Random block-coordinate methods for inconsistent convex optimisation problems

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2023; no. 1; pp. 14 - 38
• Main Authors: Staudigl, Mathias, Jacquot, Paulin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 06.11.2023; Springer Nature B.V; SpringerOpen
• Subjects: AC-OPF; Algorithms; Analysis; Applications of Mathematics; Complexity; Convex optimisation; Convexity; Differential Geometry; Distributed optimisation; Linear algebra; Linear programming; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Methods; Nesterov acceleration; Optimization; Optimization and Real World Applications; Random block-coordinate method; Topology; 90B10; 90C25; Random block-coordinate method; Nesterov acceleration; Distributed optimisation; 90C06; AC-OPF; Convex optimisation
• ISSN: 2730-5422; 2730-5422
• DOI: 10.1186/s13663-023-00751-0

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 / 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 as dual variables in our optimisation framework.