The new algorithm form of the Fletcher – Reeves Conjugate gradient algorithm

Abstract

In this paper, we propose a new spectral form of the Fletcher – Reeves conjugate gradient algorithm for solving unconstrained optimization problems which has sufficient descent direction. We prove the global convergent of these algorithms under Wolf line search conditions. We presented some numerical result and comparison with Fletcher – Reeves algorithm.

Keywords

• Conjugate gradient,Line search

References

1. [1] Fletcher, R. (1989),' Practical Method of Optimization '(2nd Edition), (John Wiley and Sons, New York ).
2. [2] Polak, E. and Ribiere, G. (1969),' Note for Convergence Direction Conjugate ' Revue Francaise Informant, Reserche. Opertionelle, pp. 35-43.
3. [3] Hestenes, M. R. and Stiefel, E. L. (1952), ' Method of conjugate gradients for Solving linear systems ' Journal National Standards 49, pp. 409-436.
4. [4] Fletcher, R. and Reeves C. (1964),' Function minimization by conjugate gradients' Computer Journal 7, pp. 149-154.
5. [5] Zoutendijk, G. (1970), ' Nonlinear programming computational algorithms '. In : Integer and Nonlinear programming, Abadie, J. (ED). North-Holland, Amsterdam, ISBN: 044410008, PP. 37-86.
6. [6] Al-Baali, M. (1985), ' Descent property and global convergence of Flecher-Reeves with inexact line search '. IMA, J. Anal. 5, pp. 121-124.
7. [7] Dai, Y. and Yuan, Y. (1999),' A nonlinear conjugate gradient method with a strong global convergence property ' SIAM J. optimization 10, pp. 177-182.
8. [8] Birgin, E. and Martinez, J. M. (2001),' A spectral conjugate gradient method for unconstrained optimization ' App. Math. Optim. 43, pp. 117-128.

9. [9] Raydan, M. (1997),' The Barzilain and Borwein gradient method for the large unconstrained minimization problem ' SIAM J. Optim. 7, pp. 26-33.
10. [10] Li, Z. , Weijun, Z. and Donghui, L. (2006),' Global convergence of a modified Fletcher Reeves conjugate gradient method with Armijo-type line ' pp. 561-572.
11. [11] Dai, Y. and Liao, Z. (2001),' New conjugate conditions and related nonlinear conjugate gradient methods ' Appl. Math. Optim. 43, pp. 87-101.
12. [12] Li, Z. (2009),' New versions of the Hestenes-Stiefel nonlinear conjugate gradient method based on the secant condition for optimization ' pp. 111-133.