In this paper, we develop an Inexact Proximal-indefinite Stochastic ADMM (abbreviated as IPS-ADMM) for solving a class of separable convex optimization problems whose objective functions consist of two parts: one is an average of many smooth convex functions and another is a convex but possibly nonsmooth function. The involved smooth subproblem is tackled by an inexact accelerated stochastic gradient method based on an adaptive expansion step to avoid the case that the sample size can be huge so that computing the objective function value or its gradient is much expensive. The restulting nonsmooth subproblem is solved inexactly under a relative error criterion to avoid the case that the proximal operator is potentially unavailable. Since the dual variable updates twice, it allows a more flexible and larger stepsize region compared with
standard deterministic and stochastic ADMMs. By a variational analysis, we characterize the generated iterates as a variational inequality and finally establish the sublinear convergence rate of this IPS-ADMM in terms of the objective function gap and constraint violation. The efficacy of our IPS-ADMM is demonstrated by comparing with several state-of-the-art methods for solving the three-dimensional (3D) CT reconstruction problem in medical imaging.
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