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：本文推导了在紧致基本半代数集上多项式全局极小化的矩-平方和层次结构（带相关稀疏性）的收敛速度上界。主要结论是，基于Schmüdgen和Putinar正定性定理的稀疏层次结构均具有多项式收敛速度，该速度取决于稀疏图中最大团的大小，而非环境维度。有趣的是，当最大团尺寸相较于环境维度足够小，且以给定精度获取全局最小值边界所需的内点法运行时间来衡量性能时，稀疏界优于当前可用的稠密层次结构最优界。