In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, $\phi(x_1,\ldots,x_p,y)$, subject to linear equality constraints that couple $x_1,\ldots,x_p,y$, where $p\ge 1$ is an integer. Our ADMM sequentially updates the primal variables in the order $x_1,\ldots,x_p,y$, followed by updating the dual variable. We separate the variable $y$ from $x_i$'s as it has a special role in our analysis. #R##N#The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, $\ell_q$ quasi-norm, Schatten-$q$ quasi-norm ($0<q<1$) and SCAD, as well as the indicator functions of compact smooth manifolds (e.g., spherical, Stiefel, and Grassman manifolds). By applying our analysis, we show, for the first time, that several ADMM algorithms applied to solve nonconvex models in statistical learning, optimization on manifold, and matrix decomposition are guaranteed to converge. #R##N#Our results provide sufficient conditions for ADMM to converge on (convex or nonconvex) monotropic programs with three or more blocks, as they are special cases of our model. #R##N#ADMM has been regarded as a variant to the augmented Lagrangian method (ALM). We present a simple example to illustrate how ADMM converges but ALM diverges. Indicated from this example and other analysis in this paper, ADMM might be a better choice than ALM for nonconvex \emph{nonsmooth} problems, because ADMM is not only easier to implement, it is also more likely to converge.
['Yu Wang', 'Wotao Yin', 'Jinshan Zeng']
Global Convergence of ADMM in Nonconvex Nonsmooth Optimization