  This paper proposes a partially inexact alternating direction method of
multipliers for computing approximate solution of a linearly constrained convex
optimization problem. This method allows its first subproblem to be solved
inexactly using a relative approximate criterion, whereas a proximal term is
added to its second subproblem in order to simplify it. A stepsize parameter is
included in the updating rule of the Lagrangian multiplier to improve its
computational performance.
  Pointwise and ergodic interation-complexity bounds for the proposed method
are established. To the best of our knowledge, this is the first time that
complexity results for an inexact ADMM with relative error criteria has been
analyzed.
  Some preliminary numerical experiments are reported to illustrate the
advantages of the new method.
