Modified Hestenes-Steifel conjugate gradient coefficient for unconstrained optimization

Abstract

Conjugate gradient methods play an important role in unconstrained optimization. Numerous studies and modifications have been devoted recently to improve this method. In this paper we propose a new conjugate gradient coefficient (β _k ) by modifying the already proven Hestenes-Steifel formula. In this new β _k we introduce a new formula for the denominator and retain the numerator of the Hestenes-Steifel formula. Numerical results have shown that the new formula for β _k performs far better than the original Hestenes-Steifel, but still possesses global convergence properties. This new method also outperforms the other conjugate gradient methods.

Keywords and phrases:

• Conjugate gradient method
• conjugate gradient coefficient
• convergence