Fast Krasnoselskii-Mann Method with Overrelaxation and Preconditioners

TL;DR

This paper introduces an accelerated Krasnoselskii-Mann method with overrelaxation and preconditioners, offering improved convergence proofs, flexible parameter tuning, and unification of acceleration mechanisms for splitting methods.

Contribution

It develops a generalized fast Krasnoselskii-Mann method with a new convergence proof, enabling larger relaxation parameters and unifying inertial and anchoring accelerations.

Findings

Allows relaxation parameters up to twice as large.

Unifies inertial and anchoring acceleration mechanisms.

Establishes weak convergence with degenerate preconditioners.

Abstract

We study accelerated Krasnoselskii-Mann-type methods with preconditioners in both continuous and discrete time. From a continuous-time model, we derive a generalized fast Krasnoselskii-Mann method, providing a new yet simple proof of convergence that leads to unprecedented flexibility in parameter tuning, allowing up to twice larger relaxation parameters. Our analysis unifies inertial and anchoring acceleration mechanisms and offers a broad range of parameter choices, which prove beneficial in practice. Additionally, we extend our analysis to the case where preconditioners are allowed to be degenerate, unifying the treatment of various splitting methods and establishing the weak convergence of shadow sequences.