Sum-of-squares chordal decomposition of polynomial matrix inequalities

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 Positivstellensätze. First, we establish a matrix P ( x ) with chordal sparsity is semidefinite all $$x\in \mathbb {R}^n$$ x ? R n if only there exists sum-of-squares (SOS) $$\sigma (x)$$ ? ( ) such P$$ P sum SOS matrices. Second, show setting (x)=(x_1^2 + \cdots x_n^2)^\nu $$ = 1 2 + ? ? some integer $$\nu suffices homogeneous definite globally. Third, on compact semialgebraic set $$\mathcal {K}=\{x:g_1(x)\ge 0,\ldots ,g_m(x)\ge 0\}$$ K { : g ? 0 , … m } satisfying Archimedean condition, then $$P(x) = S_0(x) g_1(x)S_1(x) g_m(x)S_m(x)$$ S $$S_i(x)$$ i are sums Finally, {K}$$ not or does satisfy obtain similar $$(x_1^2 P(x)$$ \ge 0$$ when $$g_1,\ldots ,g_m$$ even degree. Using these results, find representation polynomials quadratic correlatively in subset variables, construct new convergent hierarchies reformulations convex optimization problems large inequalities. Numerical examples demonstrate have significantly lower computational complexity than traditional ones.