Composite optimization models consist of the minimization of the sum
of a smooth (not necessarily convex) function and a non-smooth convex function.
Such models arise in many applications where, in addition to the composite nature
of the objective function, a hierarchy of models is readily available. It is common
to take advantage of this hierarchy of models by first solving a low fidelity model
and then using the solution as a starting point to a high fidelity model. We adopt
an optimization point of view and show how to take advantage of the availability of
a hierarchy of models in a consistent manner. We do not use the low fidelity model
just for the computation of promising starting points but also for the computa-
tion of search directions. We establish the convergence and convergence rate of
the proposed algorithm. Our numerical experiments on large scale image restora-
tion problems and the transition path problem suggest that, for certain classes of
problems, the proposed algorithm is significantly faster than the state of the art.