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Approximate Douglas–Rachford algorithm for two-sets convex feasibility problems

Approximate Douglas–Rachford algorithm for two-sets convex feasibility problems In this paper, we propose a new algorithm combining the Douglas–Rachford (DR) algorithm and the Frank–Wolfe algorithm, also known as the conditional gradient (CondG) method, for solving the classic convex feasibility problem. Within the algorithm, which will be named Approximate Douglas–Rachford (ApDR) algorithm, the CondG method is used as a subroutine to compute feasible inexact projections on the sets under consideration, and the ApDR iteration is defined based on the DR iteration. The ApDR algorithm generates two sequences, the main sequence, based on the DR iteration, and its corresponding shadow sequence. When the intersection of the feasible sets is nonempty, the main sequence converges to a fixed point of the usual DR operator, and the shadow sequence converges to the solution set. We provide some numerical experiments to illustrate the behaviour of the sequences produced by the proposed algorithm. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Global Optimization Springer Journals

Approximate Douglas–Rachford algorithm for two-sets convex feasibility problems

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References (31)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0925-5001
eISSN
1573-2916
DOI
10.1007/s10898-022-01264-7
Publisher site
See Article on Publisher Site

Abstract

In this paper, we propose a new algorithm combining the Douglas–Rachford (DR) algorithm and the Frank–Wolfe algorithm, also known as the conditional gradient (CondG) method, for solving the classic convex feasibility problem. Within the algorithm, which will be named Approximate Douglas–Rachford (ApDR) algorithm, the CondG method is used as a subroutine to compute feasible inexact projections on the sets under consideration, and the ApDR iteration is defined based on the DR iteration. The ApDR algorithm generates two sequences, the main sequence, based on the DR iteration, and its corresponding shadow sequence. When the intersection of the feasible sets is nonempty, the main sequence converges to a fixed point of the usual DR operator, and the shadow sequence converges to the solution set. We provide some numerical experiments to illustrate the behaviour of the sequences produced by the proposed algorithm.

Journal

Journal of Global Optimization – Springer Journals

Published: Jul 1, 2023

Keywords: Convex feasibility problem; Douglas–Rachford algorithm; Frank–Wolfe algorithm; Conditional gradient method; Inexact projections; 65K05; 90C30; 90C25

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