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Global convergence of ADMM in nonconvex nonsmooth optimization. (English) Zbl 07042437
The authors consider the following optimization problem: \[ \left\{ \begin{array}{l}\underset{x_{0},\ldots ,x_{p},y}{\mathrm{min}}\mathrm{\quad }\phi (x_{0},\ldots ,x_{p},y) \\ \text{subject to }\, A_{0}x_{0}+A_{1}x_{1}+\ldots +A_{p}x_{p}+By=0 \end{array}\right. \] where \(\phi \) is a continuous (possibly nonconvex, nonsmooth) function, \(x_{i}\in \mathbb{R}^{n_{i}}\), \(y\in \mathbb{R}^{q}\) and \(A_{i}\), \(B\) are \(m\times n_{i}\) (resp. \(m\times q\)) matrices. The authors propose an ADMM (Alternating Direction Method of Multipliers) type algorithm (multi-block version) that updates sucessively each of the primal variables \(x_{0}, \dots, x_{p}, y\) followed by an update of the dual variable. Under certain assumptions they establish global convergence. This approach applies to problems arising in matrix decomposition, sparse recovery, machine learning and optimization on compact smooth manifolds, covering cases that were not previously covered by any convergence theory. They also discuss relations with the ADL (Augmented Lagrangian Method) and provide an example in which ADMM converges but the latter diverges.

65K05 Numerical mathematical programming methods
90C26 Nonconvex programming, global optimization
Full Text: DOI arXiv
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