We study the problem of maximizing the geometric mean of $d$ low-degree non-negative forms on the real or complex sphere in $n$ variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree $O(d)$ on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial optimization problem requires solving a semidefinite program (SDP) of size $n^{O(d)}$, with multiplicative approximation guarantees of $\Omega(\frac{1}{n})$. We exploit the compact representation of this polynomial to introduce a SDP relaxation of size polynomial in $n$ and $d$, and prove that it achieves a constant factor multiplicative approximation when maximizing the geometric mean of non-negative quadratic forms. We also show that this analysis is asymptotically tight, with a sequence of instances where the gap between the relaxation and true optimum approaches this constant factor as $d \rightarrow \infty$. Next we propose a series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation. Finally we show that this approach can be generalized for relaxations of products of non-negative forms of any degree.

中文翻译：

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球上非负形式乘积的半定松弛

我们研究了在$ n $变量中最大化实数或复数球面上$ d $低度非负形式的几何平均数的问题。我们证明，即使形式是二次的，这个高度非凸的问题也是NP-困难的，它等效于优化球面上度数$ O（d）$的齐次多项式。针对此多项式优化问题的基于平方和的标准凸松弛要求求解大小为$ n ^ {O（d）} $的半定程序（SDP），并具有$ \ Omega（\ frac {1}的乘法近似保证{n}）$。我们利用该多项式的紧凑表示形式在$ n $和$ d $中引入大小多项式的SDP松弛，并证明当最大化非负二次形式的几何平均值时，它可以实现常数因子乘法逼近。我们还表明，这种分析是渐近严格的，在一系列实例中，松弛和真实最优之间的差距接近该常数，为d \ rightarrow \ infty $。接下来，我们提出一系列复杂度不断增加的中间松弛，这些内插可插值到整个Sum-of-Squares松弛，以及一种舍入算法，可从任何中间松弛的解中找到一个近似解。最后，我们证明了该方法可以推广到任意程度的非负形式的乘积松弛。以及从任何中间松弛的解中找到近似解的舍入算法。最后，我们证明了该方法可以推广到任意程度的非负形式的乘积松弛。以及从任何中间松弛的解中找到近似解的舍入算法。最后，我们证明了该方法可以推广到任意程度的非负形式的乘积松弛。