  In this paper, we propose the inexact alternating minimization algorithm
(inexact AMA), which allows inexact iterations in the algorithm, and its
accelerated variant, called the inexact fast alternating minimization algorithm
(inexact FAMA). We show that inexact AMA and inexact FAMA are equivalent to the
inexact proximal-gradient method and its accelerated variant applied to the
dual problem. Based on this equivalence, we derive complexity upper-bounds on
the number of iterations for the inexact algorithms. We apply inexact AMA and
inexact FAMA to distributed optimization problems, with an emphasis on
distributed MPC applications, and show the convergence properties for this
special case. By employing the complexity upper-bounds on the number of
iterations, we provide sufficient conditions on the inexact iterations for the
convergence of the algorithms. We further study the special case of quadratic
local objectives in the distributed optimization problems, which is a standard
form in distributed MPC. For this special case, we allow local computational
errors at each iteration. By exploiting a warm-starting strategy and the
sufficient conditions on the errors for convergence, we propose an approach to
certify the number of iterations for solving local problems, which guarantees
that the local computational errors satisfy the sufficient conditions and the
inexact distributed optimization algorithm converges to the optimal solution.
