  Fixed point iterations play a central role in the design and the analysis of
a large number of optimization algorithms. We study a new iterative scheme in
which the update is obtained by applying a composition of quasinonexpansive
operators to a point in the affine hull of the orbit generated up to the
current iterate. This investigation unifies several algorithmic constructs,
including Mann's mean value method, inertial methods, and multi-layer
memoryless methods. It also provides a framework for the development of new
algorithms, such as those we propose for solving monotone inclusion and
minimization problems.
