  Separable multi-block convex optimization problem appears in many
mathematical and engineering fields. In the first part of this paper, we
propose an inertial proximal ADMM to solve a linearly constrained separable
multi-block convex optimization problem, and we show that the proposed inertial
proximal ADMM has global convergence under mild assumptions on the
regularization matrices. Affine phase retrieval arises in holography, data
separation and phaseless sampling, and it is also considered as a
nonhomogeneous version of phase retrieval that has received considerable
attention in recent years. Inspired by convex relaxation of vector sparsity and
matrix rank in compressive sensing and by phase lifting in phase retrieval, in
the second part of this paper, we introduce a compressive affine phase
retrieval via lifting approach to connect affine phase retrieval with
multi-block convex optimization, and then based on the proposed inertial
proximal ADMM for multi-block convex optimization, we propose an algorithm to
recover sparse real signals from their (noisy) affine quadratic measurements.
Our numerical simulations show that the proposed algorithm has satisfactory
performance for affine phase retrieval of sparse real signals.
