  Multilayer networks refer to systems composed of several interacting
networks. This approach provides a generalized framework for exploring
real-world and engineering systems in more depth and with greater accuracy than
the traditional single-layer network scheme. Despite the wide set of results in
distributed optimization in single-layer networks, there is a lack of
developments in multilayer systems. In this paper, we propose two algorithms
for distributed optimization problems using the supra-Laplacian matrix of a
multiplex network and its diffusion dynamics. The algorithms are a distributed
primal-dual saddle-point algorithm and a distributed gradient descent
algorithm. We establish a set of saddle-point equations based on the relation
between consensus and diffusion dynamics via the graph Laplacian and impose an
analogous restriction to the traditional Laplacian in our multiplex distributed
optimization problem to obtain the multiplex supra-Laplacian matrix. We extend
the distributed gradient descent algorithm for multiplex networks using this
supra-Laplacian matrix. We propose several theoretical results to analyze the
convergence of both algorithms, demonstrating that they converge to the unique
optimizer. We validate the proposed algorithms using numerical examples. We
also explore the existence of a phase transition in the consensus time due to
the interlayer diffusion constant in the numerical examples. Finally, a
coordinated dispatch for interdependent infrastructure networks (energy-gas) is
presented to illustrate the application of the proposed framework to real
engineering problems.
