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Consider the minimization of a nonconvex differentiable function over apolyhedron. A popular primal-dual first-order method for this problem is toperform a gradient projection iteration for the augmented Lagrangian functionand then update the dual multiplier vector using the constraint residual.However, numerical examples show that this approach can exhibit "oscillation"and may not converge. In this paper, we propose a proximal alternatingdirection method of multipliers for the multi-block version of this problem. Adistinctive feature of this method is the introduction of a "smoothed" (i.e.,exponentially weighted) sequence of primal iterates, and the inclusion, at eachiteration, to the augmented Lagrangian function a quadratic proximal termcentered at the current smoothed primal iterate. The resulting proximalaugmented Lagrangian function is inexactly minimized (via a gradient projectionstep) at each iteration while the dual multiplier vector is updated using theresidual of the linear constraints. When the primal and dual stepsizes arechosen sufficiently small, we show that suitable "smoothing" can stabilize the"oscillation", and the iterates of the new proximal ADMM algorithm converge toa stationary point under some mild regularity conditions. Furthermore, when theobjective function is quadratic, we establish the linear convergence of thealgorithm. Our proof is based on a new potential function and a novel use oferror bounds.

A Proximal Alternating Direction Method of Multiplier for Linearly Constrained Nonconvex Minimization