Computing Fixed Points by Averaging

Abstract

Averaging methods for solving fixed point problems combine the underlying fixed point map T with some “well-behaved” map g. The map g might, for example, be contractive or might be a nonexpansive map whose fixed points include those of the original map T. One class of averaging methods (inside averaging) averages any current iterate with its image under the map g, and then applies the map T. Another averaging method (outside averaging) first applies the maps T and g and then takes averages. When g is the identity map, outside averaging averages a given point with its image under the map T. In this paper we summarize a number of known results concerning these averaging methods, including (i) a general averaging framework that approximates the original fixed point problem with a trajectory of averaged (parametrized) fixed point subproblems, and (ii) a procedure for following these trajectories approximately to ease the computations.