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Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 243290, 16 pages
http://dx.doi.org/10.1155/2009/243290
Research Article

Nonlinear Conjugate Gradient Methods with Sufficient Descent Condition for Large-Scale Unconstrained Optimization

1Institute of Applied Mathematics, College of Mathematics and Information Science, Henan University, Kaifeng 475000, China
2Department of Mathematics, Nanjing University, Nanjing 210093, China
3College of Mathematics and Information Science, Guangxi University, Nanning 530004, China

Received 3 January 2009; Revised 21 February 2009; Accepted 1 May 2009

Academic Editor: Joaquim J. Júdice

Copyright © 2009 Jianguo Zhang et al. 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. 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
  2. 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 · View at MathSciNet
  3. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradient for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 409–436, 1952. View at Google Scholar
  4. E. Polak and G. Ribière, “Note sur la convergence de directions conjugées,” Revue Francaise d'Informatique et de Recherche Operationnelle, vol. 16, pp. 35–43, 1969. View at Google Scholar
  5. 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 · View at MathSciNet
  6. 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 · View at MathSciNet
  7. W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pacific Journal of Optimization, vol. 2, pp. 35–58, 2006. View at Google Scholar · View at Zentralblatt MATH
  8. 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 · View at MathSciNet
  9. L. Grippo and S. Lucidi, “A globally convergent version of the Polak-Ribière conjugate gradient method,” Mathematical Programming, vol. 78, no. 3, pp. 375–391, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Grippo and S. Lucidi, “Convergence conditions, line search algorithms and trust region implementations for the Polak-Ribière conjugate gradient method,” Optimization Methods & Software, vol. 20, no. 1, pp. 71–98, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Sun and J. Zhang, “Global convergence of conjugate gradient methods without line search,” Annals of Operations Research, vol. 103, pp. 161–173, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Z. Wei, G. Y. Li, and L. Qi, “Global convergence of the Polak-Ribière-Polyak conjugate gradient method with an Armijo-type inexact line search for nonconvex unconstrained optimization problems,” Mathematics of Computation, vol. 77, no. 264, pp. 2173–2193, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  13. L. Zhang, W. Zhou, and D. Li, “Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search,” Numerische Mathematik, vol. 104, no. 4, pp. 561–572, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. J. D. Powell, “Nonconvex minimization calculations and the conjugate gradient method,” in Numerical Analysis (Dundee, 1983), vol. 1066 of Lecture Notes in Mathematics, pp. 122–141, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. N. Andrei, “Scaled conjugate gradient algorithms for unconstrained optimization,” Computational Optimization and Applications, vol. 38, no. 3, pp. 401–416, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. N. Andrei, “Another nonlinear conjugate gradient algorithm for unconstrained optimization,” Optimization Methods & Software, vol. 24, no. 1, pp. 89–104, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. G. Birgin and J. M. Martínez, “A spectral conjugate gradient method for unconstrained optimization,” Applied Mathematics and Optimization, vol. 43, no. 2, pp. 117–128, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y.-H. Dai and L.-Z. Liao, “New conjugacy conditions and related nonlinear conjugate gradient methods,” Applied Mathematics and Optimization, vol. 43, no. 1, pp. 87–101, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. W. W. Hager and H. Zhang, “A new conjugate gradient method with guaranteed descent and an efficient line search,” SIAM Journal on Optimization, vol. 16, no. 1, pp. 170–192, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Li, C. Tang, and Z. Wei, “New conjugacy condition and related new conjugate gradient methods for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 523–539, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z. Wei, G. Li, and L. Qi, “New quasi-Newton methods for unconstrained optimization problems,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1156–1188, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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 · View at MathSciNet
  23. L. Liu, S. Yao, and Z. Wei, “The global and superlinear convergence of a new nonmonotone MBFGS algorithm on convex objective functions,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 422–438, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Z. Wei, G. Yu, G. Yuan, and Z. Lian, “The superlinear convergence of a modified BFGS-type method for unconstrained optimization,” Computational Optimization and Applications, vol. 29, no. 3, pp. 315–332, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Y. Xiao, Z. Wei, and Z. Wang, “A limited memory BFGS-type method for large-scale unconstrained optimization,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 1001–1009, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. Zhang, N. Y. Deng, and L. H. Chen, “New quasi-Newton equation and related methods for unconstrained optimization,” Journal of Optimization Theory and Applications, vol. 102, no. 1, pp. 147–167, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. H. Yabe and M. Takano, “Global convergence properties of nonlinear conjugate gradient methods with modified secant condition,” Computational Optimization and Applications, vol. 28, no. 2, pp. 203–225, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L. Nazareth, “A conjugate direction algorithm without line searches,” Journal of Optimization Theory and Applications, vol. 23, no. 3, pp. 373–387, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. G. Zoutendijk, “Nonlinear programming, computational methods,” in Integer and Nonlinear Programming, J. Jabadie, Ed., pp. 37–86, North-Holland, Amsterdam, The Netherlands, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, vol. 1 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2006. View at MathSciNet
  31. I. Bongartz, A. R. Conn, N. Gould, and P. L. Toint, “CUTE: constrained and unconstrained testing environment,” ACM Transactions on Mathematical Software, vol. 21, no. 1, pp. 123–160, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. E. D. Dolan and J. J. Moré, “Benchmarking optimization software with performance profiles,” Mathematical Programming, vol. 91, no. 2, pp. 201–213, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet