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 $$x\in \mathbb {R}^n$$ x ∈ R n if and only if there exists a sum-of-squares (SOS) polynomial $$\sigma (x)$$ σ ( x ) such that $$\sigma P$$ σ P is a sum of sparse SOS matrices. Second, we show that setting $$\sigma (x)=(x_1^2 + \cdots + x_n^2)^\nu $$ σ ( x ) = ( x 1 2 + ⋯ + x n 2 ) ν for some integer $$\nu $$ ν suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set $$\mathcal {K}=\{x:g_1(x)\ge 0,\ldots ,g_m(x)\ge 0\}$$ K = { x : g 1 ( x ) ≥ 0 , … , g m ( x ) ≥ 0 } satisfying the Archimedean condition, then $$P(x) = S_0(x) + g_1(x)S_1(x) + \cdots + g_m(x)S_m(x)$$ P ( x ) = S 0 ( x ) + g 1 ( x ) S 1 ( x ) + ⋯ + g m ( x ) S m ( x ) for matrices $$S_i(x)$$ S i ( x ) that are sums of sparse SOS matrices. Finally, if $$\mathcal {K}$$ K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $$(x_1^2 + \cdots + x_n^2)^\nu P(x)$$ ( x 1 2 + ⋯ + x n 2 ) ν P ( x ) with some integer $$\nu \ge 0$$ ν ≥ 0 when P and $$g_1,\ldots ,g_m$$ 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.

Iterative Inner/outer Approximations for Scalable Semidefinite Programs using Block Factor-width-two Matrices

“…Choosing a suitable partition might be problem dependent; we refer the interested reader to [20] for more discussions. Here, we highlight that 1) a coarser partition normally leads to a faster convergence in Algorithms 1-2, as shown in our extensive numerical experiments in Section V; 2) a coarser partition also leads a smaller number p in (21) and (12). The latter fact is important in constructing the problem in each iteration, especially for large-scale cases.…”

“…The choice of the matrices V t+1 as the factorization of X * t in ( 14) leads to a sequence of monotonically decreasing cost values in (12). We have the following proposition.…”

“…This completes the proof. Remark 1 (Solving outer approximations): Unlike the inner approximation (12), the outer approximations (20) and (21) are not in the standard form of SDPs. Thus they cannot be solved directly using standard conic solvers.…”

“…We note that the size of PSD constraints has been reduced in (23), but the number of equality constraints is n 2 . Thus, solving (23) might not be as efficient as solving (12). Our iterative outer approximations for solving (1) is listed in Algorithm 2.…”

“…Specifically, one can try to decompose X = t i=1 Q i , where Q i 0 are nonzero only on a certain (and, ideally, small) principal submatrix. When X has a special chordal sparsity pattern, such a decomposition is equivalent [5], [11], [12]. In general, the decomposition above gives an inner approximation of the PSD cone S n + .…”