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 that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all xin 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)ge 0,ldots ,g_m(x)ge 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 + cdots + x_n^2)^nu P(x) with some integer nu ge 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.