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The conjugate gradient (CG) methods are iterative procedures for unconstrained optimization problem. Many variants and alterations have been done lately to develop this method. In this research, we suggest a new hybrid CG coefficient by combining a modified Hestenes-Steifel (HS) formula with Fletcher and Reeves (FR). Theoretic proves has shown that the new technique achieves sufficient descent condition if inexact line search (strong Wolfe-Powell) is used. Besides, most of the numerical outcomes show that our technique is very efficient when compared to HS, FR and some famous hybrid CG for given standard test problems. The numerical outcomes also displayed that the new coefficient performs better than the original HS and FR methods.

This research aims to introduce some of the main ideas of differential geometry. The research deals with the main concepts needed to understand this work. In this research properties of curves in 2 R are studied. The research is built on using the curvature of a curve in 2 R to derive a parametric formula for the velocity and acceleration. Also the geometry of focal points has been discussed. Examples are built to support the aim of this research.