  Composite minimization involves a collection of functions which are
aggregated in a nonsmooth manner. It covers, as a particular case, smooth
approximation of minimax games, minimization of max-type functions, and simple
composite minimization problems, where the objective function has a nonsmooth
component. We design a higher-order majorization algorithmic framework for
fully composite problems (possibly nonconvex). Our framework replaces each
component with a higher-order surrogate such that the corresponding error
function has a higher-order Lipschitz continuous derivative. We present
convergence guarantees for our method for composite optimization problems with
(non)convex and (non)smooth objective function. In particular, we prove
stationary point convergence guarantees for general nonconvex (possibly
nonsmooth) problems and under Kurdyka-Lojasiewicz (KL) property of the
objective function we derive improved rates depending on the KL parameter. For
convex (possibly nonsmooth) problems we also provide sublinear convergence
rates.
