  In this paper, we propose a communication- and computation-efficient
algorithm to solve a convex consensus optimization problem defined over a
decentralized network. A remarkable existing algorithm to solve this problem is
the alternating direction method of multipliers (ADMM), in which at every
iteration every node updates its local variable through combining neighboring
variables and solving an optimization subproblem. The proposed algorithm,
called as COmmunication-censored Linearized ADMM (COLA), leverages a
linearization technique to reduce the iteration-wise computation cost of ADMM
and uses a communication-censoring strategy to alleviate the communication
cost. To be specific, COLA introduces successive linearization approximations
to the local cost functions such that the resultant computation is first-order
and light-weight. Since the linearization technique slows down the convergence
speed, COLA further adopts the communication-censoring strategy to avoid
transmissions of less informative messages. A node is allowed to transmit only
if the distance between the current local variable and its previously
transmitted one is larger than a censoring threshold. COLA is proven to be
convergent when the local cost functions have Lipschitz continuous gradients
and the censoring threshold is summable. When the local cost functions are
further strongly convex, we establish the linear (sublinear) convergence rate
of COLA, given that the censoring threshold linearly (sublinearly) decays to 0.
Numerical experiments corroborate with the theoretical findings and demonstrate
the satisfactory communication-computation tradeoff of COLA.
