Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 305643, 12 pages
http://dx.doi.org/10.1155/2014/305643
Research Article

Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

Received 22 October 2013; Accepted 15 December 2013; Published 12 January 2014

Academic Editor: Zhongxiao Jia

Copyright © 2014 San-Yang Liu 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. 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
  2. M. E. El-Hawary, Optimal Power Flow: Solution Techniques, Requirement and Challenges, IEEE Service Center, Piscataway, NJ, USA, 1996.
  3. A. J. Wood and B. F. Wollenberg, Power Generations, Operations, and Control, John Wiley & Sons, New York, NY, USA, 1996.
  4. Y.-B. Zhao and D. Li, “Monotonicity of fixed point and normal mappings associated with variational inequality and its application,” SIAM Journal on Optimization, vol. 11, no. 4, pp. 962–973, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. V. Solodov and B. F. Svaiter, “A globally convergent inexact Newton method for systems of monotone equations,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, Eds., vol. 22, pp. 355–369, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. Zhou and K. C. Toh, “Superlinear convergence of a Newton-type algorithm for monotone equations,” Journal of Optimization Theory and Applications, vol. 125, no. 1, pp. 205–221, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. W.-J. Zhou and D.-H. Li, “A globally convergent BFGS method for nonlinear monotone equations without any merit functions,” Mathematics of Computation, vol. 77, no. 264, pp. 2231–2240, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Zhou and D. Li, “Limited memory BFGS method for nonlinear monotone equations,” Journal of Computational Mathematics, vol. 25, no. 1, pp. 89–96, 2007. View at Google Scholar · View at MathSciNet
  9. Y. Xiao and H. Zhu, “A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing,” Journal of Mathematical Analysis and Applications, vol. 405, no. 1, pp. 310–319, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. W. Cheng, “A PRP type method for systems of monotone equations,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 15–20, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Q.-R. Yan, X.-Z. Peng, and D.-H. Li, “A globally convergent derivative-free method for solving large-scale nonlinear monotone equations,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 649–657, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Li and D.-H. Li, “A class of derivative-free methods for large-scale nonlinear monotone equations,” IMA Journal of Numerical Analysis, vol. 31, no. 4, pp. 1625–1635, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Ahookhosh, K. Amini, and S. Bahrami, “Two derivative-free projection approaches for systems of large-scale nonlinear monotone equations,” Numerical Algorithms, vol. 64, no. 1, pp. 21–42, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Li, “A LS type method for solving large-scale nonlinear monotone equations,” Numerical Functional Analysis and Optimization, 2013. View at Publisher · View at Google Scholar
  15. W. Cheng, Y. Xiao, and Q.-J. Hu, “A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 11–19, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. La Cruz and M. Raydan, “Nonmonotone spectral methods for large-scale nonlinear systems,” Optimization Methods & Software, vol. 18, no. 5, pp. 583–599, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L. Zhang and W. Zhou, “Spectral gradient projection method for solving nonlinear monotone equations,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 478–484, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Z. Yu, J. Lin, J. Sun, Y. Xiao, L. Liu, and Z. Li, “Spectral gradient projection method for monotone nonlinear equations with convex constraints,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2416–2423, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Y. H. Dai, “Nonlinear conjugate gradient methods,” in Wiley Encyclopedia of Operations Research and Management Science, J. J. Cochran, L. A. Cox Jr, P. Keskinocak, J. P. Kharoufeh, and J. C. Smith, Eds., John Wiley & Sons, 2011. View at Google Scholar · View at Zentralblatt MATH
  20. 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
  21. W. W. Hager and H. Zhang, “Algorithm 851: CG DESCENT , 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 · View at MathSciNet
  22. G. Yu, L. Guan, and W. Chen, “Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization,” Optimization Methods & Software, vol. 23, no. 2, pp. 275–293, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. W. Cheng and Q. Liu, “Sufficient descent nonlinear conjugate gradient methods with conjugacy condition,” Numerical Algorithms, vol. 53, no. 1, pp. 113–131, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. E. Polak and G. Ribière, “Note sur la convergence de méthodes de directions conjuguées,” Revue Francaise Dinformatique et Derecherche Opérationnelle, Série Rouge, vol. 3, no. 16, pp. 35–43, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. 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 Zentralblatt MATH · View at MathSciNet
  27. W. W. Hager and H. Zhang, “The limited memory conjugate gradient method,” SIAM Journal on Optimization, vol. 23, no. 4, pp. 2150–2168, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. 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, no. 6, pp. 409–436, 1952. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. C. Wang, Y. Wang, and C. Xu, “A projection method for a system of nonlinear monotone equations with convex constraints,” Mathematical Methods of Operations Research, vol. 66, no. 1, pp. 33–46, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. G. Yu, S. Niu, and J. Ma, “Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints,” Journal of Industrial and Management Optimization, vol. 9, no. 1, pp. 117–129, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. N. Yamashita and M. Fukushima, “Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems,” Mathematical Programming, vol. 76, no. 3, pp. 469–491, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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