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. 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.

中文翻译：

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线性形式的最大化乘积和半正定矩阵的恒久

我们研究多项式优化问题的凸松弛，最大化复杂球体上线性形式的乘积。我们表明，这个凸程序也是 Hermitian 半正定 (HPSD) 矩阵的永久松弛。通过分析构造性随机舍入算法，我们获得了改进的 HPSD 矩阵永久乘法近似因子。我们还为 HPSD 矩阵提出了范德瓦尔登猜想的模拟，其中多项式优化问题被解释为永久松弛。