Sum-of-squares chordal decomposition of polynomial matrix inequalities.

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 ∈ R n if and only if there exists a sum-of-squares (SOS) polynomial σ (x) such that σ P is a sum of sparse SOS matrices. Second, we show that setting σ (x) = (x 1 2 + ⋯ + x n 2) 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 1 (x) ≥ 0 , ... , g m (x) ≥ 0 } satisfying the Archimedean condition, then P (x) = S 0 (x) + g 1 (x) S 1 (x) + ⋯ + g m (x) S m (x) for matrices S i (x) that are sums of sparse SOS matrices. Finally, if K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x 1 2 + ⋯ + x n 2) ν P (x) with some integer ν ≥ 0 when P and g 1 , ... , 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. [ABSTRACT FROM AUTHOR]