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In this paper we present a novel derivation for an existing node-basedalgorithm for distributed optimisation termed the primal-dual method ofmultipliers (PDMM). In contrast to its initial derivation, in this workmonotone operator theory is used to connect PDMM with other first-order methodssuch as Douglas-Rachford splitting and the alternating direction method ofmultipliers thus providing insight to the operation of the scheme. Inparticular, we show how PDMM combines a lifted dual form in conjunction withPeaceman-Rachford splitting to remove the need for collaboration between nodesper iteration. We demonstrate sufficient conditions for strong primalconvergence for a general class of functions while under the assumption ofstrong convexity and functional smoothness, we also introduce a primalgeometric convergence bound. Finally we introduce a distributed method ofparameter selection in the geometric convergent case, requiring only finitetransmissions to implement regardless of network topology.

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