New multi-step three-term conjugate gradient algorithms with inexact line searches

Abstract

This work suggests several multi-step three-term conjugate gradient (CG)-algorithms that satisfies their sufficient descent property and conjugacy conditions. First, we have  considered a number of well-known three-term CG-method, and we have, therefore, suggested two new classes of this type of algorithms which was based on hestenes and stiefel (HS) and polak-ribière (PR) formulas with four different versions. Both descent and conjugacy conditions for all the proposed algorithms are satisfied, at each iteration by using the strong Wolfe line search condition and it's accelerated version. These new suggested algorithms are some sort of modifications to the original  HS and PR  methods. These CG-algorithms are considered as a sort of the  memoryless BFGS update.  All of our new suggested methods are proved to be a  global convergent and numerically, more efficient than the similar methods in same area based on our selected set of used numerical problems.