In this paper, we consider a bi-quadratic homogeneous polynomial optimization problem over two unit spheres arising in nonlinear elastic material analysis and in entanglement studies in quantum physics. The problem is equivalent to computing the largest M-eigenvalue of a fourth-order tensor. To solve the problem, we propose a practical method whose validity is guaranteed theoretically. To make the sequence generated by the method converge to a good solution of the problem, we also develop an initialization scheme. The given numerical experiments show the effectiveness of the proposed method.

Abstract. In this paper, we show that for a symmetric tensor, its best symmetric rank-1 approximation is its best rank-1 approximation. Based on this result, a positive lower bound for the best rank-1 approximation ratio of a symmetric tensor is given. Furthermore, a higher order polynomial spherical optimization problem can be reformulated as a multilinear spherical optimization problem. Then, we present a modified power algorithm for solving the homogeneous polynomial spherical optimization problem. Numerical results are presented, illustrating the effectiveness of the proposed algorithm.Key words. symmetric tensor, the best rank-1 approximation, the best symmetric rank-1 approximation, power algorithm , i 2 , . . . , i m ). Symmetric tensors arise in higher order derivatives of smooth functions, moments, and cumulants of random vectors and have wide applications in signal and image processing, blind source separation (BSS), statistics, investment science, and so on; see [1,8,17,18,26] and references therein.A tensor is said to be rank-1 if it can be expressed as an outer product of a number of vectors. Specifically, if these vectors are all equal, then the tensor is called a symmetric rank-1 tensor. Given an m-order n-dimensional square tensor A, rank-1 tensor B = λx(1) · · · x (m) is said to be its best rank-1 approximation if it minimizes the least-squares cost function A − B F over the manifold of rank-1 tensors. Similarly, symmetric rank-1 tensor C = μx m is said to be the best symmetric rank-1 approximation if it minimizes the least-squares cost function A − C F over the manifold of symmetric rank-1 tensors. From optimization theory, B and C can be obtained by solving the optimization problems

In this paper, the cubic spherical optimization problems, including the cubic one-spherical/two-spherical/three-spherical optimization problems, are discussed. We first show that the two-spherical optimization problem is a special case of the three-spherical optimization problem. Then we show that the one-spherical optimization problem and the two-spherical optimization problem have the same optimal value when the tensor is symmetric. In addition, NP-hardness of them are established. For the cubic three-spherical optimization problem, we discuss the conditions under which the problem is polynomial time solvable and if the polynomial time approximation scheme (PTAS) exists. Then we present a relative quality bound by finding the largest singular values of matrices. Finally, a practical method for solving the cubic three-spherical optimization problem is proposed and preliminary numerical results are reported.The cubic three-spherical optimization problem has the following form:where n, p, q ≥ 2 and C is a third order (n × p × q)-dimensional real tensor. These three problems arise from the best rank-one approximation to A, B and C, respectively. In signal processing, a discrete multidimensional signal is treated as an mth order tensor with m ≥ 3 and the optimal low-rank approximation of tensor is used to approximate of multidimensional signal. When m = 3, (1.3) is used as a suboptimal solution of low-rank approximation of tensor in [9]. In [15], the optimal value of (1.1) with n = 3 is used to characterize the phase of the magnetic resonance signal in biological tissues. The best rank-1 approximation of higher order tensor has some applications in image processing and wireless communication systems, etc., [3,2,5,6,7,14]. Some other applications are about the eigenvalues of tensors; see [12,13].Furthermore, these three problems are homogeneous polynomial optimization problems, which have been considered by many scholars. In [14], some computational methods for solving (1.3) were proposed. In [11], minimizing the homogeneous polynomial over multi-spheres and unit-spheres was discussed there and some bounds were presented via sum of squares (SOS) relaxation. In [4], the authors proved the NP-hardness of the multi-variate homogeneous polynomial function and proposed approximation algorithms for considered problems.Contributions. First, we show that the cubic one-spherical/two-spherical optimization problems are special cases of the cubic three-spherical optimization problem, which can be used to improve the bound presented by Theorem 4.4 in [11] when d = 2. Then we reformulate the cubic two-spherical optimization problem as an NP-hard quartic optimization problem over a unit sphere. Based on this reformulation, the NP-hardness of the cubic two-spherical/three-spherical optimization problems are established. These will be presented in Section 2.In the subsequent discussion, we only focus on the cubic three-spherical optimization problem. We show that when some matrices are simultaneously diagonalized and min{n, p, q} = ...

A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. If it is copositive, the algorithm can get a certificate for the copositivity. If it is not, the algorithm can get a point that refutes the copositivity. We show that the detection can be done by solving a finite number of semidefinite relaxations, for all matrices and tensors.2010 Mathematics Subject Classification. 15A69,15B48,90C22.

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