  Consider the minimization of a nonconvex differentiable function over a
polyhedron. A popular primal-dual first-order method for this problem is to
perform a gradient projection iteration for the augmented Lagrangian function
and 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 alternating
direction method of multipliers for the multi-block version of this problem. A
distinctive feature of this method is the introduction of a "smoothed" (i.e.,
exponentially weighted) sequence of primal iterates, and the inclusion, at each
iteration, to the augmented Lagrangian function a quadratic proximal term
centered at the current smoothed primal iterate. The resulting proximal
augmented Lagrangian function is inexactly minimized (via a gradient projection
step) at each iteration while the dual multiplier vector is updated using the
residual of the linear constraints. When the primal and dual stepsizes are
chosen sufficiently small, we show that suitable "smoothing" can stabilize the
"oscillation", and the iterates of the new proximal ADMM algorithm converge to
a stationary point under some mild regularity conditions. Furthermore, when the
objective function is quadratic, we establish the linear convergence of the
algorithm. Our proof is based on a new potential function and a novel use of
error bounds.
