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Global Convergence of ADMM in Nonconvex Nonsmooth Optimization

Yu Wang(shifwang***at***gmail.com)
Wotao Yin(wotaoyin***at***ucla.edu)
Jinshan Zeng(jinshanzeng***at***jxnu.edu.cn)

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, $\phi(x_1,\ldots,x_p,y)$, 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. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, $\ell_q$ quasi-norm, Schatten-$q$ quasi-norm ($0

Keywords: ADMM, nonconvex optimization, augmented Lagrangian method, block coordinate descent, sparse optimization

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Nonlinear Optimization (Other )

Citation: UCLA CAM Report 15-61, 2015

Download: [PDF]

Entry Submitted: 11/28/2015
Entry Accepted: 11/28/2015
Entry Last Modified: 11/28/2015

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