Consensus-based Optimization in Multiplex Networks⋆

Abstract

Multilayer networks are systems formed by several interacting networks. This framework generalizes the single-layer network scheme, developing a scenario where most real and engineering systems can be explored in a deeper and more accurately manner. 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, using the role of the supra-Laplacian matrix of a multiplex network and its diffusion dynamics, we develop a distributed primal-dual saddle-point algorithm, and a gradient descent algorithm for distributed optimization problems. Taking advantage of the relation between the consensus and diffusion dynamics via the graph Laplacian, we establish a set of saddle-point equations driven by the multiplex supra-Laplacian matrix, obtained by imposing an analogous to the traditional Laplacian restriction in our multiplex distributed optimization problem. This extension is performed to the distributed gradient descent algorithm for multiplex networks. Two theorems are proposed to study the convergence analysis for both methods, where it is demonstrated that both flows converge to the unique optimizer. We present numerical examples showing the validation of our results.