Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 147025, 14 pages
http://dx.doi.org/10.1155/2013/147025
Research Article

A Globally Convergent Hybrid Conjugate Gradient Method and Its Numerical Behaviors

1Department of Mathematics, Xidian University, Xi’an 710071, China
2College of Mathematics Science, Chong Qing Normal University, Chong Qing 40047, China

Received 21 December 2012; Revised 12 March 2013; Accepted 27 March 2013

Academic Editor: Martin Weiser

Copyright © 2013 Yuan-Yuan Huang 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, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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 · View at MathSciNet
  3. 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
  4. E. Polak and G. Ribiere, “Note sur la convergence de méthodes de directions conjuguées,” Revue francaise d’informatique et derecherche opérationnelle, série rouge, vol. 3, no. 16, pp. 35–43, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. B. T. Polyak, “The conjugate gradient method in extremal problems,” USSR Computational Mathematics and Mathematical Physics, vol. 9, no. 4, pp. 94–112, 1969. View at Google Scholar · View at Scopus
  6. 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
  7. W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pacific Journal of Optimization, vol. 2, no. 1, pp. 35–58, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. W. Hager and H. Zhang, “Algorithm 851: CG-DESCENT , a conjugate gradient method with guaranteed descent,” Association for Computing Machinery, vol. 32, no. 1, pp. 113–137, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Y. H. Dai and Y. Yuan, “An efficient hybrid conjugate gradient method for unconstrained optimization,” Annals of Operations Research, vol. 103, pp. 33–47, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y.-h. Dai and Q. Ni, “Testing different conjugate gradient methods for large-scale unconstrained optimization,” Journal of Computational Mathematics, vol. 21, no. 3, pp. 311–320, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Pytlak, Conjugate Gradient Algorithms in Nonconvex Optimization, vol. 89 of Nonconvex Optimization and its Applications, Springer, Berlin, Germany, 2009. View at MathSciNet
  12. Y.-H. Dai and C.-X. Kou, “A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search,” SIAM Journal on Optimization, vol. 23, no. 1, pp. 296–320, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y. Dong, “A practical PR+ conjugate gradient method only using gradient,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 2041–2052, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y. Dong, “New step lengths in conjugate gradient methods,” Computers & Mathematics with Applications, vol. 60, no. 3, pp. 563–571, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. A. S. Lewis and M. L. Overton, “Nonsmooth optimization via quasi-Newton methods,” Mathematical Programming A, 2012. View at Publisher · View at Google Scholar
  16. C. Lemarechal, “A view of line search,” Optimization and Optimal Control, vol. 30, pp. 59–78, 1981, Lecture Notes on Control and Information Sciences. View at Google Scholar
  17. 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 Scopus
  18. N. I. M. Gould, D. Orban, and P. L. Toint, “CUTEr and SifDec: a constrained and unconstrained testing environment, revisited,” ACM Transactions on Mathematical Software, vol. 29, pp. 373–394, 2003. View at Google Scholar
  19. E. Spedicato and Z. Huang, “Numerical experience with Newton-like methods for nonlinear algebraic systems,” Computing, vol. 58, no. 1, pp. 69–89, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Griewank, “On automatic differentiation,” in Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, Eds., pp. 83–108, Kluwer Academic, 1989. View at Google Scholar
  21. E. D. Dolan and J. J. Moré, “Benchmarking optimization software with performance profiles,” Mathematical Programming A, vol. 91, no. 2, pp. 201–213, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. View at MathSciNet