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A Flexible ADMM Algorithm for Big Data Applications

Daniel Robinson(daniel.p.robinson***at***jhu.edu)
Rachael Tappenden(rtappen1***at***jhu.edu)

Abstract: We present a flexible Alternating Direction Method of Multipliers (F-ADMM) algorithm for solving optimization problems involving a strongly convex objective function that is separable into $n \geq 2$ blocks, subject to (non-separable) linear equality constraints. The F-ADMM algorithm uses a \emph{Gauss-Seidel} scheme to update blocks of variables, and a regularization term is added to each of the subproblems arising within F-ADMM. We prove, under common assumptions, that F-ADMM is globally convergent. We also present a special case of F-ADMM that is \emph{partially parallelizable}, which makes it attractive in a big data setting. In particular, we partition the data into groups, so that each group consists of multiple blocks of variables. By applying F-ADMM to this partitioning of the data, and using a specific regularization matrix, we obtain a hybrid ADMM (H-ADMM) algorithm: the grouped data is updated in a Gauss-Seidel fashion, and the blocks within each group are updated in a Jacobi manner. Convergence of H-ADMM follows directly from the convergence properties of F-ADMM. Also, a special case of H-ADMM can be applied to functions that are convex, rather than strongly convex. We present numerical experiments to demonstrate the practical advantages of this algorithm.

Keywords: Alternating Direction Method of Multipliers; convex optimization; Gauss-Seidel; Jacobi; regularization; separable function

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )


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Entry Submitted: 03/23/2015
Entry Accepted: 03/23/2015
Entry Last Modified: 03/23/2015

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