Random coordinate descent algorithm for open multi-agent systems with complete topology and homogeneous agents

We study the convergence in expectation of the Random Coordinate Descent algorithm (RCD) for solving optimal resource allocations problems in open multi-agent systems, i.e., multi-agent systems that are subject to arrivals and departures of agents. Assuming all local functions are strongly-convex and smooth, and their minimizers lie in a given ball, we analyse the evolution of the distance to the minimizer in expectation when the system is occasionally subject to replacements in addition to the usual iterations of the RCD algorithm. We focus on complete graphs where all agents interact with each other with the same probability, and provide conditions to guarantee convergence in open system. Finally, a discussion around the tightness of our results is provided.