About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 130682, 14 pages
http://dx.doi.org/10.1155/2012/130682
Research Article

Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems

1Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
2Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
3Mathematics Division, National Center for Theoretical Sciences (Taipei Office), Taipei 10617, Taiwan

Received 24 February 2012; Accepted 25 August 2012

Academic Editor: Malisa R. Zizovic

Copyright © 2012 Xin-He Miao and Jein-Shan Chen. 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. R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1992.
  2. R. W. Cottle, J.-S. Pang, and V. Venkateswaran, “Sufficient matrices and the linear complementarity problem,” Linear Algebra and Its Applications, vol. 114-115, pp. 231–249, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I, Springer Series in Operations Research, Springer, New York, NY, USA, 2003.
  4. M. S. Gowda, “On the extended linear complementarity problem,” Mathematical Programming, vol. 72, no. 1, pp. 33–50, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J.-S. Chen, X. Chen, and P. Tseng, “Analysis of nonsmooth vector-valued functions associated with second-order cones,” Mathematical Programming, vol. 101, no. 1, pp. 95–117, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J.-S. Chen and P. Tseng, “An unconstrained smooth minimization reformulation of the second-order cone complementarity problem,” Mathematical Programming, vol. 104, no. 2-3, pp. 293–327, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. Fukushima, Z. Q. Lou, and P. Tseng, “Smoothing functions for secondordercone complementarity problems,” SIAM Journal on Optimization, vol. 12, no. 2, pp. 436–460, 2002. View at Publisher · View at Google Scholar
  8. S. Pan and J.-S. Chen, “A damped Gauss-Newton method for the second-order cone complementarity problem,” Applied Mathematics and Optimization, vol. 59, no. 3, pp. 293–318, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. X. Chen and H. Qi, “Cartesian P-property and its applications to the semidefinite linear complementarity problem,” Mathematical Programming, vol. 106, no. 1, pp. 177–201, 2006. View at Publisher · View at Google Scholar
  10. M. S. Gowda and Y. Song, “On semidefinite linear complementarity problems,” Mathematical Programming, vol. 88, no. 3, pp. 575–587, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. D. Sun, “The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,” Mathematics of Operations Research, vol. 31, no. 4, pp. 761–776, 2006. View at Publisher · View at Google Scholar
  12. P. Tseng, “Merit functions for semi-definite complementarity problems,” Mathematical Programming, vol. 83, no. 2, pp. 159–185, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. S. Gowda, “On the continuity of the solution map in linear complementarity problems,” SIAM Journal on Optimization, vol. 2, no. 4, pp. 619–634, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. O. L. Mangasarian and T. H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,” SIAM Journal on Control and Optimization, vol. 25, no. 3, pp. 583–595, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. G. S. R. Murthy, T. Parthasarathy, and M. Sabatini, “Lipschitzian Q-matrices are P-matrices,” Mathematical Programming, vol. 74, no. 1, pp. 55–58, 1996. View at Publisher · View at Google Scholar
  16. M. S. Gowda and R. Sznajder, “On the Lipschitzian properties of polyhedral multifunctions,” Mathematical Programming, vol. 74, no. 3, pp. 267–278, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. N. D. Yen, “Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint,” Mathematics of Operations Research, vol. 20, no. 3, pp. 695–708, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. A. B. Levy, “Stability of solutions to parameterized nonlinear complementarity problems,” Mathematical Programming, vol. 85, no. 2, pp. 397–406, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. R. Balaji, T. Parthasarathy, D. Sampangi Raman, and V. Vetrivel, “On the Lipschitz continuity of the solution map in semidefinite linear complementarity problems,” Mathematics of Operations Research, vol. 30, no. 2, pp. 462–471, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. R. Balaji, “On an interconnection between the Lipschitz continuity of the solution map and the positive principal minor property in linear complementarity problems over Euclidean Jordan algebras,” Linear Algebra and Its Applications, vol. 426, no. 1, pp. 83–95, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. S. Gowda, R. Sznajder, and J. Tao, “Some P-properties for linear transformations on Euclidean Jordan algebras,” Linear Algebra and Its Applications, vol. 393, pp. 203–232, 2004. View at Publisher · View at Google Scholar
  22. J. Tao and M. S. Gowda, “Some P-properties for nonlinear transformations on Euclidean Jordan algebras,” Mathematics of Operations Research, vol. 30, no. 4, pp. 985–1004, 2005. View at Publisher · View at Google Scholar
  23. M. S. Gowda and R. Sznajder, “Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras,” Mathematics of Operations Research, vol. 31, no. 1, pp. 109–123, 2006. View at Publisher · View at Google Scholar
  24. M. S. Gowda and R. Sznajder, “Some global uniqueness and solvability results for linear complementarity problems over symmetric cones,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 461–481, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, The Clarendon Press; Oxford University Press, New York, NY, USA, 1994.
  26. L. Kong, L. Tunçel, and N. Xiu, “Vector-valued implicit Lagrangian for symmetric cone complementarity problems,” Asia-Pacific Journal of Operational Research, vol. 26, no. 2, pp. 199–233, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. I. Jeyaraman and V. Vetrivel, “On the Lipschitzian property in linear complementarity problems over symmetric cones,” Linear Algebra and Its Applications, vol. 435, no. 4, pp. 842–851, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH