Global Convergence of ADMM in Nonconvex Nonsmooth Optimization

TL;DR

This paper provides a comprehensive convergence analysis of ADMM for nonconvex, nonsmooth optimization problems with coupled constraints, demonstrating its effectiveness in statistical learning, manifold optimization, and matrix decomposition.

Contribution

It establishes the first convergence guarantees for ADMM in a broad class of nonconvex nonsmooth problems, including models with multiple blocks and complex constraints.

Findings

ADMM converges for various nonconvex functions like $ ext{l}_q$ quasi-norm and MCP.

ADMM guarantees convergence in models from statistical learning and matrix decomposition.

ADMM can outperform ALM in certain nonconvex nonsmooth scenarios.

Abstract

In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, $\phi(x_0,\ldots,x_p,y)$, subject to coupled linear equality constraints. Our ADMM updates each of the primal variables $x_0,\ldots,x_p,y$, followed by updating the dual variable. We separate the variable $y$ from $x_i$'s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, $\ell_q$ quasi-norm, Schatten-$q$ quasi-norm ($0<q<1$), minimax concave penalty (MCP), and smoothly clipped absolute deviation (SCAD) penalty. It also allows nonconvex constraints such as compact manifolds (e.g., spherical, Stiefel, and Grassman manifolds) and linear complementarity constraints. Also, the $x_0$-block can be almost any lower semi-continuous function. 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. 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. 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 with bounded penalty parameter $\beta$. Indicated by this example and other analysis in this paper, ADMM might be a better choice than ALM for some nonconvex \emph{nonsmooth} problems, because ADMM is not only easier to implement, it is also more likely to converge for the concerned scenarios.