Convergence analysis of the Halpern iteration with adaptive anchoring parameters

TL;DR

This paper introduces an adaptive Halpern iteration method with dynamically chosen parameters for fixed point problems in Hilbert spaces, proving strong convergence and improved convergence rates, supported by numerical experiments.

Contribution

It presents a novel adaptive parameter selection strategy for Halpern iteration, ensuring strong convergence and better performance than traditional methods.

Findings

Proves strong convergence of the adaptive Halpern iteration.

Establishes an asymptotic regularity rate of at least O(1/k).

Demonstrates superior performance through numerical experiments.

Abstract

We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the rate of asymptotic regularity at least O(1/k), where k is the number of iterations. Numerical experiments are also provided to show advantages and outperformance of our adaptive Halpern algorithm over the standard Halpern algorithm.