An Approximately-Zero-Gradient-Sum Algorithm For Consensus Optimization

This paper presents a set of distributed second-order methods for addressing unconstrained, smooth convex optimization over fixed networks. The proposed second-order methods, referred to as Approximately-Zero-Gradient-Sum (AZGS) algorithms, allow each node to update by combining the Hessian inverse of its local objective and the estimates of its neighbors, so that the gradient sum of the local objectives can be sufficiently close to zero at each iteration. We show that the AZGS algorithms, with properly selected parameters, enable all the nodes to asymptotically reach a consensus that can be arbitrarily close to the optimal solution. Finally, simulation results demonstrate the effectiveness of the AZGS algorithms.