Inexact accelerated high-order proximal-point methods

Abstract In this paper, we present a new framework of bi-level unconstrained minimization for development accelerated methods in Convex Programming. These use approximations the high-order proximal points, which are solutions some auxiliary parametric optimization problems. For computing these can different methods, and, particular, lower-order schemes. This opens possibility latter to overpass traditional limits Complexity Theory. As an example, obtain second-order method with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is iteration counter. better than maximal possible type as applied functions Lipschitz continuous Hessian. We also exact search procedure, have k^{-(3p+1)/ 2}\right) ( 3 p + 1 ) / 2 $$p \ge 1$$ ≥ order operator. The problem at each schemes convex.