THE PARAMETER SELECTION PROBLEM FOR MANN’S FIXED POINT ALGORITHM

Abstract

Mann's fixed point algorithm can be written as a line search method that generates a sequence $\{x_n\}$ through the recursive manner $x_{n+1}=x_n-\alpha_nv_n$, where $\alpha_n$ is the stepsize and where $v_n$ is the search direction given by $v_n=x_n-Tx_n$, with $T$ being a nonexpansive mapping. This line search method has widely been used in optimization, variational inequalities and fixed point problems. In this paper, we address the problem of selection of the sequence of parameters, $\{\alpha_n\}$, so as to have optimal convergence of this algorithm.