Linear convergence of relocated fixed-point iterations

We establish linear convergence of relocated fixed-point iterations as introduced by Atenas et al.

(2026) DOI: https://doi.org/10.1137/25M1776810 assuming the algorithmic operator satisfies a linear error bound.

In particular, this framework applies to the setting where the algorithmic operator is a contraction.

As a key application of our framework, we obtain linear convergence of the relocated Douglas--Rachford algorithm for finding a zero in the sum of two monotone operators in a setting with Lipschitz continuity and strong monotonicity assumptions.

We also apply the framework to deduce linear convergence of variable stepsize resolvent splitting algorithms for multioperator monotone inclusions.