论文标题
三角多项式的平方总和层次结构的指数收敛性
Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials
论文作者
论文摘要
我们考虑了通过下限的平方级层次结构对多元三角多项式的不受约束优化。我们首先显示$ O（1/s^2）的收敛速率，用于放松度$ s $，而没有任何对三角多项式的假设以最小化。其次，当多项式具有有限数量的全球最小化器，在这些最小化器中具有可逆的黑森斯人时，我们以显式常数显示指数的收敛速率。我们的结果还适用于最大程度地减少HyperCube上的常规多元多项式。
We consider the unconstrained optimization of multivariate trigonometric polynomials by the sum-of-squares hierarchy of lower bounds. We first show a convergence rate of $O(1/s^2)$ for the relaxation with degree $s$ without any assumption on the trigonometric polynomial to minimize. Second, when the polynomial has a finite number of global minimizers with invertible Hessians at these minimizers, we show an exponential convergence rate with explicit constants. Our results also apply to minimizing regular multivariate polynomials on the hypercube.
