Paatero [1] and Kolda and Bader [2] pointed out that optimal solutions of the best rank-R approximation of a tensor may not exist in general. This problem is also called as the degeneration problem. In this paper, we study the issue of numerical algorithms of structure preserving rank-R approximation and structure preserving CP-decomposition of partially symmetric tensors. We prove that the rank-2 approximation optimization model based on the alternating best rank-1 approximation is non-degenerate. We then propose an alternating symmetric high-order power method (a-SHOPM) for the best structure preserving rank-R approximation problem and prove the convergence of the algorithm. Furthermore, we find that if the classic BFGS algorithm is used to obtain the initial point first, then the a-SHOPM has better computational performance. So, we then propose a BFGS-a-SHOPM algorithm. Numerical examples show that the BFGS-a-SHOPM algorithm has a better success rate and less computation time for the best structure preserving rank-R approximation or structure preserving CP-decomposition.
