A multi-agent projected dual gradient method with primal convergence guarantees

We consider a class of multi-agent optimization problems, where each agent is endowed with a strongly convex (but not necessarily differentiable) loss function and is subject to individual constraints composed of linear equalities, convex inequalities, and convex set constraints. We derive a novel algorithm that allows the agents to collaboratively reach a decision that minimizes the sum of the loss functions over the intersection of the individual constraints. The algorithm is based on a projected dual gradient technique and exploits the structure of the individual constraint sets to avoid costly projections. A convergence rate analysis shows that the primal iterates produced by individual agents under our algorithm converge to the primal optimal solution at a rate that is superior to alternatives in the literature. Finally, we provide a thorough comparison of our algorithm with alternatives in terms of both theoretical and algorithmic aspects.