Abstract

In this talk, we present a generalisation of scaled diagonally dominant (SDD) matrices to block partitioned matrices as well as their applications in control theory and optimisation. Our basic definition of block SDD matrices relies on a comparison matrix, which is formed by computing particular norms of the blocks in the partitioning. If the comparison matrix is stable then partitioned matrix is stable, moreover, there exists a block-diagonal solution to Lyapunov inequality and the H infinity Riccati inequality. Furthermore, these solutions can be constructed using the combination of linear algebra and linear programming methods. We then focus on symmetric matrices and introduce a set of block factor-width-two matrices, which can also be seen as a generalisation of SDD matrices. Block factor-width-two matrices form a proper cone, which is a subset of positive semidefinite matrices. We use these cones and their duals 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 decomposition of a large semidefinite constraint into a number of smaller semidefinite constraints. As the main application of this class of matrices, we envision large-scale semidefinite feasibility optimisation programs including the sum-of-squares (SOS) programs. We present numerical examples from SOS optimisation showcasing the strengths of this decomposition.