Higher-order tensor methods for minimizing difference of convex functions

Abstract: 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.