Zheng Y, Fantuzzi G (2021)
Publication Type: Journal article
Publication year: 2021
DOI: 10.1007/s10107-021-01728-w
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all x∈ Rn if and only if there exists a sum-of-squares (SOS) polynomial σ(x) such that σP is a sum of sparse SOS matrices. Second, we show that setting σ(x)=(x12+⋯+xn2)ν for some integer ν suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set K= { x: g
APA:
Zheng, Y., & Fantuzzi, G. (2021). Sum-of-squares chordal decomposition of polynomial matrix inequalities. Mathematical Programming. https://dx.doi.org/10.1007/s10107-021-01728-w
MLA:
Zheng, Yang, and Giovanni Fantuzzi. "Sum-of-squares chordal decomposition of polynomial matrix inequalities." Mathematical Programming (2021).
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