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} = ...
