## Taiwanese Journal of Mathematics

### CONVERGENCE ANALYSIS OF A HYBRID RELAXED-EXTRAGRADIENT METHOD FOR MONOTONE VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS

#### Abstract

In this paper we introduce a hybrid relaxed-extragradient method for finding a common element of the set of common fixed points of $N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The hybrid relaxed-extragradient method is based on two well-known methods: hybrid and extragradient. We derive a strong convergence theorem for three sequences generated by this method. Based on this theorem, we also construct an iterative process for finding a common fixed point of $N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the rest $N$ mappings are nonexpansive.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 9 (2008), 2549-2568.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405195

Digital Object Identifier
doi:10.11650/twjm/1500405195

Mathematical Reviews number (MathSciNet)
MR2479071

Zentralblatt MATH identifier
1220.47085

#### Citation

Ceng, Lu-Chuan; Kien, B. T.; Wong, N. C. CONVERGENCE ANALYSIS OF A HYBRID RELAXED-EXTRAGRADIENT METHOD FOR MONOTONE VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS. Taiwanese J. Math. 12 (2008), no. 9, 2549--2568. doi:10.11650/twjm/1500405195. https://projecteuclid.org/euclid.twjm/1500405195

#### References

• [1.] A. S. Antipin, Methods for solving variational inequalities with related constraints, Comput. Math. Math. Phys., 40 (2000), 1239-1254.
• [2.] A. S. Antipin and F. P. Vasiliev, Regularized prediction method for solving variational inequalities with an inexactly given set, Comput. Math. Math. Phys., \bf44 (2004), 750-758.
• [3.] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.
• [4.] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
• [5.] R. S. Burachik, J. O. Lopes, and B. F. Svaiter, An outer approximation method for the variational inequality problem, SIAM J. Control Optim., 43 (2005), 2071-2088.
• [6.] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.
• [7.] K. Geobel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.
• [8.] 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. Butnariu, Y. Censor, and S. Reich, eds., Kluwer Academic, Dordrecht, The Netherlands, 2001, pp. 473-504.
• [9.] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
• [10.] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.
• [11.] B.-S. He, Z.-H. Yang, and X.-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374.
• [12.] H. Iiduka and W. Takahashi, Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications, Adv. Nonlinear Var. Inequal., 9 (2006), 1-10.
• [13.] H. Iiduka, W. Takahashi, and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Panamer. Math. J., 14 (2004), 49-61.
• [14.] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
• [15.] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747-756.
• [16.] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
• [17.] F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344.
• [18.] K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
• [19.] L. C. Zeng, N. C. Wong and J. C. Yao, Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theory Appl., 132 (2007), 51-69.
• [20.] L. C. Ceng and J. C. Yao, Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions, J. Comput. Appl. Math., (2007), doi: 10.1016/j.cam.2007.01.034.
• [21.] N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241.
• [22.] L. C. Ceng and J. C. Yao, On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators, J. Comput. Appl. Math., (2007), doi:10.1016/j.cam.2007.02.010.
• [23.] L. C. Ceng and J. C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., (2007), doi:10.1016/j.amc.2007.01.021.
• [24.] M. V. Solodov, Convergence rate analysis of iterative algorithms for solving variational inequality problem, Math. Program., 96 (2003), 513-528.
• [25.] L. C. Zeng, N. C. Wong and J. C. Yao, Convergence of hybrid steepest-descent methods for generalized variational inequalities, Acta Math. Sinica English Ser., 22(1) (2006), 1-12.
• [26.] M. V. Solodov and B. F. Svaiter, An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res., 25 (2000), 214-230.
• [27.] L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math., 10(5) (2006), 1293-1303.
• [28.] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.
• [29.] R. P. Agarwal, Donal O' Regan and D. R. Shahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mapping, J. Nonlinear and Convex Analysis, 8 (2007), 61-79.