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Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng's F-B four-operator splitting method for solving monotone inclusions

M. Marques Alves(maicon.alves***at***ufsc.br)
Marina Geremia(marinageremia***at***yahoo.com.br)

Abstract: In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng's forward-backward (F-B) splitting method for solving two-operator and four-operator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. Yao, in which an inexact DRS method is derived from a special instance of the hybrid proximal extragradient (HPE) method of Solodov and Svaiter, while the latter one combines the proposed inexact DRS method (used as an outer iteration) with a Tseng's F-B splitting type method (used as an inner iteration) for solving the corresponding subproblems. We prove iteration complexity bounds for both algorithms in the pointwise (non-ergodic) as well as in the ergodic sense by showing that they admit two different iterations: one that can be embedded into the HPE method, for which the iteration complexity is known since the work of Monteiro and Svaiter, and another one which demands a separate analysis. Finally, we perform simple numerical experiments to show the performance of the proposed methods when compared with other existing algorithms.

Keywords: Inexact Douglas-Rachford method; splitting; monotone operators; HPE method; complexity; Tseng's forward-backward method.

Category 1: Convex and Nonsmooth Optimization

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Entry Submitted: 11/30/2017
Entry Accepted: 11/30/2017
Entry Last Modified: 11/30/2017

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