  This paper considers decentralized consensus optimization problems where
nodes of a network have access to different summands of a global objective
function. Nodes cooperate to minimize the global objective by exchanging
information with neighbors only. A decentralized version of the alternating
directions method of multipliers (DADMM) is a common method for solving this
category of problems. DADMM exhibits linear convergence rate to the optimal
objective but its implementation requires solving a convex optimization problem
at each iteration. This can be computationally costly and may result in large
overall convergence times. The decentralized quadratically approximated ADMM
algorithm (DQM), which minimizes a quadratic approximation of the objective
function that DADMM minimizes at each iteration, is proposed here. The
consequent reduction in computational time is shown to have minimal effect on
convergence properties. Convergence still proceeds at a linear rate with a
guaranteed constant that is asymptotically equivalent to the DADMM linear
convergence rate constant. Numerical results demonstrate advantages of DQM
relative to DADMM and other alternatives in a logistic regression problem.
