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A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems

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A Correction to this article was published on 05 September 2018

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Abstract

In this paper, we introduce a parallel extragradient-like projection algorithm for finding a common solution of a system of unrelated variational inequalities and fixed point problems corresponding to different feasible domains in a real Hilbert space. The main idea of the paper, based on the 2006 hybrid extragradient method of Nadezhkina and Takahashi, is to combine three methods including the extragradient method, the Mann iteration method and the projection method with the parallel splitting-up technique. After computing the parallel extragradient iteration point, the next iteration point is modified by projecting a given initial point on the intersect of suitable convex sets to get a strong convergence property under certain assumptions by suitable choice parameters. Finally, a numerical example is developed to illustrate the behavior of our algorithm.

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Change history

  • 05 September 2018

    In the original article the affiliation of the corresponding author Pham Ngoc Anh was wrongly published.

References

  1. Anh, P.N.: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 154, 303–320 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, P.N., Hai, T.N., Tuan, P.M.: On ergodic algorithms for equilibrium problems. J. Global Optim. 64, 179–195 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anh, P.N., Phuong, N.X.: Linesearch methods for variational inequalities involving strict pseudocontractions. Optim. 64, 1841–1854 (2015)

    Article  MATH  Google Scholar 

  4. Ansari, Q.H., Yao, J.C.: A fixed point theorem and its applications to a system of variational inequalities. Bull. Aust. Math. Soc. 59, 433–442 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antipin, A.S.: On a method for convex programs using a symmetri cal modification of the Lagrange function. Ekonomika i Matematicheskie Metody 12(6), 1164–1173 (1976)

    Google Scholar 

  6. Bauschke, H.H., Combettes, P.L., Luke, D.R.: A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. J. Approx. Theory 141, 63–69 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent direct method for monotone variational inequalities in Hilbert spaces. Numer. Funct. Anal. Optim. 30, 23–36 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set Valued Var. Anal. 20, 229–247 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementary problems. Springer, NewYork (2003)

    MATH  Google Scholar 

  11. Harker, P.T., Pang, J.S.: A damped-newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Konnov, I.V.: On systems of variational inequalities. Russ. Math. 41, 79–88 (1997)

    MathSciNet  Google Scholar 

  13. Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer, Berlin (2000)

    MATH  Google Scholar 

  14. Korpelevich, G.M.: Extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  15. Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  16. Marino-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400–2411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM Optim. 16, 1230–1241 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nakajo, N., Takahashi, W.: Strong convergence theorem for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mapping. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peng, J.W.: Iterative algorithms for mixed equilibrium problems, strict pseudocontractions and monotone mappings. J. Optim. Theory Appl. 144, 107–119 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Reich, S., Zalas, R.: A modular string averaging procedure for solving the common fixed point problem for quasi-nonexpansive mappings in Hilbert space. Num. Algorithm 72, 297–323 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zeng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 10, 1293–1303 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Ton Duc Thang University, Hanoi city, Vietnam

    Pham Ngoc Anh

  2. Department of Mathematics, University of Fire Fighting and Prevention, Hanoi city, Vietnam

    Ngo Xuan Phuong

Authors
  1. Pham Ngoc Anh
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  2. Ngo Xuan Phuong
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Correspondence to Pham Ngoc Anh.

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Anh, P.N., Phuong, N.X. A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems. J. Fixed Point Theory Appl. 20, 74 (2018). https://doi.org/10.1007/s11784-018-0554-1

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  • DOI: https://doi.org/10.1007/s11784-018-0554-1

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