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In this paper, we introduce a \textit{Bi-level OPTimization} (BiOPT) framework for minimizing the sum of two convex functions, where both can be nonsmooth. The BiOPT framework involves two levels of methodologies. At the upper level of BiOPT, we first regularize the objective by a $(p+1)$th-order proximal term and then develop the generic inexact high-order proximal-point scheme and its acceleration using the standard estimation sequence technique. At the lower level, we solve the corresponding $p$th-order proximal auxiliary problem inexactly either by one iteration of the $p$th-order tensor method or by a lower-order non-Euclidean composite gradient scheme with the complexity $\mathcal{O}(\log \tfrac{1}{\varepsilon})$, for the accuracy parameter $\varepsilon>0$. Ultimately, if the accelerated proximal-point method is applied at the upper level, and the auxiliary problem is handled by a non-Euclidean composite gradient scheme, then we end up with a $2q$-order method with the convergence rate $\mathcal{O}(k^{-(p+1)})$, for $q=\lfloor p/2 \rfloor$, where $k$ is the iteration counter.

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.