# 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.

## Keywords

Variational Inequality Average Method Variational Inequality Problem Point Solution Function Iteration## Preview

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