A new family of high-order directions for unconstrained optimization inspired by Chebyshev and Shamanskii methods

The 1669-1670 Newton-Raphson’s method is still used to solve equations systems and unconstrained optimization problems. Since this method, some other algorithms inspired by Newton’s have been proposed: in 1839 Chebyshev developped a high order cubical convergence algorithm, and in 1967 Shamanskii proposed an acceleration of Newton’s method. By considering a Newton-type methods as displacement directions, we introduce in this paper new high order algorithms extending these famous methods. We provide convergence order results and per iteration complexity analysis to predict the efficiency of such iterative processes. Preliminary numerous examples confirm the applicability of our analysis. keywords : Unconstrained optimization; Newton method; Chebyshev method; automatic differentiation; Shamanskii method.