Distributed Convex Optimization “Over-the-Air” in Dynamic Environments

This paper presents a decentralized algorithm for solving distributed convex optimization problems in dynamic networks with time-varying objectives. The unique feature of the algorithm lies in its ability to accommodate a wide range of communication systems, including previously unsupported ones, by abstractly modeling the information exchange in the network. Specifically, it supports a novel communication protocol based on the “over-the-air” function computation (OTA-C) technology, that is designed for an efficient and truly decentralized implementation of the consensus step of the algorithm. Unlike existing OTA-C protocols, the proposed protocol does not require the knowledge of network graph structure or channel state information, making it particularly suitable for decentralized implementation over ultra-dense wireless networks with time-varying topologies and fading channels. Furthermore, the proposed algorithm synergizes with the “superiorization” methodology, allowing the development of new distributed algorithms with enhanced performance for the intended applications. The theoretical analysis establishes sufficient conditions for almost sure convergence of the algorithm to a common time-invariant solution for all agents, assuming such a solution exists. Our algorithm is applied to a real-world distributed random field estimation problem, showcasing its efficacy in terms of convergence speed, scalability, and spectral efficiency. Furthermore, we present a superiorized version of our algorithm that achieves faster convergence with significantly reduced energy consumption compared to the unsuperiorized algorithm.