Optimal parameter selections for a general Halpern iteration

Abstract

Let C be a closed affine subset of a real Hilbert space H and T : C -> C be a nonexpansive mapping. In this paper, for any fixed u is an element of C, a general Halpern iteration process: {x(0) is an element of C, x(n+1) = t(n)u + (1 - t(n))Tx(n), n >= 0, is considered for finding a fixed point of T nearest to u, where the parameter sequence {t(n)} is selected in the real number field, R. The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities.