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Viscosity Self-Adaptive Method for Generalized Split System of Variational Inclusion Problem

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Abstract

In this paper, we propose a viscosity-type and self-adaptive iterative algorithm resulting in strong convergence theorem for finding the common solution of generalized split system of variational inclusion problem. The self-adaptive technique presented in this paper is a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. As an application of our result, we study the strong convergence theorem to approximate the solution of generalized forms of multiple-set split feasibility problem, split system of equilibrium problem and split system of minimization problem. Our result generalize, improve and unify various results in the existing literature.

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References

  1. Abbas, M., AlShahrani, M., Ansari, Q.H., Iyiola, O.S., Shehu, Y.: Iterative methods for solving proximal split minimization problems. Numer. Algorithms 78, 193–215 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Ansari, Q.H., Rehan, A.: An iterative method for split hierarchical monotone variational inclusions. Fixed Point Theory Appl. 1, 121 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Aubin, J.P.: Optima and Equilibria: An Introduction to Nonlinear Analysis, Graduate Texts in Mathematics. (1993)

  4. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, vol. 408. Springer, New York (2011)

    MATH  Google Scholar 

  5. Brezis, H.: New York, vol. 5. Elsevier, New York (1973)

    Google Scholar 

  6. Byrne, C.: A unified treatment of some iterative methods in signal processing and image reconstruction. Inverse Prob. 20, 103–1020 (2004)

    MATH  Google Scholar 

  7. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Byrne, C., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J Nonlinear Convex Anal. 13, 759–775 (2011)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Chang, S.S., Kim, J., Wang, X.: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. J. 1, 869684 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Che, H., Li, M.: The conjugate gradient method for split variational inclusion and constrained convex minimization problems. Appl. Math. Comput. 290, 426–438 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Chuang, C.S.: Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem. Optimization 65, 859–876 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Chuang, C.S.: Simultaneous subgradient algorithms for the generalized split variational inclusion problem in Hilbert spaces. Numer. Func. Anal. Opt. 38, 306–326 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Chuang, C.S.: Strong convergence theorems for the split variational inclusion problem inHilbert spaces. Fixed Point Theory Appl. 2013, 350 (2013)

    MATH  Google Scholar 

  19. Chuang, C.S.: Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization 66, 777–792 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal. 6, 117–36 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Combettes, P.L.: The convex feasibility problem in image recovery. Adv Imaging Electron Phys. 95, 155–270 (1996)

    Google Scholar 

  22. Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Prob. 27, 015007 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Deepho, J., Kumam, P.: The hybrid steepest descent method for split variational inclusion and constrained convex minimization problems. Abstract and Applied Analysis. Article ID 365203, 13 (2014)

    MATH  Google Scholar 

  24. Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, T.M.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12, 87–102 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Gebrie, A.G., Wangkeeree, R.: Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 1, 5 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Gebrie, A.G., Wangkeeree, R.: Proximal method of solving split system of minimization problem. J. Appl. Math. Comput. 1, 1–26 (2019)

    MATH  Google Scholar 

  27. Gebrie, A.G., Wangkeeree, R.: Parallel proximal method of solving split system of fixed point set constraint minimization problems. RACSAM Rev. R. Acad. A. 114, 13 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Hendrickx, J.M., Olshevsky, A.: Matrix p-norms are NP-hard to approximate If \(p\ne 1,2,\infty \). SIAM J. Matrix Anal. Appl. 31, 2802–2812 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Lemaire, B.: Which fixed point does the iteration method select? In Recent Advances in optimization, pp. 154–167. Springer, Berlin, Heidelberg (1997)

    MATH  Google Scholar 

  30. López, G., Martín-Márquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Prob. 28, 085–004 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Maingé, P.E.: Inertial iterative process for fixed points of certain quasi-nonexpansive mappings Set-Valued Anal. 15, 67–79 (2007)

    MathSciNet  Google Scholar 

  32. Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Majee, P., Nahak, C.: On inertial proximal algorithm for split variational inclusion problems. Optimization 67, 1701–1716 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Moudafi, A., Thakur, S.: Solving proximal split feasibility problems without prior knowledge of operator norms. Optim. Lett. 8, 2099–2110 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Phairatchatniyom, P., Kumam, P., Cho, Y.J., Jirakitpuwapat, W., Sitthithakerngkiet, K.: The modified inertial iterative algorithm for solving split variational inclusion problem for multi-valued quasi nonexpansive mappings with some applications. Mathematics 7, 560 (2019)

    Google Scholar 

  37. Rockafellar, R.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    MathSciNet  MATH  Google Scholar 

  38. Shehu, Y., Iyiola, O.S.: Nonlinear iteration method for proximal split feasibility problems. Math. Method Appl. Sci. 41, 781–802 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Shehu, Y., Iyiola, O.S.: Strong convergence result for proximal split feasibility problem in Hilbert spaces. Optimization 66, 2275–2290 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Sitthithakerngkiet, K., Deepho, J., Martínez-Moreno, J., Kumam, P.: Convergence analysis of a general iterative algorithm for finding a common solution o split variational inclusion and optimization problems. Numer. Algorithms 79, 801–824 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Tang, J., Chang, S.S., Yuan, F.: A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces. Fixed Point Theory Appl. 1, 36 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Tang, Y.: Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. J. Ind. Manag. Optim. 1, 9–26 (2019)

    Google Scholar 

  43. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    MathSciNet  MATH  Google Scholar 

  44. Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Prob. 26, 105018 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Yu, X., Shahzad, N., Yao, Y.: Implicit and explicit algorithms for solving the split feasibility problem. Optim. Lett. 6, 1447–1462 (2012)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Computational and Natural Science, Debre Berhan University, PO. Box 445, Debre Berhan, Ethiopia

    Anteneh Getachew Gebrie

  2. Department of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, PO. Box 1176, Addis Ababa, Ethiopia

    Berhanu Bekele

Authors
  1. Anteneh Getachew Gebrie
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  2. Berhanu Bekele
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Correspondence to Anteneh Getachew Gebrie.

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Communicated by Behzad Djafari-Rouhani.

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Gebrie, A.G., Bekele, B. Viscosity Self-Adaptive Method for Generalized Split System of Variational Inclusion Problem. Bull. Iran. Math. Soc. 47, 897–917 (2021). https://doi.org/10.1007/s41980-020-00418-1

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  • DOI: https://doi.org/10.1007/s41980-020-00418-1

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