On Hager and Zhang’s conjugate gradient method with guaranteed descent

Abstract

In this paper, we point out that Hager–Zhang family of conjugate gradient methods are based on the spectral scaling secant equation, are really members of Dai–Liao type of gradient methods with common constant parameters. We propose a possible efficient parameter for Hager–Zhang’s conjugate gradient method with guaranteed descent (CG_DESCENT) and present a conjugate gradient method CG_DESCENT1. The CG_DESCENT1 method preserves the same global convergence properties as CG_DESCENT method of Hager and Zhang, and is numerically tested compared with CG_DESCENT method and other conjugate gradient methods. Considering the difference of the search directions approximation to the search direction of the memoryless BFGS method, we give the reason why the proposed method is more efficient than CG_DESCENT method under some conditions, and further find an optimal conjugate gradient method and an approximation optimal conjugate method in Hager–Zhang family of conjugate gradient methods.