  Majorization-minimization algorithms consist of successively minimizing a
sequence of upper bounds of the objective function so that along the iterations
the objective function decreases. Such a simple principle allows to solve a
large class of optimization problems, even nonconvex and nonsmooth. We propose
a general higher-order majorization-minimization algorithmic framework for
minimizing an objective function that admits an approximation (surrogate) such
that the corresponding error function has a higher-order Lipschitz continuous
derivative. We present convergence guarantees for our new method for general
optimization problems with (non)convex and/or (non)smooth objective function.
For convex (possibly nonsmooth) problems we provide global sublinear
convergence rates, while for problems with uniformly convex objective function
we obtain locally faster superlinear convergence rates. We also prove global
stationary point guarantees for general nonconvex (possibly nonsmooth) problems
and under Kurdyka-Lojasiewicz property of the objective function we derive
local convergence rates ranging from sublinear to superlinear for our
majorization-minimization algorithm. Moreover, for unconstrained nonconvex
problems we derive convergence rates in terms of first- and second-order
optimality conditions.
