  This paper studies the problem of finding best rank-1 approximations for both
symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent
to optimizing homogeneous polynomials over unit spheres; for nonsymmetric
tensors, this is equivalent to optimizing multi-quadratic forms over
multi-spheres. We propose semidefinite relaxations, based on sum of squares
representations, to solve these polynomial optimization problems. Their
properties and structures are studied. In applications, the resulting
semidefinite programs are often large scale. The recent Newton-CG augmented
Lagrangian method by Zhao, Sun and Toh is suitable for solving these
semidefinite relaxations. Extensive numerical experiments are presented to show
that this approach is practical in getting best rank-1 approximations.
