Variants of the A-HPE and large-step A-HPE algorithms for strongly convex problems with applications to accelerated high-order tensor methods

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 Oleft(k^,-kleft(fracp-1p+1right)right) global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter p≥ 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 Oleft(k^,-kleft(fracp-1p+1right)right). In particular, for p=2, we obtain an inexact Newton-proximal algorithm with fast global Oleft(k,-k/3right) convergence rate.