An Efficient Hybrid Conjugate Gradient Method with the Strong Wolfe-Powell Line Search

Ahmad Alhawarat,Zabidin Salleh,Mohd Rivaie,Mustafa Mamat

doi:10.1155/2015/103517

Abstract

Conjugate gradient (CG) method is an interesting tool to solve optimization problems in many fields, such as design, economics, physics, and engineering. In this paper, we depict a new hybrid of CG method which relates to the famous Polak-Ribière-Polyak (PRP) formula. It reveals a solution for the PRP case which is not globally convergent with the strong Wolfe-Powell (SWP) line search. The new formula possesses the sufficient descent condition and the global convergent properties. In addition, we further explained about the cases where PRP method failed with SWP line search. Furthermore, we provide numerical computations for the new hybrid CG method which is almost better than other related PRP formulas in both the number of iterations and the CPU time under some standard test functions.

Highlights

The nonlinear conjugate gradient (CG) method is a useful tool to find the minimum value for unconstrained optimization problems
We provide numerical computations for the new hybrid CG method which is almost better than other related PRP formulas in both the number of iterations and the CPU time under some standard test functions
Where gk = g(xk) and βk is known as CG method, formula, or coefficient

Summary

Introduction

The nonlinear conjugate gradient (CG) method is a useful tool to find the minimum value for unconstrained optimization problems. Al-Baali [13] proved that FR method is globally convergent with the SWP line search when σ < 1/2. The global convergence of PRP method (10) with the exact line search was proved by Elijah and Ribiere in [10]. Powell [15] gave out a counterexample showing that there exists nonconvex function, where PRP method does not converge globally, even when the exact line search is used. There is no guarantee that PRP+ is convergent with SWP line search for general nonlinear functions. This paper is organized as follows; in Section 2 we will present the current problem for PRP and nonnegative PRP method with SWP line search.

Motivation and the Hybrid Formula

Numerical Results and Discussions

Conclusion