Oberseminar "Dynamische Systeme und Kontrolltheorie" - Prof. Dr. rer. nat. Björn Rüffer

The Douglas-Rachford method was conceived as a matrix splitting method to solve partial differential equations numerically, but has since proven itself to be a robust go-to method to solve very generic non-convex optimisation problems, including feasibility problems. In the convex case it is well known that the method converges weakly if the problem has a solution, and even if it does not, the method provides insights into how far away from a solution the problem is. The elusive non-convex case has seen several problem-specific convergence proofs based on Lyapunov’s direct method, which have subsequently inspired approaches to accelerate the algorithm. This talk will provide an introduction to the Douglas-Rachford algorithm and the problems it can solve, as well as attempt to give an overview of some of the Lyapunov function constructions and the benefits the Lyapunov perspective provides.