A Convex Feasibility Problem for Which Douglas-Rachford Converges Only Weakly

Jan Quan, Panagiotis Patrinos · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28414 · cited by 0 works (OpenAlex) · open-access PDF

Russian

Мы предоставляем два закрытых, выпуклых набора с непустым пересечением и начальной точкой, для которой алгоритм Дугласа-Рачфорда сближается слабо, но не сильно. Это решает открытый вопрос, как отмечалось ранее Арагоном Артахо, Борвеном и Там (2014, Примечание 2.2). Наша конструкция является простым подъемом недавнего строительства Баушке и Тунгом относительно твердо нерасширяющихся операторов.

Machine translation (Meta M2M100); the English original follows.

Abstract

We provide two closed, convex sets with nonempty intersection and an initial point for which the Douglas-Rachford algorithm converges weakly but not strongly. This resolves an open question as noted earlier by Aragón Artacho, Borwein and Tam (2014, Remark 2.2). Our construction is a simple lift of a recent construction by Bauschke and Tung concerning firmly nonexpansive operators.

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Topics: Mathematics, Intersection (aeronautics), Regular polygon, Lift (data mining), Simple (philosophy), Point (geometry)

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