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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 515902, 10 pages
The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities
1School of Computer Science, Civil Aviation Flight University of China, Guanghan 618307, China
2School of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Received 22 June 2013; Accepted 19 August 2013
Academic Editor: Xu Minghua
Copyright © 2013 Haiwen Xu. 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.
- M. S. Gowda and Y. Song, “On semidefinite linear complementarity problems,” Mathematical Programming, vol. 88, no. 3, pp. 575–587, 2000.
- I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Bumariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
- F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998.
- 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.
- L. C. Zeng, N. C. Wong, and J. C. Yao, “Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities,” Journal of Optimization Theory and Applications, vol. 132, no. 1, pp. 51–69, 2007.
- L. C. Zeng, “On a general projection algorithm for variational inequalities,” Journal of Optimization Theory and Applications, vol. 97, no. 1, pp. 229–235, 1998.
- X. P. Ding, Y. C. Lin, and J. C. Yao, “Three-step relaxed hybrid steepest-descent methods for variational inequalities,” Applied Mathematics and Mechanics, vol. 28, no. 8, pp. 1029–1036, 2007.
- B. S. He, “A new method for a class of linear variational inequalities,” Mathematical Programming, vol. 66, no. 2, pp. 137–144, 1994.
- B. S. He and M. H. Xu, “A general framework of contraction methods for monotone variational inequalities,” Pacific Journal of Optimization, vol. 4, no. 2, pp. 195–212, 2008.
- B. S. He, “PPA-based contraction methods for general linearly constrained convex optimization,” Lectures of Contraction Methods for Convex Optimization and Monotone Variational Inequalities, 06C, 2012, http://math.nju.edu.cn/~hebma/.
- N. J. Huang, X. X. Huang, and X. Q. Yang, “Connections among constrained continuous and combinatorial vector optimization,” Optimization, vol. 60, no. 1-2, pp. 15–27, 2011.
- P. T. Harker and J. S. Pang, “A damped-Newton method for the linear complementarity problem,” in Computational Solution of Nonlinear Systems of Equations, vol. 26, pp. 265–284, American Mathematical Society, Providence, RI, USA, 1990.
- T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.
- H. W. Xu, E. B. Song, H. P. Pan, H. Shao, and L. M. Sun, “The modified and relaxed hybrid steepestdescent methods for variational inequalities,” in Proceedings of the 1st International Conference on Modelling and Simulation, vol. 2, pp. 169–174, World Academic Press, 2008.
- H. W. Xu, H. Shao, and Q. C. Zhang, “The Prediction-correction and relaxed hybrid steepest-descent method for variational inequalities,” in Proceedings of the International Symposium on Education and Computer Science, vol. 1, pp. 252–256, IEEE Computer Society and Academy, 2009.
- H. W. Xu, “Efficient implementation of a modified and relaxed hybrid steepest-descent method for a type of variational inequality,” Journal of Inequalities and Applications, vol. 2012, article 93, 2012.
- J. H. Hammond, Solving asymmetric variational inequality problems and systems of equations with generalized nonlinear programming algorithms [Ph.D. dissertation], Department of Mathematics, MIT, Cambridge, Mass, USA, 1984.
- P. Tseng, “Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,” Mathematical Programming, vol. 48, no. 2, pp. 249–263, 1990.
- R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
- D. F. Sun, “A projection and contraction method for generalized nonlinear complementarity problems,” Mathematica Numerica Sinica, vol. 16, no. 2, pp. 183–194, 1994.
- Y. Gao and D. F. Sun, “Calibrating least squares covariance matrix problems with equality and inequality constraints,” Tech. Rep., Department of Mathematics, National University of Singapore, 2008.
- M. A. Noor, “Some recent advances in variational inequalities. I. Basic concepts,” New Zealand Journal of Mathematics, vol. 26, no. 1, pp. 53–80, 1997.
- M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000.
- Y. Yao, M. A. Noor, R. Chen, and Y.-C. Liou, “Strong convergence of three-step relaxed hybrid steepest-descent methods for variational inequalities,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 175–183, 2008.
- “Advances in Equilibrium Modeling, Analysis, and Computation,” in Annals of Operations Research, A. Nagurney, Ed., vol. 44, J. C. Baltzer AG Scientific Publishing, Basel, Switzerland, 1993.