## Abstract and Applied Analysis

### The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Haiwen Xu

#### Abstract

We establish the strong convergence of prediction-correction and relaxed hybrid steepest-descent method (PRH method) for variational inequalities under some suitable conditions that simplify the proof. And it is to be noted that the proof is different from the previous results and also is not similar to the previous results. More importantly, we design a set of practical numerical experiments. The results demonstrate that the PRH method under some descent directions is more slightly efficient than that of the modified and relaxed hybrid steepest-descent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 515902, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393450486

Digital Object Identifier
doi:10.1155/2013/515902

Mathematical Reviews number (MathSciNet)
MR3121487

Zentralblatt MATH identifier
1291.90267

#### Citation

Xu, Haiwen. The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 515902, 10 pages. doi:10.1155/2013/515902. https://projecteuclid.org/euclid.aaa/1393450486

#### References

• 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/$\sim\,\!$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. \endinput