Linear convergence of the Douglas–Rachford method for two closed sets

Abstract

In this paper, we investigate the Douglas–Rachford method (DR) for two closed (possibly nonconvex) sets in Euclidean spaces. We show that under certain regularity conditions, the DR converges locally with -linear rate. In convex settings, we prove that the linear convergence is global. Our study recovers recent results on the same topic.

Keywords:

• affine-hull reduction
• Douglas–Rachford method
• Fejér monotonicity
• R-linear convergence
• linear regularity
• strong regularity
• superregularity

AMS Subject Classifications:

• Primary: 49M27
• 65K10
• Secondary: 47H09
• 49J52
• 49M37
• 65K05
• 90C26

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author was partially supported an NSERC accelerator grant of Heinz Bauschke (University of British Columbia, Kelowna, Canada) and by an internal grant of University of Massachusetts Lowell. The author is grateful to Heinz Bauschke and Xianfu Wang (UBC Kelowna, Canada) for their support. The author also thanks the editors and the referees for their constructive comments.