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Strong convergence of inertial algorithms for solving equilibrium problems

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Abstract

In this paper, we introduce several inertial-like algorithms for solving equilibrium problems (EP) in real Hilbert spaces. The algorithms are constructed using the resolvent of the EP associated bifunction and combines the inertial and the Mann-type technique. Under mild and standard conditions imposed on the cost bifunction and control parameters strong convergence of the algorithms is established. We present several numerical examples to illustrate the behavior of our schemes and emphasize their convergence advantages compared with some related methods.

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References

  1. Antipin, A.S.: Second order controlled differential grradient methods for equilibrium problems. Differ. Equ. 35, 592–601 (1999)

    MATH  Google Scholar 

  2. Antipin, A.S.: Equilibrium programming: proximal methods. Comput. Math. Phys. 37, 1285–1296 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Antipin, A.S.: Second-order proximal differential systems with feedback control. Differ. Equ. 29, 1597–1607 (1993)

    Google Scholar 

  4. Anh, P.N., Hieu, D.V.: Multi-step algorithms for solving EPs. Math. Model. Anal. 23, 453–472 (2018)

    Article  MathSciNet  Google Scholar 

  5. Balhag, A., Chbani, Z., Riahi, H.: Existence and continuousdiscrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evol. Equ. Control Theory 7, 373–401 (2018)

    Article  MathSciNet  Google Scholar 

  6. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  Google Scholar 

  7. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Program. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Chbani, Z., Riahi, H.: Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities. Optim. Lett. 7, 185–206 (2013)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)

  11. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2002)

    MATH  Google Scholar 

  12. Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    Article  MathSciNet  Google Scholar 

  13. Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67(11), 2003–2029 (2018). https://doi.org/10.1080/02331934.2018.1505886

    Article  MathSciNet  MATH  Google Scholar 

  14. Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Meth. Oper. Res. 88(3), 399–415 (2018). https://doi.org/10.1007/s00186-018-0640-6

    Article  MathSciNet  MATH  Google Scholar 

  15. Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algor. 80(4), 1413–1436 (2018). https://doi.org/10.1007/s11075-018-0532-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Hieu, D.V.: Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems. Math. Model. Anal. 24, 1–19 (2019)

    Article  MathSciNet  Google Scholar 

  17. Hieu, D.V., Strodiot, J.J.: Strong conve rgence theorems for equilibrium problems and fixed point problems in Banach spaces. J. Fixed Point Theory Appl. 20, 131 (2018)

    Article  Google Scholar 

  18. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput. Optim. Appl. 73, 913–932 (2019)

    Article  MathSciNet  Google Scholar 

  19. Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Golden ratio algorithms with new stepsize rules for variational inequalities. Math. Meth. Appl. Sci. (2019). https://doi.org/10.1002/mma.5703

    Article  MathSciNet  MATH  Google Scholar 

  20. Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problems in Hilbert spaces. Optimization 59, 1259–1274 (2010)

    Article  MathSciNet  Google Scholar 

  21. Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)

    Article  MathSciNet  Google Scholar 

  22. Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  23. Konnov, I.V., Ali, M.S.S.: Descent methods for monotone equilibrium problems in Banach spaces. J. Comput. Appl. Math. 188, 165–179 (2006)

    Article  MathSciNet  Google Scholar 

  24. Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)

    Article  MathSciNet  Google Scholar 

  25. Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér. Anal. Numér. 4, 154–159 (1970)

    MATH  Google Scholar 

  26. Moudafi, A.: Proximal point algorithm extended to equilibrum problem. J. Nat. Geom. 15, 91–100 (1999)

    MathSciNet  MATH  Google Scholar 

  27. Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4, Art. 18 (2003)

  28. Mordukhovich, B., Panicucci, B., Pappalardo, M., Passacantando, M.: Hybrid proximal methods for equilibrium problems. Optim. Lett. 6, 1535–1550 (2012)

    Article  MathSciNet  Google Scholar 

  29. Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)

    Article  MathSciNet  Google Scholar 

  30. Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash–Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)

    Article  MathSciNet  Google Scholar 

  31. Muu, L.D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18, 1159–1166 (1992)

    Article  Google Scholar 

  32. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  33. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  Google Scholar 

  34. Saewan, S., Kumam, P.: A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces. Comput. Math. Appl. 62, 1723–1735 (2011)

    Article  MathSciNet  Google Scholar 

  35. Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    Article  MathSciNet  Google Scholar 

  37. Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)

    Article  MathSciNet  Google Scholar 

  38. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)

    Article  MathSciNet  Google Scholar 

  39. Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65, 109–113 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the Associate Editor and the two anonymous referees for their valuable comments and suggestions which helped in improving the original version of this paper. This paper is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.315.

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Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Dang Van Hieu

  2. Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  3. The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838, Haifa, Israel

    Aviv Gibali

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  1. Dang Van Hieu
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  2. Aviv Gibali
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Correspondence to Aviv Gibali.

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Van Hieu, D., Gibali, A. Strong convergence of inertial algorithms for solving equilibrium problems. Optim Lett 14, 1817–1843 (2020). https://doi.org/10.1007/s11590-019-01479-w

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