Distributed Subgradient-Based Multiagent Optimization With More General Step Sizes

A wider selection of step sizes is explored for the distributed subgradient algorithm for multigent optimization problems with time-varying and balanced communication topologies. The square summable requirement of the step sizes commonly adopted in the literature is removed. The step sizes are only required to be positive, vanishing, and nonsummable, which provides the possibility for better convergence rates. Both unconstrained and constrained optimization problems are considered. It is proved that the agents' estimates reach a consensus and converge to the minimizer of the global objective function with the more general choice of step sizes. The best convergence rate is shown to be the reciprocal of the square root of iterations for the best record of the function value at the average of the agents' estimates for the unconstrained case with the wider selection of step sizes. A simulation example is provided to show the effectiveness of the results.