An Optimal High-Order Tensor Method for Convex Optimization

This paper is concerned with finding an optimal algorithm for minimizing a composite convex objective function. The basic setting that the sum of two functions: first function smooth up to dth-order derivative information available, and second possibly nonsmooth, but its proximal tensor mappings can be computed approximately in efficient manner. problem find—in setting—the best possible (optimal) iteration complexity optimization. Along line, case (without nonsmooth part objective), Nesterov proposed first-order methods ([Formula: see text]) [Formula: text], whereas high-order algorithms (using general information) text] were recently established. In this paper, we propose new case, which matches lower bound as previously established hence optimal. Our approach based on accelerated hybrid extragradient (A-HPE) framework by Monteiro Svaiter, where bisection procedure installed each A-HPE iteration. At step, subproblem solved, total number steps per shown bounded logarithmic factor precision required.