  This work studies the problem of maximizing a higher degree real homogeneous
multivariate polynomial over the unit sphere. This problem is equivalent to
finding the leading eigenvalue of the associated symmetric tensor of higher
order, which is nonconvex and NP-hard. Recent advances show that semidefinite
relaxation is quite effective to find a global solution. However, the solution
methods involve full/partial eigenvalue decomposition during the iterates,
which heavily limits its efficiency and scalability. On the other hand, for odd
degree (odd order) cases, the order has to be increased to even, which
potentially reduces the efficiency.
  To find the global solutions, instead of convexifying the problem, we
equivalently reformulate the problem as a nonconvex matrix program based on an
equivalence property between symmetric rank-1 tensors and matrices of any
order, which is a generalization of the existing results. The program is
directly solved by a vanilla alternating direction method, which only involves
the computation of leading eigenvalue/singular value of certain matrices,
benefiting from the special structure of the program. Although being nonconvex,
under certain hypotheses, it is proved that the algorithm converges to a
leading eigenvalue of the associated tensor. Numerical experiments on different
classes of tensors demonstrate that the proposed approach has a significant
improvement in efficiency and scalability, while it can keep the effectiveness
of semidefinite relaxation as much as possible. For instance, the proposed
method finds the leading eigenpair of a third-order 500 dimensional Hilbert
tensor in a personal computer within 100 seconds.
