  Higher-order tensor methods were recently proposed for minimizing smooth
convex and nonconvex functions. Higher-order algorithms accelerate the
convergence of the classical first-order methods thanks to the higher-order
derivatives used in the updates. The purpose of this paper is twofold. Firstly,
to show that the higher-order algorithmic framework can be generalized and
successfully applied to (nonsmooth) difference of convex functions, namely,
those that can be expressed as the difference of two smooth convex functions
and a possibly nonsmooth convex one. We also provide examples when the
subproblem can be solved efficiently, even globally. Secondly, to derive a
complete convergence analysis for our higher-order difference of convex
functions (HO-DC) algorithm. In particular, we prove that any limit point of
the HO-DC iterative sequence is a critical point of the problem under
consideration, the corresponding objective value is monotonically decreasing
and the minimum value of the norms of its subgradients converges globally to
zero at a sublinear rate. The sublinear or linear convergence rates of the
iterations are obtained under the Kurdyka-Lojasiewicz property.
