A note on semidefinite programming relaxations for polynomial optimization over a single sphere
We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank 1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose Einstein condensates (BEC) whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth order best rank 1 tensor approximation. We show that the BEC problem is NP hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.