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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 780107, 5 pages
http://dx.doi.org/10.1155/2013/780107
Research Article

A New Smoothing Nonlinear Conjugate Gradient Method for Nonsmooth Equations with Finitely Many Maximum Functions

1College of Mathematics, Qingdao University, Qingdao 266071, China
2School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 8 March 2013; Accepted 25 March 2013

Academic Editor: Yisheng Song

Copyright © 2013 Yuan-yuan Chen and Shou-qiang Du. 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. Qi and P. Tseng, “On almost smooth functions and piecewise smooth functions,” Nonlinear Analysis, vol. 67, no. 3, pp. 773–794, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1, Springer, New York, NY, USA, 2003. View at MathSciNet
  3. D. Sun and J. Han, “Newton and quasi-Newton methods for a class of nonsmooth equations and related problems,” SIAM Journal on Optimization, vol. 7, no. 2, pp. 463–480, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Y. Gao, “Newton methods for solving two classes of nonsmooth equations,” Applications of Mathematics, vol. 46, no. 3, pp. 215–229, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S.-Q. Du and Y. Gao, “A parametrized Newton method for nonsmooth equations with finitely many maximum functions,” Applications of Mathematics, vol. 54, no. 5, pp. 381–390, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. N. Yamashita and M. Fukushima, “Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems,” Mathematical Programming B, vol. 76, no. 3, pp. 469–491, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. Q. Qi and J. Sun, “A nonsmooth version of Newton's method,” Mathematical Programming A, vol. 58, no. 3, pp. 353–367, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Elfoutayeni and M. Khaladi, “Using vector divisions in solving the linear complementarity problem,” Journal of Computational and Applied Mathematics, vol. 236, no. 7, pp. 1919–1925, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. X. Chen and W. Zhou, “Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization,” SIAM Journal on Imaging Sciences, vol. 3, no. 4, pp. 765–790, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. X. Chen, “Smoothing methods for nonsmooth, nonconvex minimization,” Mathematical Programming B, vol. 134, no. 1, pp. 71–99, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. M. C. Ferris, O. L. Mangasarian, and J. S. Pang, Complementarity: Applications, Algorithms and Extensions, Kluwer Academic, Dordrecht, The Netherlands, 2001. View at MathSciNet
  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, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Z. Dai and F. Wen, “Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search,” Numerical Algorithms, vol. 59, no. 1, pp. 79–93, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C.-Y. Wang, Y.-Y. Chen, and S.-Q. Du, “Further insight into the Shamanskii modification of Newton method,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 46–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Y. Chen and S. Q. Du, “Nonlinear conjugate gradient methods with Wolfe type line search,” Abstract and Applied Analysis, vol. 2013, Article ID 742815, 5 pages, 2013. View at Publisher · View at Google Scholar
  16. Y. Xiao, H. Song, and Z. Wang, “A modified conjugate gradient algorithm with cyclic Barzilai-Borwein steplength for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 236, no. 13, pp. 3101–3110, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Zhou and Z. Zou, “A relaxation scheme for Hamilton-Jacobi-Bellman equations,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 806–813, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X. Chen, L. Qi, and D. Sun, “Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities,” Mathematics of Computation, vol. 67, no. 222, pp. 519–540, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. C. Ferris and J. S. Pang, “Engineering and economic applications of complementarity problems,” SIAM Review, vol. 39, no. 4, pp. 669–713, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet