Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2011 (2011), Article ID 161853, 12 pages
http://dx.doi.org/10.1155/2011/161853
Research Article

The Levenberg-Marquardt-Type Methods for a Kind of Vertical Complementarity Problem

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

Received 30 August 2011; Accepted 12 October 2011

Academic Editor: Chong Lin

Copyright © 2011 Shou-qiang Du and Yan Gao. 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. C. Kanzow and M. Fukushima, “Equivalence of the generalized complementarity problem to differentiable unconstrained minimization,” Journal of Optimization Theory and Applications, vol. 90, no. 3, pp. 581–603, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J.-S. Pang and L. Q. Qi, “Nonsmooth equations: motivation and algorithms,” SIAM Journal on Optimization, vol. 3, no. 3, pp. 443–465, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. L. Qi and P. Tseng, “On almost smooth functions and piecewise smooth functions,” Nonlinear Analysis, Theory, Methods & Applications, vol. 67, no. 3, pp. 773–794, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. Q. Qi and J. Sun, “A nonsmooth version of Newton's method,” Mathematical Programming, vol. 58, no. 3, pp. 353–367, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, New York, NY, USA, 1982.
  6. 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
  7. F. Facchinei and C. Kanzow, “A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems,” Mathematical Programming, vol. 76, no. 3, pp. 493–512, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. Fischer, V. Jeyakumar, and D. T. Luc, “Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems,” Journal of Optimization Theory and Applications, vol. 110, no. 3, pp. 493–513, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Q. Du and Y. Gao, “A modified Levenberg-Marquardt method for nonsmooth equations with finitely many maximum functions,” Mathematical Problems in Engineering, vol. 2008, Article ID 942391, 10 pages, 2008. View at Publisher · View at Google Scholar
  10. 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
  11. M. J. Śmietański, “An approximate Newton method for non-smooth equations with finite max functions,” Numerical Algorithms, vol. 41, no. 3, pp. 219–238, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. C. Ma and J. Tang, “The quadratic convergence of a smoothing Levenberg-Marquardt method for nonlinear complementarity problem,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 566–581, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. C. Kanzow, N. Yamashita, and M. Fukushima, “Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 375–397, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. N. Yamashita and M. Fukushima, “On the rate of convergence of the Levenberg-Marquardt method,” Computing, vol. 15, pp. 227–238, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. Fan and J. Pan, “Convergence properties of a self-adaptive Levenberg-Marquardt algorithm under local error bound condition,” Computational Optimization and Applications, vol. 34, no. 1, pp. 47–62, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. K. Ueda and N. Yamashita, “On a Global Complexity Bound of the Levenberg-Marquardt Method,” Journal of Optimization Theory and Applications, vol. 147, no. 3, pp. 443–453, 2010. View at Publisher · View at Google Scholar
  17. R. A. Polyak, “Regularized Newton method for unconstrained convex optimization,” Mathematical Programming, vol. 120, no. 1, pp. 125–145, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. K. Ueda and N. Yamashita, “Convergence properties of the regularized Newton method for the unconstrained nonconvex optimization,” Applied Mathematics and Optimization, vol. 62, no. 1, pp. 27–46, 2010. View at Publisher · View at Google Scholar
  19. H. Yin, Z.-H. Huang, and L. Qi, “The convergence of a Levenberg-Marquardt method for nonlinear inequalities,” Numerical Functional Analysis and Optimization, vol. 29, no. 5-6, pp. 687–716, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. O. L. Mangasarian and M. V. Solodov, “Nonlinear complementarity as unconstrained and constrained minimization,” Mathematical Programming, vol. 62, no. 2, pp. 277–297, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. J. S. Chen and S. Pan, “A family of NCP functions and a descent method for the nonlinear complementarity problem,” Computational Optimization and Applications, vol. 40, no. 3, pp. 389–404, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH