Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 264965, 19 pages
http://dx.doi.org/10.1155/2014/264965
Research Article

Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

1Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan
3Department of Information Management, and Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li 32003, Taiwan
4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 4 May 2014; Accepted 31 July 2014; Published 31 August 2014

Academic Editor: Jong Kyu Kim

Copyright © 2014 Lu-Chuan Ceng 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. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, France, 1969. View at MathSciNet
  2. G. M. Korpelevich, “The extragradient method for finding saddle points and other problems,” Matecon, vol. 12, pp. 747–756, 1976. View at Google Scholar
  3. L. C. Zeng and J. C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006. View at Google Scholar · View at MathSciNet · View at Scopus
  4. L. Ceng, Q. H. Ansari, and J. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112–1123, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. L. C. Ceng, Q. H. Ansari, N. C. Wong, and J. C. Yao, “An extragradient-like approximation method for variational inequalities and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, article 22, 18 pages, 2011. View at Google Scholar · View at MathSciNet
  7. L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. J. Peng and J. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008. View at Google Scholar · View at MathSciNet · View at Scopus
  9. Y. Yao, Y. C. Liou, and G. Marino, “Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems,” Journal of Applied Mathematics and Computing, vol. 31, no. 1-2, pp. 433–445, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L. Ceng and J. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. K. Goebel and W. A. Kirk, Topics on Metric Fixed-Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, 1976. View at MathSciNet
  15. N. Huang, “A new completely general class of variational inclusions with noncompact valued mappings,” Computers & Mathematics with Applications, vol. 35, no. 10, pp. 9–14, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. L. Ceng, Q. H. Ansari, M. M. Wong, and J. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  17. L. C. Zeng, S. M. Guu, and J. C. Yao, “Characterization of H-monotone operators with applications to variational inclusions,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 329–337, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus