  A class of algorithms comprised by certain semismooth Newton and active-set
methods is able to solve convex minimization problems involving
sparsity-inducing regularizers very rapidly; the speed advantage of methods
from this class is a consequence of their ability to benefit from the sparsity
of the corresponding solutions by solving smaller inner problems than
conventional methods. The convergence properties of such conventional methods
(e.g., the forward-backward and the proximal-Newton ones) can be studied very
elegantly under the framework of iterations of scalar-averaged operators - this
is not the case for the aforementioned class. However, we show in this work
that by instead considering operator-averaged operators, one can indeed study
methods of that class, and also to derive algorithms outside of it that may be
more convenient to implement than existing ones. Additionally, we present
experiments whose results suggest that methods based on operator-averaged
operators achieve substantially faster convergence than conventional ones.
