Decentralized quadratically approximated alternating direction method of multipliers

This paper considers an optimization problem that components of the objective function are available at different nodes of a network and nodes are allowed to only exchange information with their neighbors. The decentralized alternating method of multipliers (DADMM) is a well-established iterative method for solving this category of problems; however, implementation of DADMM requires solving an optimization subproblem at each iteration for each node. This procedure is often computationally costly for the nodes. We introduce a decentralized quadratic approximation of ADMM (DQM) that reduces computational complexity of DADMM by minimizing a quadratic approximation of the objective function. Notwithstanding that DQM successively minimizes approximations of the cost, it converges to the optimal arguments at a linear rate which is identical to the convergence rate of DADMM. Further, we show that as time passes the coefficient of linear convergence for DQM approaches the one for DADMM. Numerical results demonstrate the effectiveness of DQM.