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An Explicit Parallel Algorithm for Variational Inequalities

  • Dang Van HieuEmail author
Article

Abstract

The paper proposes an explicit parallel iterative algorithm for solving variational inequalities over the intersection of fixed point sets of finitely many demicontractive mappings. The algorithm combines the steepest descent method with the Krasnosel’skii–Mann type method and the splitting-up technique. The strongly convergent theorem is established under fewer restrictions imposed on parameters and mappings. Several applications of the algorithm to other problems are presented. We also illustrate the convergence of the algorithm and compare it with others by considering some preliminary numerical experiments.

Keywords

Variational inequality Demicontractive mapping Nonexpansive mapping Parallel computation 

Mathematics Subject Classification

65J15 47H05 47J25 47J20 

Notes

Acknowledgements

The author would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in presenting and improving the original version of this paper. The guidance of Profs. P. K. Anh and L. D. Muu is gratefully acknowledged.

References

  1. 1.
    Anh, P.K., Buong, Ng, Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93(10), 2136–2157 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi \(\phi \)-nonexpansive mappings. J. Appl. Math. Comput. 48, 241–263 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44(2), 351–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Program. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bnouhachem, A., Ansari, Q.H., Yao, J.C.: Stromg convergence algorithm for hierarchical fixed point problems of a finite family of nonexpansive mappings. Fixed Point Theory 17(1), 1–17 (2016)Google Scholar
  6. 6.
    Bnouhachem, A., Al-Homidan, S., Ansari, Q.H.: New descent LQP alternating direction methods for solving a class of structured variational inequalities. Fixed Point Theory Appl. 2015(137), 1–11 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Buong, Ng, Duong, L.T.: An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 151, 513–524 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burger, M., Kaltenbacher, B.: Regularizing Newton–Kaczmarz methods for nonlinear ill-posed problems. SIAM J. Numer. Anal. 44, 153–182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chidume, C.E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234, 861–882 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cezaro, A.D., Haltmeier, M., Leitao, A., Scherzer, O.: On steepest-descent-Kaczmarz method for regularizing systems of nonlinear ill-posed equations. Appl. Math. Comput. 202, 596–607 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dafermos, S.C., McKelvey, S.C.: Partitionable variational inequalities with applications to network and economic equilibria. J. Optim. Theory Appl. 73, 243–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dafermos, S.: Traffic equilibria and variational inequalities. Transp. Sci. 14, 42–54 (1980)CrossRefGoogle Scholar
  13. 13.
    Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65(2), 443–465 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eslamian, M.: Strong convergence of a new multi-step algorithm for strict pseudo-contractive mappings and Ky Fan inequality. Mediterr. J. Math. 12(3), 1161–1176 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)zbMATHGoogle Scholar
  16. 16.
    Haltmeier, M., Kowar, R., Leitao, A., Scherzer, O.: Kaczmarz methods for regularizing nonlinear ill-posed equations. Inverse Probl. Imaging 1(2–3), 289–298 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hartman, P., Stampacchia, G.: On some non-linear elliptic diferential-functional equations. Acta Math. 115, 271–310 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hieu, D.V.: A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space. J. Korean Math. Soc. 52, 373–388 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model Anal. 21, 478–501 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hieu, D.V.: Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. J. Appl. Math. Comput. 53(1), 531–554 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66(1), 75–96 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jung, J.S.: A general iterative scheme for \(\kappa \)-strictly pseudo-contractive mappings and optimization problems. Appl. Math. Comput. 217(2), 5581–5588 (2011)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  25. 25.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)zbMATHGoogle Scholar
  26. 26.
    Maingé, P.E.: Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. Numer. Funct. Anal. Optim. 29(7–8), 820–834 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for Variational Inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vilenkin, N.Ya., Gorin, E.A., Kostyuchenko, A.G., Krasnosel’skii, M.A., Krein, S.G., Maslov, V.P., Mityagin, B.S., Petunin, YuI, et al.: Functional Analysis. Wolters-Noordhoff, Groningen (1972)zbMATHGoogle Scholar
  31. 31.
    Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, pp. 473–504. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  32. 32.
    Zeng, L.C., Wong, N.C., Yao, JCh.: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51–69 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhou, H., Wang, P.: A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 161, 716–727 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam

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