Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global Convergence of ADMM in Nonconvex Nonsmooth Optimization   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.

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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: eess