Subgradient averaging for multi-agent optimisation with different constraint sets
Licio Romao, Kostas Margellos, Giuseppe Notarstefano, Antonis Papachristodoulou · arXiv (Cornell University) · 2019
We consider a multi-agent setting with agents exchanging information over a possibly time-varying network, aiming at minimising a separable objective function subject to constraints. To achieve this objective we propose a novel subgradient averaging algorithm that allows for non-differentiable objective functions and different constraint sets per agent. Allowing different constraints per agent simultaneously with a time-varying communication network constitutes a distinctive feature of our approach, extending existing results on distributed subgradient methods. To highlight the necessity of dealing with a different constraint set within a distributed optimisation context, we analyse a problem instance where an existing algorithm does not exhibit a convergent behaviour if adapted to account for different constraint sets. For our proposed iterative scheme we show asymptotic convergence of the iterates to a minimum of the underlying optimisation problem for step sizes of the form $ \\frac{\\eta}{k+1} $, $ \\eta > 0 $. We also analyse this scheme under a step size choice of $ \\frac{\\eta}{\\sqrt{k+1}} $, $ \\eta > 0 $, and establish a convergence rate of $ \\mathcal{O}(\\frac{\\ln k}{\\sqrt{k}}) $ in objective value. To demonstrate the efficacy of the proposed method, we investigate a robust regression problem and an $ \\ell_2 $ regression problem with regularisation.