Convergence rates for sums-of-squares hierarchies with correlative sparsity
研究了具有相关稀疏性的矩-平方和层级结构在紧致基本半代数集上多项式全局最小化的收敛速度上界，发现稀疏层级结构的收敛速度取决于稀疏图中最大团的大小，而非环境维度。
Abstract 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 Schmüdgen and Putinar Positivstellensätze 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.