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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 507102, 6 pages
http://dx.doi.org/10.1155/2014/507102
Research Article

The Hybrid BFGS-CG Method in Solving Unconstrained Optimization Problems

1School of Applied Sciences and Foundation, Infrastructure University Kuala Lumpur, 43000 Kajang, Malaysia
2Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Tembila Campus, 22200 Besut, Malaysia
3Department of Mathematics, Faculty of Science and Technology, Universiti Malaysia Terengganu (UMT), 21030 Kuala Terengganu, Malaysia
4Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), 43400 Serdang, Malaysia

Received 22 April 2013; Accepted 23 January 2014; Published 4 March 2014

Academic Editor: Lucas Jodar

Copyright © 2014 Mohd Asrul Hery Ibrahim 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. L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific Journal of Mathematics, vol. 16, pp. 1–3, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. Wolfe, “Convergence conditions for ascent methods,” SIAM Review, vol. 11, no. 2, pp. 226–235, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. P. Wolfe, “Convergence conditions for ascent methods. II: some corrections,” SIAM Review, vol. 13, no. 2, pp. 185–188, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. A. Goldstein, “On steepest descent,” Journal of the Society for Industrial and Applied Mathematics A, vol. 3, no. 1, pp. 147–151, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z.-J. Shi, “Convergence of quasi-Newton method with new inexact line search,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 120–131, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. L. Han and M. Neumann, “Combining quasi-Newton and Cauchy directions,” International Journal of Applied Mathematics, vol. 12, no. 2, pp. 167–191, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” The Computer Journal, vol. 7, no. 2, pp. 149–154, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. N. Andrei, “Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization,” European Journal of Operational Research, vol. 204, no. 3, pp. 410–420, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. E. Polak and G. Ribière, “Note on the convergence of methods of conjugate directions,” Revue Française d’Informatique et de Recherche Opérationnelle, vol. 3, pp. 35–43, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z.-J. Shi and J. Shen, “Convergence of the Polak-Ribiére-Polyak conjugate gradient method,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 6, pp. 1428–1441, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. G. Yu, L. Guan, and Z. Wei, “Globally convergent Polak-Ribière-Polyak conjugate gradient methods under a modified Wolfe line search,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 3082–3090, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. M. R. Hestenes and E. Stiefel, “Method of conjugate gradient for solving linear equations,” Journal of Research of the National Bureau of Standards, vol. 49, no. 6, pp. 409–436, 1952. View at Publisher · View at Google Scholar
  13. A. Ludwig, “The Gauss-Seidel-quasi-Newton method: a hybrid algorithm for solving dynamic economic models,” Journal of Economic Dynamics and Control, vol. 31, no. 5, pp. 1610–1632, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. Y.-Z. Luo, G.-J. Tang, and L.-N. Zhou, “Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method,” Applied Soft Computing, vol. 8, no. 2, pp. 1068–1073, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. R. H. Byrd and J. Nocedal, “A tool for the analysis of quasi-Newton methods with application to unconstrained minimization,” SIAM Journal on Numerical Analysis, vol. 26, no. 3, pp. 727–739, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. H. Byrd, J. Nocedal, and Y.-X. Yuan, “Global convergence of a class of quasi-Newton methods on convex problems,” SIAM Journal on Numerical Analysis, vol. 24, no. 5, pp. 1171–1191, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Mathematical Programming, vol. 12, no. 1, pp. 241–254, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. N. Andrei, “An unconstrained optimization test functions collection,” Advanced Modeling and Optimization, vol. 10, no. 1, pp. 147–161, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer, Berlin, Germany, 1996. View at MathSciNet
  20. J. J. Moré, B. S. Garbow, and K. E. Hillstrom, “Testing unconstrained optimization software,” ACM Transactions on Mathematical Software, vol. 7, no. 1, pp. 17–41, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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 · View at Scopus