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Parallel modified methods for pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings

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Abstract

The paper considers the problem of finding common solutions of a system of pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. The problem covers various mathematical models of convex feasibility problems and the problems whose constraints are expressed by the intersection of fixed point sets of mappings. The main purpose of the paper is to design and improve computations over each step and weaken several assumptions imposed on bifunctions and mappings. Two parallel algorithms for finding of a particular solution of the problem are proposed in Hilbert spaces where each subproblem in the family can be computed simultaneously. The first one is a modified hybrid method which combines three methods including the generalized gradient-like projection method, the Mann’s iteration and the hybrid (outer approximation) method. This algorithm improves the hybrid extragradient method at each computational step where only one optimization problem is solved for each equilibrium subproblem in the family and the hybrid step does not deal with the feasible set of the considered problem. The strong convergence of the algorithm comes from the hybrid method under the Lipschitz-type condition of bifunctions. The second algorithm is a viscosity-like method with a linesearch procedure that aims to avoid the Lipschitz-type condition imposed on bifunctions. With the incorporated viscosity technique, the algorithm also provides strong convergence. Several numerical experiments are performed to illustrate the efficiency of the proposed algorithms and also to compare them with known parallel hybrid extragradient methods.

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Notes

  1. We randomly chose \(\lambda _{1k}^i\in [-m,0],~\lambda _{2k}^i\in [1,m],~ k=1,\ldots ,m,~i=1\ldots ,N\). Set \(\widehat{Q}_1^i\), \(\widehat{Q}_2^i\) as two diagonal matrixes with eigenvalues \(\left\{ \lambda _{1k}^i\right\} _{k=1}^m\) and \(\left\{ \lambda _{2k}^i\right\} _{k=1}^m\), respectively. Then, we make a positive definite matrix \(Q_i\) and a negative semidefinite matrix \(T_i\) by using random orthogonal matrixes with \(\widehat{Q}_2^i\) and \(\widehat{Q}_1^i\), respectively. Finally, set \(P_i=Q_i-T_i\).

References

  1. Antipin, A.S.: On convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence. Comp. Maths. Math. Phys. 35, 539–551 (1995)

    MATH  Google Scholar 

  2. Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62, 271–283 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44, 351–374 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Chen, J.: A projection method for approximating fixed points of quasi nonexpansive mappings without the usual demiclosedness condition. J. Nonlinear Convex Anal. 15, 129–135 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fej\(\rm \acute{e}\)r-monotone methods in Hilbert space. Math. Oper. Res. 26, 248–264 (2001)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  10. Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 95, pp. 155–270. Academic Press, New York (1996)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. Hieu, D.V., Gibali, A.: Strong convergence of inertial algorithms for solving equilibrium problems. Optim. Lett. (2019). https://doi.org/10.1007/s11590-019-01479-w

    Article  MATH  Google Scholar 

  15. Hieu, D.V., Quy, P.K., Vy, L.V.: Explicit iterative algorithms for solving equilibrium problems. Calcolo 56, 11 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67, 2003–2029 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Meth. Oper. Res. 88, 399–415 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algor. 80, 1413–1436 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Hieu, D.V., Strodiot, J.J.: Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces. J. Fixed Point Theory Appl. 20, 1–32 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Hieu, D.V., Strodiot, J.J., Muu, L.D.: Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems. J. Comput. Appl. Math. 376, 112844 (2020). https://doi.org/10.1016/j.cam.2020.112844

    Article  MathSciNet  MATH  Google Scholar 

  21. Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)

    MathSciNet  Google Scholar 

  22. 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)

    MathSciNet  MATH  Google Scholar 

  23. Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58, 251–261 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)

    MATH  Google Scholar 

  25. Konnov, I.V.: Combined relaxation methods for solving vector equilibrium problems. Russ. Math. (Iz. VUZ) 39(12), 51–59 (1995)

    MATH  Google Scholar 

  26. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  27. Kumam, W., Piri, H., Kumam, P.: Solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings. Appl. Math. Comput. 248, 441–455 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Malitsky, YuV, Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Maingé, P.E., Moudafi, A.: Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems. J. Nonlinear Convex Anal. 9, 283–294 (2008)

    MathSciNet  MATH  Google Scholar 

  31. Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., et al. (eds.) Equilibrium Problems and Variational Models, pp. 289–298. Kluwer Academic Publishers, Dordrecht (2003)

    MATH  Google Scholar 

  32. Muu, L.D.: Stability property of a class of variational inequalities. Optimization. 15, 347–351 (1984)

    MathSciNet  MATH  Google Scholar 

  33. Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space. J. Optim. Theory Appl. 160, 809–831 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Phuangphoo, P., Kumam, P.: Approximation theorems for solving common solution of a system of mixed equilibrium problems and variational inequality problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense. Appl. Math. Comput. 219, 837–855 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    MathSciNet  MATH  Google Scholar 

  36. Shehu, Y.: Hybrid iterative scheme for fixed point problem, infinite systems of equilibrium and variational inequality problems. Comput. Math. Appl. 63, 1089–1103 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Solodov, M.V., Svaiter, B.F.: A new projection method for variational i nequality problem. SIAM J. Control Optim. 37, 765–776 (1999)

    MathSciNet  MATH  Google Scholar 

  38. Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. 56, 373–397 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Tiel, J.: Convex Analysis. An Introductory Text. Wiley, Chichester (1984)

    MATH  Google Scholar 

  40. 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)

    MathSciNet  MATH  Google Scholar 

  41. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64, 429–451 (2015)

    MathSciNet  MATH  Google Scholar 

  42. 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, Elsevier, Amsterdam, 473–504 (2001)

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Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project: 101.01-2020.06.

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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 Basic Sciences, College of Air Force, Nha Trang, Vietnam

    Bui Huu Thai

  3. Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok, 10140, Thailand

    Poom Kumam

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  1. Dang Van Hieu
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  2. Bui Huu Thai
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Correspondence to Dang Van Hieu.

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Communicated by Elias Katsoulis.

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Van Hieu, D., Thai, B.H. & Kumam, P. Parallel modified methods for pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. Adv. Oper. Theory 5, 1684–1717 (2020). https://doi.org/10.1007/s43036-020-00081-7

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