Distributed Optimisation Via The Generalised Primal-Dual Method of Multipliers Under Unreliable and Quantised Communication

Abstract

We study distributed nonlinear optimisation under unreliable and quantised communication. The primal-dual method of multipliers (PDMM), originally developed for equality-constrained optimisation, has recently been extended to handle cone constraints [1], resulting in the generalised primal–dual method of multipliers (GPDMM). This algorithm enables the distributed solution of a broad class of optimisation problems, including semidefinite programs with partially separable structure, without reliance on interior-point methods. In this work, we focus on the robustness of GPDMM under practical communication constraints. We show that resilience to transmission failures can be achieved by interpreting the method within the framework of randomised coordinate descent, thereby removing the need for acknowledgment-based communication protocols. In addition, communication efficiency can be improved through data quantisation. The resulting update dynamics can be modelled via inexact Krasnosel’skiǐ–Mann iterations, which guarantees convergence despite quantisation errors. These findings underline the suitability of GPDMM for large-scale distributed optimisation in realistic networked environments.