Distributed Subgradient-based Multi-agent Optimization with More General Step Sizes

TL;DR

This paper broadens the class of step sizes in distributed subgradient algorithms for multi-agent optimization, proving convergence under more general conditions for both static and dynamic communication networks.

Contribution

It introduces a more general step size condition, removing the square summability requirement, and proves convergence for both unconstrained and constrained problems in various network topologies.

Findings

Agents' estimates reach consensus and converge to the optimal solution.

Convergence holds for both static and time-varying communication graphs.

Weighted averages of estimates approach the optimal solution.

Abstract

A wider selection of step sizes is explored for the distributed subgradient algorithm for multi-agent optimization problems, for both time-invariant and time-varying 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 non-summable. It is proved that in both unconstrained and constrained optimization problems, the agents' estimates reach consensus and converge to the optimal solution with the more general choice of step sizes. The idea is to show that a weighted average of the agents' estimates approaches the optimal solution, but with different approaches. In the unconstrained case, the optimal convergence of the weighted average of the agents' estimates is proved by analyzing the distance change from the weighted average to the optimal solution and showing that the weighted average is arbitrarily close to the optimal solution. In the constrained case, this is achieved by analyzing the distance change from the agents' estimates to the optimal solution and utilizing the boundedness of the constraints. Then the optimal convergence of the agents' estimates follows because consensus is reached in both cases. These results are valid for both a strongly connected time-invariant graph and time-varying balanced graphs that are jointly strongly connected.