Convergence analysis of the Halpern iteration with adaptive anchoring parameters

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 ) O(1/k) , where k 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.