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For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro and B.F. Svaiter [An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods, SIAM J. Optim. 23 (2013), pp. 1092–1125.]. We prove  linear  and the  superlinearglobal rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameterappears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our results to high-order tensor methods, obtaining a new inexact (relative-error) tensor method for (smooth) strongly convex optimization with iteration-complexity. In particular, for  p  = 2, we obtain an inexact proximal-Newton algorithm with fast globalconvergence rate.

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