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.
翻译:我们研究如何在实际或复杂领域以美元变量在实际或复杂领域最大限度地使用美元这一低度非负面形式的几何平均值的问题。我们表明,这一高度非混凝土问题即使形式是四面形的,也是NP-硬的。这相当于优化该领域以美元(d)美元为单位的同质多元度宽度。基于共度的平方标准总和对于这一多元优化问题而言,需要解决一个规模为$ ⁇ O(d)美元(SDP)的半成品(SDP)半成份的半成品度非负负偏差(SDP)方案。我们利用这一极非混凝土质问题的缩略近似保证是NPO(frag)$(frac{{1}n)$。我们利用这一多面形形形形的缩影代表来引入SDP以美元和美元为单位的单一度宽度宽度宽度宽度宽度宽度宽度宽度宽度宽度宽度宽度的SDP,并且证明,在尽可能扩大非负偏差的平方形平方形平方形平方形的平方形平方形平方形的平方程方法中,这样一小平整平整平整平整的平整的平整的平整平整的平整的平整的平整的平整式平整式平整式平整的平整式平整。