Achieving Linear Convergence for Distributed Optimization with Zeno-Like-Free Event-Triggered Communication Scheme

This paper develops a distributed algorithm for solving a class of optimization problems which are defined over undirected connected networks of N agents where each function f(i) is held privately by agent i. The communication between agents in the network is limited: each agent can only interact with its neighboring agents at some independent event-triggered sampling time instants. The algorithm uses a doubly stochastic mixing matrix and employs a fixed step-size and, yet, exactly drives all agents' states to a global and consensual minimizer. Under some fairly standard assumptions on objective functions, i.e., strong convexity and smoothness, we show that the algorithm converges to the optimal solution at a linear rate as long as the constant step-size do not exceed some upper bound and the convergence rate can be explicitly characterized. The Zeno-like behavior is rigorously excluded, that is, the difference between any two successive sampling time instants of each agent is at least two, reducing the communication cost by at least one half comparing with traditional methods. We provide a numerical experiment to demonstrate the efficacy of the proposed algorithm and to validate the theoretical findings.