Augmented Lagrange algorithms for distributed optimization over multi-agent networks via edge-based method

In this paper, the augmented Lagrange (AL) algorithm for distributed optimization is studied. Compared with the existing results, this paper uses different techniques, including the factorization of weighted Laplacian and the spectral decomposition technique, to prove the linear convergence of the AL algorithm, and simultaneously provides a novel description on the convergence rate. First, by using an important factorization of weighted Laplacian, it is proved that the linear convergence of the AL algorithm can be achieved via a simplified analysis procedure. Within this framework, a novel quantitative description on the convergence rate is then provided based on spectral decomposition technique. Meanwhile, by determining the monotonicity of an auxiliary function, a connection between convergence rate, step size and edge weights is established. Finally, simulation examples illustrate the theoretical results.