Derivation and Analysis of the Primal-Dual Method of Multipliers Based on Monotone Operator Theory

In this paper we present a novel derivation for an existing node-based algorithm for distributed optimisation termed the primal-dual method of multipliers (PDMM). In contrast to its initial derivation, in this work monotone operator theory is used to connect PDMM with other first-order methods such as Douglas-Rachford splitting and the alternating direction method of multipliers thus providing insight to the operation of the scheme. In particular, we show how PDMM combines a lifted dual form in conjunction with Peaceman-Rachford splitting to remove the need for collaboration between nodes per iteration. We demonstrate sufficient conditions for strong primal convergence for a general class of functions while under the assumption of strong convexity and functional smoothness, we also introduce a primal geometric convergence bound. Finally we introduce a distributed method of parameter selection in the geometric convergent case, requiring only finite transmissions to implement regardless of network topology.