Relaxed hybrid consensus ADMM for distributed convex optimisation with coupling constraints

In this study, the solution of a convex distributed optimisation problem with a global coupling inequality constraint is considered. By using the Lagrange duality framework, the problem is transformed into a distributed consensus optimisation problem and then based on the recently proposed Hybrid Alternating Direction Method of Multipliers (H-ADMM), which merges distributed and centralised optimisation concepts problems, a novel distributed algorithm is developed. In particular, the authors offer a reformulation of the original H-ADMM in an operator theoretical framework, which exploits the known relationship between ADMM and Douglas–Rachford splitting. In addition, the authors' formulation allows us to generalise the H-ADMM by including a relaxation constant, not present in the original design of the algorithm. Moreover, an adaptive penalty parameter selection scheme that consistently improves the practical convergence properties of the algorithm is proposed. Finally, the convergence results of the proposed algorithm are discussed and moreover, in order to present the effectiveness and the major capabilities of the proposed algorithm in off-line and on-line scenarios, distributed quadratic programming and distributed model predictive control problems are considered in the simulation section.