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Journal of Applied Mathematics
Volume 2012, Article ID 932980, 14 pages
http://dx.doi.org/10.1155/2012/932980
Research Article

The Global Convergence of a New Mixed Conjugate Gradient Method for Unconstrained Optimization

School of Mathematics, Beihua University, Jilin 132013, China

Received 14 June 2012; Revised 7 September 2012; Accepted 13 September 2012

Academic Editor: Hak-Keung Lam

Copyright © 2012 Yang Yueting and Cao Mingyuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” The Computer Journal, vol. 7, pp. 149–154, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. B. T. Poliak, “The conjugate gradient method in extreme problems,” USSR Computational Mathematics and Mathematical Physics, vol. 9, pp. 94–112, 1969. View at Publisher · View at Google Scholar
  3. Y. H. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM Journal on Optimization, vol. 10, no. 1, pp. 177–182, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 409–436, 1952. View at Google Scholar · View at Zentralblatt MATH
  5. Z. Wei, S. Yao, and L. Liu, “The convergence properties of some new conjugate gradient methods,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1341–1350, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. G. Yuan, “Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems,” Optimization Letters, vol. 3, no. 1, pp. 11–21, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. G. Yuan, X. Lu, and Z. Wei, “A conjugate gradient method with descent direction for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 519–530, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Al-Baali, “Descent property and global convergence of the Fletcher-Reeves method with inexact line search,” IMA Journal of Numerical Analysis, vol. 5, no. 1, pp. 121–124, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. G. H. Liu, J. Y. Han, and H. X. Yin, “Global convergence of the Fletcher-Reeves algorithm with inexact linesearch,” Applied Mathematics B, vol. 10, no. 1, pp. 75–82, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. H. Dai and Y. Yuan, “Convergence properties of the Fletcher-Reeves method,” IMA Journal of Numerical Analysis, vol. 16, no. 2, pp. 155–164, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. C. Gilbert and J. Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” SIAM Journal on Optimization, vol. 2, no. 1, pp. 21–42, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. H. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM Journal on Optimization, vol. 10, no. 1, pp. 177–182, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. X. Zheng, Z. Tian, and L. W. Song, “The global convergence of a mixed conjugate gradient method with the Wolfe line search,” OR Transactions, vol. 13, no. 2, pp. 18–24, 2009. View at Google Scholar · View at Zentralblatt MATH
  14. D. Touati-Ahmed and C. Storey, “Efficient hybrid conjugate gradient techniques,” Journal of Optimization Theory and Applications, vol. 64, no. 2, pp. 379–397, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Z. F. Dai and L. P. Chen, “A mixed conjugate gradient method by HS and DY,” Journal of Computational Mathematics, vol. 27, no. 4, pp. 429–436, 2005. View at Google Scholar
  16. Y. H. Dai and Q. Ni, Testing Different Nonlinear Conjugate Gradient Methods, Institude of ComputationalMathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijin, China, 1998.
  17. W. W. Hager and H. Zhang, “Algorithm 851: CGDESCENT, a conjugate gradient method with guaranteed descent,” ACM Transactions on Mathematical Software, vol. 32, no. 1, pp. 113–137, 2006. View at Publisher · View at Google Scholar
  18. L. Zhang, W. Zhou, and D.-H. Li, “A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence,” IMA Journal of Numerical Analysis, vol. 26, no. 4, pp. 629–640, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. Zoutendijk, “Nonlinear programming, computational methods,” in Integer and Nonlinear Programming, pp. 37–86, North-Holland, Amsterdam, The Netherlands, 1970. View at Google Scholar · View at Zentralblatt MATH
  20. N. I. M. Gould, D. Orban, and P. L. Toint, “GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization,” ACM Transactions on Mathematical Software, vol. 29, no. 4, pp. 353–372, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH