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We consider convex optimization problems with the objective function havingLipshitz-continuous $p$-th order derivative, where $p\geq 1$. We propose a newtensor method, which closes the gap between the lower$O\left(\varepsilon^{-\frac{2}{3p+1}} \right)$ and upper$O\left(\varepsilon^{-\frac{1}{p+1}} \right)$ iteration complexity bounds forthis class of optimization problems.  We also consider uniformly convex functions, and show how the proposed methodcan be accelerated under this additional assumption. Moreover, we introduce a$p$-th order condition number which naturally arises in the complexity analysisof tensor methods under this assumption.  Finally, we make a numerical study of the proposed optimal method and showthat in practice it is faster than the best known accelerated tensor method. Wealso compare the performance of tensor methods for $p=2$ and $p=3$ and showthat the 3rd-order method is superior to the 2nd-order method in practice.

The global rate of convergence for optimal tensor methods in smooth convex optimization