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Multilayer networks refer to systems composed of several interactingnetworks. This approach provides a generalized framework for exploringreal-world and engineering systems in more depth and with greater accuracy thanthe traditional single-layer network scheme. Despite the wide set of results indistributed optimization in single-layer networks, there is a lack ofdevelopments in multilayer systems. In this paper, we propose two algorithmsfor distributed optimization problems using the supra-Laplacian matrix of amultiplex network and its diffusion dynamics. The algorithms are a distributedprimal-dual saddle-point algorithm and a distributed gradient descentalgorithm. We establish a set of saddle-point equations based on the relationbetween consensus and diffusion dynamics via the graph Laplacian and impose ananalogous restriction to the traditional Laplacian in our multiplex distributedoptimization problem to obtain the multiplex supra-Laplacian matrix. We extendthe distributed gradient descent algorithm for multiplex networks using thissupra-Laplacian matrix. We propose several theoretical results to analyze theconvergence of both algorithms, demonstrating that they converge to the uniqueoptimizer. We validate the proposed algorithms using numerical examples. Wealso explore the existence of a phase transition in the consensus time due tothe interlayer diffusion constant in the numerical examples. Finally, acoordinated dispatch for interdependent infrastructure networks (energy-gas) ispresented to illustrate the application of the proposed framework to realengineering problems.

Consensus-based Distributed Optimization for Multi-agent Systems over Multiplex Networks