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A Tseng extragradient method for solving variational inequality problems in Banach spaces

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Abstract

This paper presents an inertial Tseng extragradient method for approximating a solution of the variational inequality problem. The proposed method uses a single projection onto a half space which can be easily evaluated. The method considered in this paper does not require the knowledge of the Lipschitz constant as it uses variable stepsizes from step to step which are updated over each iteration by a simple calculation. We prove a strong convergence theorem of the sequence generated by this method to a solution of the variational inequality problem in the framework of a 2-uniformly convex Banach space which is also uniformly smooth. Furthermore, we report some numerical experiments to illustrate the performance of this method. Our result extends and unifies corresponding results in this direction in the literature.

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References

  1. Alakoya, T.O., Jolaoso, L.O., Mewomo, O.T.: Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems. Optimization, 1–30 (2020)

  2. Alber, Y.I.: Metric and generalized projection operators in banach spaces: properties and applications Kartsatos, A.G. (ed.) . Dekker, New York (1996)

  3. Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekon. Mat. Metody. 12, 1164–1173 (1976)

    Google Scholar 

  4. Aoyama, K., Kohsaka, F.: Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings. Fixed Point Theory Appl. 95, 13 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Apostol, R.Y., Grymenko, A.A., Semenov, V.V.: Iterative algorithms for monotone bilevel variational inequalities. J. Comp. Appl. Math. 107, 3–14 (2012)

    Google Scholar 

  6. Aubin, J.P., Ekeland, I.: Applied nonlinear analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  7. Avetisyan, K., Djordjevi’ĉ, O., Pavlovi’ĉ, M.: Littlewood-palew inequalities in uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 336(1), 31–43 (2007)

    Article  MathSciNet  Google Scholar 

  8. Censor, Y., Lent, A.: An iterative row-action method for interval complex programming. J. Optim. Theory Appl. 34, 321–353 (1981)

    Article  MathSciNet  Google Scholar 

  9. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2012)

    Article  MathSciNet  Google Scholar 

  10. Censor, Y., Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323–339 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Chidume, C.E., Nnakwe, M.O.: Convergence theorems of subgradient extragradient algorithm for solving variational inequalities and a convex feasibility problem. Fixed Point Theory and Appl. 2018, 16 (2018). https://doi.org/10.1186/s13663-018-0641-4

    Article  MathSciNet  Google Scholar 

  14. Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems. Kluwer, Dordrecht (1990)

    Book  Google Scholar 

  15. Goldstein, A.A.: Convex programming in Hilbert space. Bull. Amer. Math. Soc. 70(5), 709–910 (1964)

    Article  MathSciNet  Google Scholar 

  16. Gibali, A., Reich, S., Zalas, R.: Iterative methods for solving variational inequalities in Euclidean space. J. Fixed Point Theory Appl. 17, 775–811 (2015)

    Article  MathSciNet  Google Scholar 

  17. Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017)

    Article  MathSciNet  Google Scholar 

  18. Gibali, A.: Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces. J. Nonlinear Variat. Anal. 2, 49–61 (2017)

    MATH  Google Scholar 

  19. Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York and Basel (1984)

    MATH  Google Scholar 

  20. Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2002)

    Article  MathSciNet  Google Scholar 

  21. Kanzow, C., Shehu, Y.: Strong convergence of a double-type method for monotone variational inequalities in Hilbert spaces. J. Fixed Point Theory Appl. 20(1). Art. 51, p. 24 (2018)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  Google Scholar 

  25. Mashregi, J., Nasri, M.: Forcing strong convergence of Korpelevich’s method in Banach spaces with applications in game theory. Nonlinear Anal. 72, 2086–2099 (2010)

    Article  MathSciNet  Google Scholar 

  26. Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory. Appl. 128(1), 191–201 (2006)

    Article  MathSciNet  Google Scholar 

  27. Ogwo, G.N., Izuchukwu, C., Aremu, K.O., Mewomo, O.T.: A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space. Bull. Belg. Math. Soc. Simon Stevin 27, 1–26 (2020)

    Article  MathSciNet  Google Scholar 

  28. Lin, L.J., Yang, M.F., Ansari, Q.H., Kassay, G.: Existence results for Stampacchia and Minty type implicit variational inequalities with multivalued maps. Nonlinear Anal. Theory Methods Appl. 61, 1–19 (2005)

    Article  MathSciNet  Google Scholar 

  29. Cholamjiak, P., Thong, D.V., Cho, Y.J.: A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl. Math., 1–29 (2019)

  30. Reem, D., Reich, S., De Pierro, A.: Re-examination of Bregman functions and new properties of their divergences. Optimization 68, 279–348 (2019)

    Article  MathSciNet  Google Scholar 

  31. Reich, S.: A weak convergence theorem for alternating method with Bregman distance. In: Kartsatos, A.G. (ed.) Theory and applications of nonlinear operators of accretive and monotone type, pp 313–318. Marcel Dekker, New York (1996)

  32. Rockfellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–808 (1977)

    Article  MathSciNet  Google Scholar 

  33. Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operator in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)

    Article  MathSciNet  Google Scholar 

  34. Shehu, Y., Iyiola, O.S.: On a modified extragradient method for variational inequality problem with application to industrial electricity production. J. Ind. Manag. Optim. 15(1), 319–342 (2019)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Thong, D.V., Vihn, 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)

    Article  MathSciNet  Google Scholar 

  37. Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

    Article  MathSciNet  Google Scholar 

  38. Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algor. 80, 741–752 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author acknowledges with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF CoE-MaSS) Doctoral Bursary. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF and CoE-MaSS.

The authors would like to thank the editor and anonymous reviewers for their invaluable suggestions which have greatly improved the manuscript.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    O. K. Oyewole, H. A. Abass & A. A. Mebawondu

  2. DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    O. K. Oyewole, H. A. Abass & A. A. Mebawondu

  3. Department of Mathematics, Usmanu Danfodiyo University Sokoto, P.M.B. 2346, Sokoto state, Nigeria

    K. O. Aremu

  4. Department of Mathematics and Applied of Mathematics, Sefako Magkatho Health Sciences University, Ga-Rankuwa, P.O. Box 60, 0204, South Africa

    K. O. Aremu

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  1. O. K. Oyewole
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  2. H. A. Abass
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  3. A. A. Mebawondu
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  4. K. O. Aremu
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Correspondence to O. K. Oyewole.

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Oyewole, O.K., Abass, H.A., Mebawondu, A.A. et al. A Tseng extragradient method for solving variational inequality problems in Banach spaces. Numer Algor 89, 769–789 (2022). https://doi.org/10.1007/s11075-021-01133-6

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  • DOI: https://doi.org/10.1007/s11075-021-01133-6

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