Article

Numerische Mathematik

, Volume 104, Issue 4, pp 561-572

First online:

Global convergence of a modified Fletcher–Reeves conjugate gradient method with Armijo-type line search

  • Li ZhangAffiliated withCollege of Mathematics and Computational Science, Changsha University of Science and Technology
  • , Weijun ZhouAffiliated withCollege of Mathematics and Computational Science, Changsha University of Science and Technology Email author 
  • , Donghui LiAffiliated withCollege of Mathematics and Econometrics, Hunan University

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Abstract

In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. It is well-known that the direction generated by a conjugate gradient method may not be a descent direction of the objective function. In this paper, we take a little modification to the Fletcher–Reeves (FR) method such that the direction generated by the modified method provides a descent direction for the objective function. This property depends neither on the line search used, nor on the convexity of the objective function. Moreover, the modified method reduces to the standard FR method if line search is exact. Under mild conditions, we prove that the modified method with Armijo-type line search is globally convergent even if the objective function is nonconvex. We also present some numerical results to show the efficiency of the proposed method.

Mathematics Subject Classification (2000)

90C30 65K05