A New Linesearch Algorithm for Split Equilibrium Problems


In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the ε-subdifferential. We obtain a strong convergence theorem for the new algorithm.

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  1. 1.

    Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. European J. Oper. Res. 227, 1–11 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127–149 (1994)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441–453 (2002)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 18, 103–120 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

  7. 7.

    Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems 21, 2071–2084 (2005)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Censor, Y., Motova, X. A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Combettes, P. L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dinh, B. V., Kim, D. S.: Projection algolithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106–117 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dinh, B.V., Son, D.X., Jiao, L., Kim, D.S.: Linesearch algorithms for split equilibrium problems and nonexpansive mappings. Fixed Point Theory Appl. 2016, 27 (2016). https://doi.org/10.1186/s13663-016-0518-3

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Dinh, B. V., Kim, D. S.: Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings. Optim Lett. 11, 537–553 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    He, Z.: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012, 162 (2012). https://doi.org/10.1186/1029-242X-2012-162

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Iusem, A. N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. 52, 621–635 (2003)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Takahashi, W.: Nonlinear variational inequalities and fixed point theorems. J. Math. Soc. Japan 28, 168–181 (1976)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Xu, H.K.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Problems 2010, 26 (2010). https://doi.org/10.1088/0266-5611/26/10/105018

    MathSciNet  Article  Google Scholar 

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The first author would like to thank the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program and Naresuan University for supporting his research via Grant No. PHD/0032/2555.

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Correspondence to Somyot Plubtieng.

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Dedicated to Professor Hoang Tuy

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Yuying, T., Plubtieng, S. A New Linesearch Algorithm for Split Equilibrium Problems. Acta Math Vietnam 45, 397–409 (2020). https://doi.org/10.1007/s40306-020-00361-7

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  • Split equilibrium problem
  • Line search rule
  • Projected methods
  • Strong convergence

Mathematics Subject Classification (2010)

  • 47H09
  • 47J25
  • 65K10
  • 65K15
  • 90C99