On the rate of convergence of distributed relaxed-ADMM algorithms in distributed optimization

In this paper, we present a preliminary analysis of the convergence rate of the relaxed ADMM-based algorithm recently introduced in the literature for distributed optimization problems. It is well known that the relaxed ADMM is a more general algorithm than the classical ADMM. Indeed, while the performance of the latter typically depends on one parameter, say p, the performance of the former depends on two parameters, say p and a, and the relaxed ADMM reduces to the classical ADMM when α = 1/2. Interestingly, from a computational point of view, the presence of the additional parameter a does not significantly complicate the updating steps of the algorithm. Nevertheless, the relaxed ADMM is much less considered in distributed optimization problems than the classical ADMM. In this paper, we show that by restricting to quadratic functions with the same convexity and to communication graphs that are regular connected graphs, it is possible to analytically compute the eigenvalues of the matrix that governs the dynamics of the algorithm based on the relaxed ADMM. Based on these eigenvalues, it is possible to design an efficient numerical procedure to evaluate the rate of convergence and to optimise it with respect to both α and p. Our results show that, by properly tuning the a parameter, the relaxed ADMM can can achieve superior convergence rates compared to its classical ADMM counterpart; in particular, for the family of graphs we considered, the optimal a typically tends to 1 as the number of agents increases. The analysis we present is preliminary, but suggests that the use of the relaxed ADMM can significantly improve the performance of the classical ADMM when applied to distributed optimization problems, at the price of a slight increase in computational complexity.