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Separable multi-block convex optimization problem appears in manymathematical and engineering fields. In the first part of this paper, wepropose an inertial proximal ADMM to solve a linearly constrained separablemulti-block convex optimization problem, and we show that the proposed inertialproximal ADMM has global convergence under mild assumptions on theregularization matrices. Affine phase retrieval arises in holography, dataseparation and phaseless sampling, and it is also considered as anonhomogeneous version of phase retrieval that has received considerableattention in recent years. Inspired by convex relaxation of vector sparsity andmatrix rank in compressive sensing and by phase lifting in phase retrieval, inthe second part of this paper, we introduce a compressive affine phaseretrieval via lifting approach to connect affine phase retrieval withmulti-block convex optimization, and then based on the proposed inertialproximal ADMM for multi-block convex optimization, we propose an algorithm torecover sparse real signals from their (noisy) affine quadratic measurements.Our numerical simulations show that the proposed algorithm has satisfactoryperformance for affine phase retrieval of sparse real signals.

Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval