In this paper, we consider a new pattern of sparsity for SOS Programming named by cross sparsity patterns. We use matrix decompositions for a class of PSD matrices with chordal sparsity patterns to construct sets of supports for a sparse SOS decomposition. The method is applied to the certificate of the nonnegativity of sparse polynomials and unconstrained sparse polynomial optimization problems. Various numerical experiments are given. It turns out that our method can dramatically reduce the computational cost and can handle really huge polynomials, for example, polynomials with $10$ variables, of degree $40$ and more than $5000$ terms. 查看全文>>