Subgradient averaging for multi-agent optimisation with different constraint sets

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.