  In this paper, we propose and analyze an inexact version of the symmetric
proximal alternating direction method of multipliers (ADMM) for solving
linearly constrained optimization problems. Basically, the method allows its
first subproblem to be solved inexactly in such way that a relative approximate
criterion is satisfied. In terms of the iteration number $k$, we establish
global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic
convergence rates of the method for a domain of the acceleration parameters,
which is consistent with the largest known one in the exact case. Since the
symmetric proximal ADMM can be seen as a class of ADMM variants, the new
algorithm as well as its convergence rates generalize, in particular, many
others in the literature. Numerical experiments illustrating the practical
advantages of the method are reported. To the best of our knowledge, this work
is the first one to study an inexact version of the symmetric proximal ADMM.
