The Douglas--Rachford Algorithm Converges Only Weakly

TL;DR

This paper proves that the Douglas--Rachford algorithm's convergence is inherently weak and cannot be strengthened to strong convergence, highlighting fundamental limitations in its convergence properties.

Contribution

It establishes that the weak convergence of the Douglas--Rachford algorithm cannot be improved to strong convergence, and similarly for the method of partial inverses.

Findings

Weak convergence cannot be improved to strong convergence for Douglas--Rachford.

Strong convergence can fail for the method of partial inverses.

Fundamental limitations in convergence properties of these algorithms.

Abstract

We show that the weak convergence of the Douglas--Rachford algorithm for finding a zero of the sum of two maximally monotone operators cannot be improved to strong convergence. Likewise, we show that strong convergence can fail for the method of partial inverses.