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EXTRAGRADIENT METHOD FOR VARIATIONAL INEQUALITIES

Year 2011, Volume: 40 Issue: 6, 839 - 854, 01.06.2011

Abstract

References

  • Aoyama, K., Kimura, Y., Takahashi, W. and Toyoda, M. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal- ysis 67, 2350–2360, 2007.
  • Bnouhachem, A., Noor, A. M. and Hao, Z. Some new extragradient iterative methods for variational inequalities, Nonlinear Anal. 70 (3), 1321–1329, 2009.
  • Chen, J., Zhang, L. and Fan, T. Viscosity approximation methods for nonexpansive map- pings and monotone mappings, J. Math. Anal. Appl. 334, 1450–1461, 2007.
  • Giannessi, F., Maugeri, A. and Pardalos, P. M. Equilibrium Problems: Nonsmooth opti- mization and variational inequality models(Kluwer Academic Press, Dordrecht, Holland, 2001).
  • Glowinski, R., Lions, J. L. and Tremolieres, R. Numerical Analysis of Variational Inequali- ties(North-Holland, Amsterdam, Holland, 1981).
  • Goebel, K. and Kirk, W. A. Topics on Metric Fixed-Point Theory (Cambridge University Press, Cambridge, England, 1990).
  • Harker, P. T. and Pang, J. S. Finite-dimensional variational inequality and nonlinear com- plementarity problems: A Survey of Theory, Algorithms and Applications, Mathematical Programming 48, 161–220, 1990.
  • He, B. S. and Liao, L. Z. Improvement of some projection methods for monotone variational inequalities, J. Optim. Theory Appl. 112, 111–128, 2002.
  • He, B. S., Yang, Z. H. and Yuan, X. M. An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl. 300 (2), 362–374, 2004.
  • Iiduka, H. and Takahashi, W. Strong convergence theorems for nonexpansine mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61, 341–350, 2005.
  • Kinderlehrer, D. and Stampacchia, G. An Introduction to Variational Inequalities and their Applications(SIAM, Philadelphia, 2000).
  • Korpelevich, G. M. The extragradient method for finding saddle points and other problems, Matecon 12, 747–756, 1976.
  • Lions, J. L. and Stampacchia, G. Variational inequalities, Comm. Pure Appl. Math. 20, 493–512, 1967.
  • Marino, G. and Xu, H. K. Convergence of generalized proximal point algorithms, Commu- nications on Pure and Applied Analysis 3, 791–808, 2004.
  • Moudafi, A. Viscosity approximating methods for fixed point problems, J. Math. Anal. Appl. 241, 46–55, 2000.
  • Nadezhkina, N. and Takahashi, W. Weak convergence theorem by an extragradient method for nonexpansive and monotone mappings, J. Optim. Theory Appl. 128, 191–201, 2006.
  • Noor, M. A. General variational inequalities, Appl. Math. Letters 1, 119–121, 1988.
  • Noor, M. A. New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251, 217–229, 2000.
  • Noor, M. A. New extragradient-type methods for general variational inequalities, J. Math. Anal. Appl. 277, 379–395, 2003.
  • Noor, M. A. Some developments in general variational inequalities, Appl. Math. Computa- tion 152, 199–277, 2004.
  • Noor, M. A. Projection methods for nonconvex variational inequalities, Optim. Letetrs 3, 411–418, 2009.
  • Noor, M. A. Extended general variational inequalities, Appl. Math. Letters 22, 182–185, 2009.
  • Noor, M. A., Noor, K. I. and Rassias, Th. M. Some aspects of variational inequalities, J. Comput. Appl. Math. 47, 285–312, 1993.
  • Opial, Z. Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73, 591–597, 1967.
  • Pardalos, P. M., Rassias, T. M. and Khan, A. A. Nonlineaqr Analysis and Variational Prob- lems(Springer, New York, 2010).
  • Patriksson, M. Nonlinear Programming and Variational Inequality Problems: A Unified Approach(Kluwer Academic Publishers, Dordrecht, Holland, 1999).
  • Rockafellar, R. T. On the maximality of sums nonlinear monotone operators, SIAM Trans. Amer. Math. Soc 149, 75–88, 1970.
  • Stampacchia, G. Formes bilineaires coercitivies sur les ensembles convexes, C. R. Acad. Sciences, Paris 258, 4413–4416, 1964.
  • Suzuki, T. Strong convergence of Krasnoselskii and Mann’s type sequences for one- parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305, 227–239, 2005.
  • Takahashi, W. and Toyoda, M. Weak convergence theorems for nonexpansive mapping, J. Optim. Theory Appl. 118, 417–428, 2003.
  • Xu, H. K. Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298, 279–291, 2004.
  • Zeng, L. C. and Yao, J. C. Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics 10(5), 1293–1303, 2006.

Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT

Year 2011, Volume: 40 Issue: 6, 839 - 854, 01.06.2011

Abstract

In this paper, we suggest and analyze a new extragradient iterative method, which is suggested by combining a modified extragradient method with the viscosity approximation method, for finding the common element of the set of fixed points of a countable family of nonexpansive mappings, and the solution set of the variational inequality in a Hilbert space. This new method includes the extragradient and viscosity methods as special cases. We also consider the strong convergence of the proposed method under some mild conditions. Several special
cases are also discussed. Results proved in this paper may be viewed as an improvement and refinement of the previously known results.

References

  • Aoyama, K., Kimura, Y., Takahashi, W. and Toyoda, M. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal- ysis 67, 2350–2360, 2007.
  • Bnouhachem, A., Noor, A. M. and Hao, Z. Some new extragradient iterative methods for variational inequalities, Nonlinear Anal. 70 (3), 1321–1329, 2009.
  • Chen, J., Zhang, L. and Fan, T. Viscosity approximation methods for nonexpansive map- pings and monotone mappings, J. Math. Anal. Appl. 334, 1450–1461, 2007.
  • Giannessi, F., Maugeri, A. and Pardalos, P. M. Equilibrium Problems: Nonsmooth opti- mization and variational inequality models(Kluwer Academic Press, Dordrecht, Holland, 2001).
  • Glowinski, R., Lions, J. L. and Tremolieres, R. Numerical Analysis of Variational Inequali- ties(North-Holland, Amsterdam, Holland, 1981).
  • Goebel, K. and Kirk, W. A. Topics on Metric Fixed-Point Theory (Cambridge University Press, Cambridge, England, 1990).
  • Harker, P. T. and Pang, J. S. Finite-dimensional variational inequality and nonlinear com- plementarity problems: A Survey of Theory, Algorithms and Applications, Mathematical Programming 48, 161–220, 1990.
  • He, B. S. and Liao, L. Z. Improvement of some projection methods for monotone variational inequalities, J. Optim. Theory Appl. 112, 111–128, 2002.
  • He, B. S., Yang, Z. H. and Yuan, X. M. An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl. 300 (2), 362–374, 2004.
  • Iiduka, H. and Takahashi, W. Strong convergence theorems for nonexpansine mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61, 341–350, 2005.
  • Kinderlehrer, D. and Stampacchia, G. An Introduction to Variational Inequalities and their Applications(SIAM, Philadelphia, 2000).
  • Korpelevich, G. M. The extragradient method for finding saddle points and other problems, Matecon 12, 747–756, 1976.
  • Lions, J. L. and Stampacchia, G. Variational inequalities, Comm. Pure Appl. Math. 20, 493–512, 1967.
  • Marino, G. and Xu, H. K. Convergence of generalized proximal point algorithms, Commu- nications on Pure and Applied Analysis 3, 791–808, 2004.
  • Moudafi, A. Viscosity approximating methods for fixed point problems, J. Math. Anal. Appl. 241, 46–55, 2000.
  • Nadezhkina, N. and Takahashi, W. Weak convergence theorem by an extragradient method for nonexpansive and monotone mappings, J. Optim. Theory Appl. 128, 191–201, 2006.
  • Noor, M. A. General variational inequalities, Appl. Math. Letters 1, 119–121, 1988.
  • Noor, M. A. New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251, 217–229, 2000.
  • Noor, M. A. New extragradient-type methods for general variational inequalities, J. Math. Anal. Appl. 277, 379–395, 2003.
  • Noor, M. A. Some developments in general variational inequalities, Appl. Math. Computa- tion 152, 199–277, 2004.
  • Noor, M. A. Projection methods for nonconvex variational inequalities, Optim. Letetrs 3, 411–418, 2009.
  • Noor, M. A. Extended general variational inequalities, Appl. Math. Letters 22, 182–185, 2009.
  • Noor, M. A., Noor, K. I. and Rassias, Th. M. Some aspects of variational inequalities, J. Comput. Appl. Math. 47, 285–312, 1993.
  • Opial, Z. Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73, 591–597, 1967.
  • Pardalos, P. M., Rassias, T. M. and Khan, A. A. Nonlineaqr Analysis and Variational Prob- lems(Springer, New York, 2010).
  • Patriksson, M. Nonlinear Programming and Variational Inequality Problems: A Unified Approach(Kluwer Academic Publishers, Dordrecht, Holland, 1999).
  • Rockafellar, R. T. On the maximality of sums nonlinear monotone operators, SIAM Trans. Amer. Math. Soc 149, 75–88, 1970.
  • Stampacchia, G. Formes bilineaires coercitivies sur les ensembles convexes, C. R. Acad. Sciences, Paris 258, 4413–4416, 1964.
  • Suzuki, T. Strong convergence of Krasnoselskii and Mann’s type sequences for one- parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305, 227–239, 2005.
  • Takahashi, W. and Toyoda, M. Weak convergence theorems for nonexpansive mapping, J. Optim. Theory Appl. 118, 417–428, 2003.
  • Xu, H. K. Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298, 279–291, 2004.
  • Zeng, L. C. and Yao, J. C. Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics 10(5), 1293–1303, 2006.
There are 32 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Mathematics
Authors

A. Bnouhachem This is me

M.a. Noor This is me

Muhammad Aslam Noor This is me

E. Al-said This is me

M. Khalfaoui This is me

 s. Zhaohan This is me

Publication Date June 1, 2011
Published in Issue Year 2011 Volume: 40 Issue: 6

Cite

APA Bnouhachem, A., Noor, M., Noor, M. A., Al-said, E., et al. (2011). Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT. Hacettepe Journal of Mathematics and Statistics, 40(6), 839-854.
AMA Bnouhachem A, Noor M, Noor MA, Al-said E, Khalfaoui M, Zhaohan . Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT. Hacettepe Journal of Mathematics and Statistics. June 2011;40(6):839-854.
Chicago Bnouhachem, A., M.a. Noor, Muhammad Aslam Noor, E. Al-said, M. Khalfaoui, and  s. Zhaohan. “Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT”. Hacettepe Journal of Mathematics and Statistics 40, no. 6 (June 2011): 839-54.
EndNote Bnouhachem A, Noor M, Noor MA, Al-said E, Khalfaoui M, Zhaohan  (June 1, 2011) Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT. Hacettepe Journal of Mathematics and Statistics 40 6 839–854.
IEEE A. Bnouhachem, M. Noor, M. A. Noor, E. Al-said, M. Khalfaoui, and  . Zhaohan, “Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT”, Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 6, pp. 839–854, 2011.
ISNAD Bnouhachem, A. et al. “Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT”. Hacettepe Journal of Mathematics and Statistics 40/6 (June 2011), 839-854.
JAMA Bnouhachem A, Noor M, Noor MA, Al-said E, Khalfaoui M, Zhaohan . Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT. Hacettepe Journal of Mathematics and Statistics. 2011;40:839–854.
MLA Bnouhachem, A. et al. “Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT”. Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 6, 2011, pp. 839-54.
Vancouver Bnouhachem A, Noor M, Noor MA, Al-said E, Khalfaoui M, Zhaohan . Extragradient Method for Variational Inequalities  ABSTRACT  |  FULL TEXT. Hacettepe Journal of Mathematics and Statistics. 2011;40(6):839-54.