Block Factor-width-two Matrices and Their Applications to Semidefinite and Sum-of-squares Optimization

TL;DR

This paper introduces block factor-width-two matrices to improve the scalability and accuracy of semidefinite and sum-of-squares optimization, enabling more efficient solutions for large-scale problems.

Contribution

It proposes a new hierarchy of inner and outer approximations of the PSD cone using block factor-width-two matrices, extending existing methods and balancing scalability with solution quality.

Findings

Enhanced inner-approximation of the PSD cone.

Scalable approach for large-scale SOS problems.

Numerical experiments confirm improved performance.

Abstract

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a trade-off between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings.