Sum-of-squares chordal decomposition of polynomial matrix inequalities - Zheng, Yang; Fantuzzi, Giovanni (Mathematical programming, 2023)
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.