Global convergence properties of conjugate gradient methods for optimization. (English) Zbl 0767.90082

Gilbert, Jean Charles; Nocedal, Jorge

SIAM J. Optim. 2, No. 1, 21-42 (1992).

This paper studies the convergence properties of nonlinear conjugate gradient methods without restarts, and with practical line searches for the problem \(min_{x\in\mathbb{R}^ n}f(x)\). Iterations of the search directions and new points under study are chosen as: \[ x_{k+1}=x_ k+\alpha_ kd_ k,\text{ where } d_ k=\begin{cases} -g_ k, & \text{ for } k=1,\\ -g_ k+\beta_ kd_{k-1} & \text{ for } k\geq 2, \end{cases} \] Various choices of \(\beta_ k\) and inexact line searches that result in global convergence are considered. The analysis is closely related to the methods of Fletcher-Reeves and Polak-Ribière. Numerical experiments are presented.

Reviewer: X.Q.Yang (Kensington)

MSC:

90C52	Methods of reduced gradient type
90C30	Nonlinear programming
90-08	Computational methods for problems pertaining to operations research and mathematical programming
65K05	Numerical mathematical programming methods

Keywords:

global convergence; unconstrained optimization; convergence properties; nonlinear conjugate gradient methods