  For solving strongly convex optimization problems, we propose and study the
global convergence of variants of the A-HPE and large-step A-HPE algorithms of
Monteiro and Svaiter. We prove linear and the superlinear
$\mathcal{O}\left(k^{\,-k\left(\frac{p-1}{p+1}\right)}\right)$ global rates for
the proposed variants of the A-HPE and large-step A-HPE methods, respectively.
The parameter $p\geq 2$ appears in the (high-order) large-step condition of the
new large-step A-HPE algorithm. We apply our results to high-order tensor
methods, obtaning a new inexact (relative-error) tensor method for (smooth)
strongly convex optimization with iteration-complexity
$\mathcal{O}\left(k^{\,-k\left(\frac{p-1}{p+1}\right)}\right)$. In particular,
for $p=2$, we obtain an inexact Newton-proximal algorithm with fast global
$\mathcal{O}\left(k^{\,-k/3}\right)$ convergence rate.
