Maximizing products of linear forms, and the permanent of positive semidefinite matrices

We study the convex relaxation of a polynomial optimization problem, maximizing product linear forms over complex sphere. show that this program is also permanent Hermitian positive semidefinite (HPSD) matrices. By analyzing constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to HPSD matrices, as well computationally efficient certificates for approximation. propose analog van der Waerden's conjecture where problem interpreted permanent.