The present paper considers leveraging network topology information to improve the convergence rate of ADMM for decentralized optimization, where networked nodes work collaboratively to minimize the objective. Such problems can be solved efficiently using ADMM via decomposing the objective into easier subproblems. Properly exploiting network topology can significantly improve the algorithm performance. Hybrid ADMM explores the direction of exploiting node information by taking into account node centrality but fails to utilize edge information. This paper fills the gap by incorporating both node and edge information and provides a novel convergence rate bound for decentralized ADMM that explicitly depends on network topology. Such a novel bound is attainable for certain class of problems, thus tight. The explicit dependence further suggests possible directions to optimal design of edge weights to achieve the best performance. Numerical experiments show that simple heuristic methods could achieve better performance, and also exhibits robustness to topology changes.
 Distributed optimization has gained popularity in many areas because it can to cope with the increasing volume of data, and the corresponding demand for computational resources. A popular distributed solver is the alternating direction method of multipliers (ADMM). Fully decentralized (D) ADMM arises when node-to-node communications have to abide by the underlying network connectivity. DADMM’s convergence however, slows down as the network diameter grows large. To deal with this challenge, the recently proposed hybrid (H) ADMM provides considerable performance boost over DADMM by exploiting local topology information. But HADMM only applies to unweighted graphs. The present contribution develops a weighted hybrid (WH) consensus-based ADMM approach that can deal with weighted graphs. The resultant scheme further improves the performance of HADMM through graph-aware weight tuning. Theoretical analysis offers convergence guarantees and establishes linear convergence rate, while numerical tests on various graphs demonstrate the WHADMM merits. We derive the optimal step-size and over-relaxation parameter that minimizes the convergence time of two ADMM-based algorithms for distributed averaging. Our study shows that the convergence times for given step-size and over-relaxation parameters depend on the spectral properties of the normalized Laplacian of the underlying communication graph. Motivated by this, we optimize the edge-weights of the communication graph to improve the convergence speed even further. The performance of the ADMM algorithms with our parameter selection are compared with alternatives from the literature in extensive numerical simulations on random graphs.