This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schm\"udgen and Putinar Positivstellens\"atze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.

翻译：摘要：本研究针对基础半代数集合上的多项式全局极小化，导出了带有相关稀疏性的动量-平方和层次结构的收敛速度上限。主要结论是，当最大团的规模足够小时，基于Shu-mu-teng 和 Pu-ti-na两位数学家在Positivstellens\"atze中提出的相关稀疏层次结构，其多项式收敛率多项式，不仅取决于环境维度，而且在效率上优于目前最佳可获得的密集层次结构的收敛速度。具体性能由所需运行内点算法的时间和得到目标精度的全局最小值边界来衡算。