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Zbl 1103.65074
Zhang, Li; Zhou, Weijun; Li, Donghui
Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search. (English)
[J] Numer. Math. 104, No. 4, 561-572 (2006). ISSN 0029-599X; ISSN 0945-3245/e

Summary: 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.

MSC 2000:: *65K05 Mathematical programming (numerical methods)
90C30 Nonlinear programming

Keywords: convergence; conjugate gradient methods; unconstrained optimization; Fletcher-Reeves method; Armijo-type line search; numerical results

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