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An Iterative Method for Solving the Multiple-Sets Split Variational Inequality Problem

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Abstract

In this work, we introduce a new algorithm for finding the minimum-norm solution of the multiple-sets split variational inequality problem in real Hilbert spaces. The strong convergence of the iterative sequence generated by the algorithm method is established under the condition that the map**s are monotone and Lipschitz continuous. We apply our main result to study the minimum-norm solution of the multiple-sets split feasibility problem and the split variational inequality problem. Finally, a numerical example is given to illustrate the proposed algorithm.

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References

  1. Adamu, A., Deepho, J., Ibrahim, A.H., Abubakar, A.B.: Approximation of zeros of sum of monotone map**s with applications to variational inequality and image restoration problems. Nonlinear Funct. Anal. and Appl. 26(2), 411–432 (2021)

    MATH  Google Scholar 

  2. Afassinou, K., Narain, O.K., Otunuga, O.E.: Iterative algorithm for approximating solutions of Split Monotone Variational Inclusion, Variational inequality and fixed point problems in real Hilbert spaces. Nonlinear Funct. Anal. and Appl. 25(3), 491–510 (2020)

    MATH  Google Scholar 

  3. Anh, P.K., Anh, T.V., Muu, L.D.: On bilevel split pseudomonotone variational inequality problems with applications. Acta Math, Vietnam. 42, 413–429 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Anh, P.N., Muu, L.D., Strodiot, J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math, Vietnam. 34, 67–79 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Anh, T.V.: A strongly convergent subgradient Extragradient-Halpern method for solving a class of bilevel pseudomonotone variational inequalities. Vietnam J. Math. 45, 317–332 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Anh, T.V.: An extragradient method for finding minimum-norm solution of the split equilibrium problem. Acta Math, Vietnam. 42, 587–604 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Anh, T.V.: A parallel method for variational inequalities with the multiple-sets split feasibility problem constraints. J. Fixed Point Theory Appl. 19, 2681–2696 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Anh, T.V.: Linesearch methods for bilevel split pseudomonotone variational inequality problems. Numer. Algorithms 81, 1067–1087 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Anh, T.V., Muu, L.D.: A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints. Optimization 65, 1229–1243 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Buong, N.: Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces. Numer. Algorithms 76, 783–798 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75, 2116–2125 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Censor, Y., Ben-Israel, A., **ao, Y., Galvin, J.M.: On linear infeasibility arising in intensity-modulated radiation therapy inverse planning. Linear Algebra Appl. 428, 1406–1420 (2008)

    MathSciNet  MATH  Google Scholar 

  15. 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 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. 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)

    MathSciNet  MATH  Google Scholar 

  20. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827–845 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Censor, Y., Segal, A.: Iterative projection methods in biomedical inverse problems. In: Censor, Y., Jiang, M., Louis, A.K. (eds.) Mathematical Methods in Biomedical Imaging and Intensity-Modulated Therapy, pp. 65–96. IMRT, Edizioni della Norale, Pisa, Italy (2008)

    Google Scholar 

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

    Google Scholar 

  23. Dadashi, V.: Shrinking projection algorithms for the split common null point problem. Bull. Aust. Math. Soc. 99(2), 299–306 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Dinh, B.V., Muu, L.D.: Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities. Acta Math, Vietnam. 38, 529–540 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Dong, Q.L., Jiang, D., Gibali, A.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algorithms 79, 927–940 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)

    MATH  Google Scholar 

  27. Hai, N.M., Van, L.H.M., Anh, T.V.: An Algorithm for a Class of Bilevel Variational Inequalities with Split Variational Inequality and Fixed Point Problem Constraints. Acta Math, Vietnam. 46, 515–530 (2021)

    MathSciNet  MATH  Google Scholar 

  28. He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Huy, P.V., Hien, N.D., Anh, T.V.: A strongly convergent modified Halpern subgradient extragradient method for solving the split variational inequality problem. Vietnam J. Math. 48, 187–204 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Huy, P.V., Van, L.H.M., Hien, N.D., Anh, T.V.: Modified Tseng’s extragradient methods with self-adaptive step size for solving bilevel split variational inequality problems. Optimization (2020). https://doi.org/10.1080/02331934.2020.1834557

    Article  Google Scholar 

  31. Kim, J.K., Anh, P.N., Anh, T.T.H., Hien, N.D.: Projection methods for solving the variational inequalities involving unrelated nonexpansive map**s. J. Nonlinear Convex Anal. 21(11), 2517–2537 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Kim, J.K., Tuyen, T.M.: Parallel iterative method for solving the split common null point problem in Banach spaces. J. Nonlinear Convex Anal. 20(10), 2075–2093 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Kim, J.K., Tuyen, T.M., Ha, M.T.N.: Two projection methods for solving the split common fixed point problem with multiple output sets in Hilbert spaces. Numer. Funct. Anal. Optim. 42(8), 973–988 (2021)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  35. Kraikaew, R., Saejung, S.: Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces. J. Optim. Theory Appl. 163, 399–412 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Liu, B., Qu, B., Zheng, N.: A Successive Projection Algorithm for Solving the Multiple-Sets Split Feasibility Problem. Numer. Funct. Anal. Optim. 35, 1459–1466 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Malitsky, Y.V.: Projected reflected gradient methods for variational inequalities. SIAM J. Optim. 25, 502–520 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Takahashi, W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15, 1349–1355 (2014)

    MathSciNet  MATH  Google Scholar 

  40. Takahashi, W.: The split feasibility problem and the shrinking projection method in Banach spaces. J. Nonlinear Convex Anal. 16(7), 1449–1459 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Takahashi, W.: The split common null point problem in Banach spaces. Arch. Math. 104, 357–365 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Thong, D.V., Vinh, N.T., Cho, Y.J.: A strong convergence theorem for Tseng’s extragradient method for solving variational inequality problems. Optim Lett. 14, 1157–1175 (2020)

  43. Thuy, N.T.T., Hoai, P.T.T., Hoa, N.T.T.: Explicit iterative methods for maximal monotone operators in Hilbert spaces. Nonlinear Funct. Anal. Appl. 25(4), 753–767 (2020)

    MATH  Google Scholar 

  44. Tuyen, T.M., Ha, N.S., Thuy, N.T.T.: A shrinking projection method for solving the split common null point problem in Banach spaces. Numer. Algorithms 81, 813–832 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Wen, M., Peng, J.G., Tang, Y.C.: A cyclic and simultaneous iterative method for solving the multiple-sets split feasibility problem. J. Optim. Theory Appl. 166(3), 844–860 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Zhao, J.L., Yang, Q.Z.: A simple projection method for solving the multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. 21, 537–546 (2013)

    MathSciNet  MATH  Google Scholar 

  48. Zhao, J.L., Zhang, Y.J., Yang, Q.Z.: Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem. Appl. Math. Comput. 219, 1644–1653 (2012)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author would like to thank Van Lang University, Vietnam, for funding this work.

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Authors and Affiliations

  1. Faculty of Basic Sciences, Van Lang University, 69/68 Dang Thuy Tram Street, Ward 13, Binh Thanh District, Ho Chi Minh City, Vietnam

    Tran Luu Cuong

  2. Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

    Tran Viet Anh

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  1. Tran Luu Cuong
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  2. Tran Viet Anh
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Correspondence to Tran Viet Anh.

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Cuong, T.L., Anh, T.V. An Iterative Method for Solving the Multiple-Sets Split Variational Inequality Problem. Bull. Malays. Math. Sci. Soc. 45, 1737–1755 (2022). https://doi.org/10.1007/s40840-022-01283-3

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  • DOI: https://doi.org/10.1007/s40840-022-01283-3

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