Sum-of-squares chordal decomposition of polynomial matrix inequalities

TL;DR

This paper develops new sum-of-squares decomposition methods for sparse polynomial matrices with chordal sparsity, enabling more efficient convex optimization for large-scale polynomial matrix inequalities.

Contribution

It introduces sparsity-exploiting SOS decomposition theorems for polynomial matrices, extending classical Positivstellens"atze to the sparse setting and providing new hierarchies for optimization.

Findings

Hierarchies can significantly reduce computational complexity.

Decomposition theorems apply to both compact and non-compact semialgebraic sets.

Numerical examples confirm efficiency improvements.

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 Positivstellens\"atze. First, we establish that a polynomial matrix $P(x)$ with chordal sparsity is positive semidefinite for all $x\in \mathbb{R}^n$ 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 $\mathcal{K}=\{x:g_1(x)\geq 0,\ldots,g_m(x)\geq 0\}$ 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 $(x_1^2 + \ldots + x_n^2)^\nu P(x)$ with some integer $\nu\geq 0$ 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.