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

翻译：我们研究多元优化问题的共振松动,在复杂领域将线性形式产品最大化。我们显示,这一共振程序也是埃米提亚正正半无限制矩阵永久值的松动。通过分析一种建设性的随机四舍五入算法,我们获得了一种更佳的多倍近似系数,以至HPSD矩阵永久值,以及这一近似值的计算效率证书。我们还提出了一种类似范德韦尔登对HPSD矩阵的推测,其中将多元优化问题解释为永久值的松动。