On Inexact Solution of Auxiliary Problems in Tensor Methods for Convex Optimization

In this paper we study the auxiliary problems that appear in $p$-order tensor methods for unconstrained minimization of convex functions with $\nu$-H"{o}lder continuous $p$th derivatives. This type of auxiliary problems corresponds to the minimization of a $(p+\nu )$-order regularization of the $p$th order Taylor approximation of the objective. For the case $p=3$, we consider the use of Gradient Methods with Bregman distance. When the regularization parameter is sufficiently large, we prove that the referred methods take at most $\mathcal{O}(\log ({ϵ}^{-1}))$ iterations to find either a suitable approximate stationary point of the tensor model or an $ϵ$-approximate stationary point of the original objective function.