  In this paper, we study the polynomial optimization problem of multi-forms
over the intersection of the multi-spheres and the nonnegative orthants. This
class of problems is NP-hard in general, and includes the problem of finding
the best nonnegative rank-one approximation of a given tensor. A
Positivstellensatz is given for this class of polynomial optimization problems,
based on which a globally convergent hierarchy of doubly nonnegative (DNN)
relaxations is proposed. A (zero-th order) DNN relaxation method is applied to
solve these problems, resulting in linear matrix optimization problems under
both the positive semidefinite and nonnegative conic constraints. A worst case
approximation bound is given for this relaxation method. Then, the recent
solver SDPNAL+ is adopted to solve this class of matrix optimization problems.
Typically, the DNN relaxations are tight, and hence the best nonnegative
rank-one approximation of a tensor can be revealed frequently. Extensive
numerical experiments show that this approach is quite promising.
