Distributed Optimization in Discrete-time Multi-Agent Systems with Independent Step Size

In this paper, we develop a distributed algorithm for multiple agents aiming to solve an optimization problem. The communication topology among these agents is governed by an undirected graph. In our algorithm, each agent has an independent step-size, the upper bound of which are independent on network topologies. The step-sizes only depend on local objective functions. The algorithm is with different step-sizes and guarantees the convergence of the agent states to a common point. With such step-size range, the states of each agent converge to a common optimal point at a linear rate 0(kappa(-k)) where 0 < kappa < 1 and k is the number of iterations. As compared to the existing approaches, we only require that the local cost function of each agent is convex rather than strongly convex. Finally, the numerical examples are given to validate the theoretical results.