Sum-of-squares chordal decomposition of polynomial matrix inequalities
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We prove three decomposition results for sparse positive (semi-)definite polynomial matrices. First, we show 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(x)P(x) can be decomposed into a sum of sparse SOS matrices, each of which is zero outside a small principal submatrix. Second, we establish that setting σ(x)=(x_1^2 + ⋯ + x_n^2)^ν for some integer ν suffices if P(x) is even, homogeneous, and positive definite. Third, we prove a sparse-matrix version of Putinar's Positivstellensatz: if P(x) has chordal sparsity and 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, each of which is zero outside a small principal submatrix. Using these decomposition results, we obtain sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables. We also obtain new convergent hierarchies of sparsity-exploiting SOS reformulations to convex optimization problems with large and sparse polynomial matrix inequalities. Analytical examples illustrate all our decomposition results, while large-scale numerical examples demonstrate that the corresponding sparsity-exploiting SOS hierarchies have significantly lower computational complexity than traditional ones.
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