  This paper proposes and analyzes a dampened proximal alternating direction
method of multipliers (DP.ADMM) for solving linearly-constrained nonconvex
optimization problems where the smooth part of the objective function is
nonseparable. Each iteration of DP.ADMM consists of: (i) a sequence of partial
proximal augmented Lagrangian (AL) updates, (ii) an under-relaxed Lagrange
multiplier update, and (iii) a novel test to check whether the penalty
parameter of the AL function should be updated. Under a basic Slater condition
and some requirements related to the dampening factor and under-relaxation
parameter, it is shown that DP.ADMM obtains a first-order stationary point of
the constrained problem in ${\cal O}(\varepsilon^{-3})$ iterations for a given
numerical tolerance $\varepsilon>0$. One of the main novelties of the paper is
that convergence of the method is obtained without requiring any rank
assumptions on the constraint matrices.
