Global Complexity Bound of a Proximal ADMM for Linearly-Constrained Nonseperable Nonconvex Composite Programming

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.