The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz (2017). In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.

翻译：矩-平方和（moment-SOS）层级是近似计算定义在多项式（不）等式可行域上的n元多项式最小值的最著名且应用最广泛的方法之一。该层级的一个关键特征在于，在固定层级上，它可以表示为变量数n的多项式规模的半定规划。尽管这暗示其可能可在多项式时间内计算，但实际情况并非必然如此。事实上，正如O'Donnell（2017）及后来Raghavendra与Weitz（2017）所证明的，存在层级中使用的SOS表示具有指数比特复杂度的反例。我们研究了矩-SOS层级的计算复杂度，对Raghavendra与Weitz（2017）的早期工作进行了补充与拓展。特别地，我们建立了保证多项式时间计算可行的代数与几何条件。