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We consider the problem of minimizing the sum of a convex function and aconvex function composed with an injective linear mapping. For such problems,subject to a coercivity condition at fixed points of the corresponding Picarditeration, iterates of the alternating directions method of multipliersconverge locally linearly to points from which the solution to the originalproblem can be computed. Our proof strategy uses duality and strong metricsubregularity of the Douglas--Rachford fixed point mapping. Our analysis doesnot require strong convexity and yields error bounds to the set of modelsolutions. We show in particular that convex piecewise linear-quadraticfunctions naturally satisfy the requirements of the theory, guaranteeingeventual linear convergence of both the Douglas--Rachford algorithm and thealternating directions method of multipliers for this class of objectives undermild assumptions on the set of fixed points. We demonstrate this result onquantitative image deconvolution and denoising with multiresolution statisticalconstraints.

Local Linear Convergence of the ADMM/Douglas--Rachford Algorithms without Strong Convexity and Application to Statistical Imaging