Linear convergence of relocated fixed-point iterations

2025/12/15 by Atenas, Felipe, Simi, Farhana Ahmed, Tam, Matthew K
#47H04 #47H05 #47H09 #65K10 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2512.12954

Abstract

We establish linear convergence of relocated fixed-point iterations as introduced by Atenas et al. (2025) 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.

Citations

• Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting
• Linear Convergence of Resolvent Splitting with Minimal Lifting and its Application to a Primal-Dual Algorithm
• Understanding the Douglas-Rachford splitting method through the lenses of Moreau-type envelopes
• Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka-Łojasiewicz Property without Global Lipschitz Assumptions
• Frugal and Decentralised Resolvent Splittings Defined by Nonexpansive Operators
• Inertial Krasnoselskii-Mann Iterations
• Linear convergence of the Douglas-Rachford algorithm via a generic error\n bound condition
• Resolvent Splitting for Sums of Monotone Operators with Minimal Lifting
• Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions
• Non-stationary Douglas-Rachford and alternating direction method of multipliers: adaptive stepsizes and convergence
• Regular Sequences of Quasi-Nonexpansive Operators and Their Applications
• Adaptive Relaxed ADMM: Convergence Theory and Practical Implementation
• Line Search for Averaged Operator Iteration
• Gradient Descent Converges to Minimizers
• Convergence rate analysis of several splitting schemes
• Linear and strong convergence of algorithms involving averaged nonexpansive operators
• Rectangularity and paramonotonicity of maximally monotone operators
• Weak convergence on Douglas-Rachford method
• On Weak Convergence of the Douglas–Rachford Method
• Splitting Algorithms for the Sum of Two Nonlinear Operators
• Monotone Operators and the Proximal Point Algorithm

Related

• Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting
• On the Bredies-Chenchene-Lorenz-Naldi algorithm
• Linear Convergence of Resolvent Splitting with Minimal Lifting and its Application to a Primal-Dual Algorithm
• Resolvent Splitting for Sums of Monotone Operators with Minimal Lifting
• Inertial Modified S-Iteration Process for Split Monotone Inclusion and Fixed Point Problem In Real Hilbert Space