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Improved pointwise iteration-complexity of a regularized ADMM and of a regularized non-Euclidean HPE framework

Max L. N. Goncalves (maxlng***at***ufg.br)
Renato D.C. Monteiro (renato.monteiro***at***isye.gatech.edu)
Jefferson G. Melo (jefferson.ufg***at***gmail.com)

Abstract: This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity of the latter method. Our analysis is based on first presenting and establishing the pointwise iteration-complexity of a regularized non-Euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error. It is then shown that the new method is a special instance of the latter framework where the sequence of summable errors is identically zero when the ADMM stepsize is less than one or a nontrivial sequence when the stepsize is in the interval [1, (1 +\sqrt{5})/2).

Keywords: alternating direction method of multipliers, hybrid proximal extragradient method, non-Euclidean Bregman distances, convex program, pointwise iteration-complexity, first-order methods, inexact proximal point method, regularization.

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation:

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Entry Submitted: 01/05/2016
Entry Accepted: 01/05/2016
Entry Last Modified: 01/06/2017

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