Random block coordinate methods for inconsistent convex optimisation problems

TL;DR

This paper introduces a new randomized block coordinate primal-dual algorithm for non-smooth convex problems, achieving optimal convergence rates and demonstrating effectiveness on power system optimization tasks.

Contribution

The paper presents a novel randomized block coordinate primal-dual method with proven convergence and optimal complexity rates for a class of convex programs.

Findings

Achieves optimal $O(1/k)$ and $O(1/k^2)$ convergence rates.

Demonstrates effectiveness on power system optimization (AC-OPF).

Provides a distributed control approach via dual variables.

Abstract

We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the 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 optimal $O(1/k)$ and $O(1/k^{2})$ 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 \revise{as} dual variables in our optimisation framework.