# Block Factor-Width-Two Matrices in Semidefinite Programming

## TL;DR

This paper introduces block factor-width-two matrices, a new subset of positive semidefinite matrices, enabling decomposition of large semidefinite constraints for more efficient large-scale semidefinite programming.

## Contribution

It defines block factor-width-two matrices, derives their dual cones, and develops hierarchies of approximations to improve large-scale semidefinite programming.

## Key findings

- Enables decomposition of large semidefinite constraints into smaller ones
- Provides closed-form dual cone expressions
- Demonstrates effectiveness in sum-of-squares optimization

## Abstract

In this paper, we introduce a set of block factor-width-two matrices, which is a generalisation of factor-width-two matrices and is a subset of positive semidefinite matrices. The set of block factor-width-two matrices is a proper cone and we compute a closed-form expression for its dual cone. We use these cones to build hierarchies of inner and outer approximations of the cone of positive semidefinite matrices. The main feature of these cones is that they enable a decomposition of a large semidefinite constraint into a number of smaller semidefinite constraints. As the main application of these classes of matrices, we envision large-scale semidefinite feasibility optimisation programs including sum-of-squares (SOS) programs. We present numerical examples from SOS optimisation showcasing the properties of this decomposition.