Decomposition and Completion of Sum-of-Squares Matrices

TL;DR

This paper develops methods for decomposing and completing sum-of-squares matrices with chordal sparsity, enabling more efficient solutions to large-scale sparse SOS programs.

Contribution

It extends chordal decomposition and completion techniques from scalar matrices to SOS matrices with polynomial entries, facilitating scalable SOS optimization.

Findings

Decomposition of sparse SOS matrices into sums of smaller SOS matrices.

Completion of SOS matrices via principal submatrix conditions.

Numerical results show improved efficiency in large-scale SOS problems.

Abstract

This paper introduces a notion of decomposition and completion of sum-of-squares (SOS) matrices. We show that a subset of sparse SOS matrices with chordal sparsity patterns can be equivalently decomposed into a sum of multiple SOS matrices that are nonzero only on a principal submatrix. Also, the completion of an SOS matrix is equivalent to a set of SOS conditions on its principal submatrices and a consistency condition on the Gram representation of the principal submatrices. These results are partial extensions of chordal decomposition and completion of scalar matrices to matrices with polynomial entries. We apply the SOS decomposition result to exploit sparsity in matrix-valued SOS programs. Numerical results demonstrate the high potential of this approach for solving large-scale sparse matrix-valued SOS programs.