Random block-coordinate methods for inconsistent convex optimisation problems

We develop a novel randomised block-coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying midway between the celebrated Chambolle–Pock primal-dual algorithm and Tseng’s accelerated proximal gradient method, we establish global convergence of the last iterate as well as optimal O(1 / k) and O(1 / k²) complexity rates in the convex and strongly convex case, respectively, k being the iteration count. Motivated by the increased complexity in the control of distribution-level electric-power systems, we test the performance of our method on a second-order cone relaxation of an AC-OPF problem. Distributed control is achieved via the distributed locational marginal prices (DLMPs), which are obtained as dual variables in our optimisation framework.