Primal-dual interior-point methods for semidefinite programming, as implemented in widely used solvers, rely on the elegant theory of self-scaled barriers and the property that the dense symmetric positive semidefinite matrices form a symmetric convex cone. In applications it is very common that the constraints in a semidefinite program are sparse. It is natural to pose such problems as conic optimization problems over cones of sparse positive semidefinite matrices, in order to exploit theory and algorithms from sparse linear algebra. Examples of this approach are the splitting techniques based on decomposition theorems for chordal sparsity patterns, which have been applied successfully in first-order methods and interior-point algorithms. However sparse semidefinite matrix cones are not symmetric, and conic optimization algorithms for these cones lack the extensive theory behind the primal-dual methods for dense semidefinite optimization. The talk will explore new formulations of primal-dual interior-point methods for sparse semidefinite programming. It will mostly focus on the important special case of chordal sparsity patterns that define homogeneous convex cones. The talk is based on joint work with Levent Tuncel.
The Douglas-Rachford algorithm is a popular method for finding minimizers of sums of nonsmooth convex functions; or, more generally, zeros of sums of maximally monotone operators. In this talk, I will focus on classical and recent results on this method as well as challenges for the future.