{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/sum-of-squares-chordal-decomposition-of","title":"Sum-of-squares chordal decomposition of polynomial matrix inequalities","arxiv_id":"2007.11410","date":"2020-07-22","proceeding":null,"authors":["Yang Zheng","Giovanni Fantuzzi"],"abstract":"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 Positivstellens\\\"atze. First, we establish that a polynomial matrix $P(x)$ with chordal sparsity is positive semidefinite for all $x\\in \\mathbb{R}^n$ if and only if there exists a sum-of-squares (SOS) polynomial $\\sigma(x)$ such that $\\sigma P$ is a sum of sparse SOS matrices. Second, we show that setting $\\sigma(x)=(x_1^2 + \\cdots + x_n^2)^\\nu$ for some integer $\\nu$ suffices if $P$ is homogeneous and positive definite globally. Third, we prove that if $P$ is positive definite on a compact semialgebraic set $\\mathcal{K}=\\{x:g_1(x)\\geq 0,\\ldots,g_m(x)\\geq 0\\}$ satisfying the Archimedean condition, then $P(x) = S_0(x) + g_1(x)S_1(x) + \\cdots + g_m(x)S_m(x)$ for matrices $S_i(x)$ that are sums of sparse SOS matrices. Finally, if $\\mathcal{K}$ is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $(x_1^2 + \\ldots + x_n^2)^\\nu P(x)$ with some integer $\\nu\\geq 0$ when $P$ and $g_1,\\ldots,g_m$ are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.","url_abs":"https://arxiv.org/abs/2007.11410v2","url_pdf":"https://arxiv.org/pdf/2007.11410v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"sum-of-squares-chordal-decomposition-of","repo_url":"https://github.com/aeroimperial-optimization/sos-chordal-decomposition-pmi","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"sum-of-squares-chordal-decomposition-of","repo_url":"https://github.com/zhengy09/sos_csp","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}