Linear matrix inequalities with chordal sparsity patterns and applications to robust quadratic optimization

We discuss nonsymmetric interior-point methods for linear cone programs with chordal sparse matrix cone constraints. The algorithms take advantage of fast recursive algorithms for evaluating the function values and derivatives for the logarithmic barrier functions of the cone of positive semidefinite matrices with a given chordal sparsity pattern, and of the corresponding dual cone. We provide numerical results that show that our implementation can be significantly faster than general purpose semidefinite programming solvers. As a specific application, we discuss robust quadratic optimization.