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Global Convergence of ADMM in Nonconvex Nonsmooth Optimization

Yu Wang (wang.yu***at***berkeley.edu)
Wotao Yin (wotaoyin***at***ucla.edu)
Jinshan Zeng (jsh.zeng***at***gmail.com)

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

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

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 11/26/2016
Entry Accepted: 11/26/2016
Entry Last Modified: 12/06/2017

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