As in [9], it can be shown that for all k ρ (2) and the sequences {ρ k } and {ρ (2) k } are monotonically increasing.Furthermore, suppose y * is a minimizer of (17).If there exists a real integer t ∈ [k 0 , k] such that k = f s , and we can get r := rankM t (y * ) global optimizers of ( 16) (cf.[12]).It is clear to have the following results.

Theorem 1
Tensor A is a P-tensor if ρ (1) k > 0, and tensor A is a P 0 -tensor but not a P-tensor if and only if f s = 0. Furthermore, tensor A is not a P 0 -tensor if and only if f s < 0.
Based on this result, we present our numerical algorithm here.

Algorithm 1
To check the membership problem of P(P 0 )-tensor Step 0: For tensor A, write polynomial tuples f and h as in (15).Let k = ⌈ m+1 2 ⌉.
Step 1: Solve the hierarchy of (17) for k and get ρ k > 0, then A is a P-tensor and stop; if ρ (1) k = 0 and rank condition (19) with y * k holds for some t, then A is a P 0 -tensor but not P-tensor and stop; and if ρ (1) k < 0, rank condition (19) is satisfied for some t, then A is not a P 0 -tensor and stop.If rank condition (19) with y * k fails, let k := k + 1 and go to Step 1.

Theorem 2
Let A ∈ S m (R n ).Then we have: (19) is satisfied for some k.Hence, Algorithm 1 terminates in finitely many steps.
The proof can be seen from Theorem 3.1 in [13] and omitted here.

Numerical Examples
In this section, we give numerical examples for how to check whether a given tensor is P(P 0 )-tensor or not.The computation is implemented in MATLAB 7.10 in a Dell Linux Desktop with 8GB memory and Intel(R) CPU 2.8GHz.The software Gloptipoly 3 [7] is used to solve the semidefinite relaxations.For convenience, we use the following notation: for any i 1 , i 2 , . . ., i m ∈ [n], we use π(i and S π(i1i2•••im) to denote the set of all these permutations.ρ

By [5],
A is not a P 0 -tensor.Using Algorithm 1, it is obtained that f s = −100.0034.The detection takes about 0.15 seconds to assert that the tensor is not a P 0 -tensor.

The corresponding polynomial of the tensor
. This is the famous Motzkin polynomial, which is nonnegative but not a sum of squares.By Algorithms 1, it is obtained that f s = 0.The detection takes about 0.60 seconds to assert that the tensor is a P-tensor but not P 0 -tensor.This confirms the fact that the Motzkin polynomial f (x 1 , x 2 , x 3 ) is positive semi-definite from [2].By [3], -1 is the minimum H-eigenvalue of A, that is, A is not a P 0 -tensor.By Algorithm 1, it is obtained that f s = −0.2500.The detection takes about 1.8 seconds to assert that the tensor is not a P 0 -tensor.Similar to [15], a is chosen different values.In table 1, we report the results, where "times" denotes the elapsed time.

Conclusion
In this paper, we proposed a SDP relaxation method for checking whether a given symmetric tensor is P(P 0 )-tensor or not.We determine P(P 0 )-tensor by solving a finite number of SDP relaxations.As a prospect, whether such method can be applied to check a nonsymmetric P(P 0 )-tensor is our future research.