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An improved algorithm with Armijo line-search rule for solving pseudomonotone variational inequality problems in Banach spaces

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Abstract

The aim of this paper is to introduce a new subgradient extragradient algorithm for solving variational inequality problems involving pseudomonotone and uniformly continuous operator in Banach spaces. Moreover, we prove a strong convergence theorem by constructing a new line-search rule. At the same time, several numerical experimental results are given to demonstrate the performance of our proposed algorithm.

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References

  1. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. theory and applications of nonlinear operators of accretive and monotone type, 178 of Lecture Notes in pure and applied mathematics, pp. 15–50. Dekker, New York (1996)

  2. Alber, Y.I., Li, J.L.: The connection between the metric and generalized projection operators in Banach spaces. Acta Math. Sin. 23, 1109–1120 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, 463–482 (1994)

    Article  MathSciNet  Google Scholar 

  5. Cai, G.: Viscosity iterative algorithm for variational inequality problems and fixed point problems in a real q-uniformly smooth Banach space. Fixed Point Theory Appl. 2015, 67 (2015)

    Article  MathSciNet  Google Scholar 

  6. Cai, G., Bu, S.: Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces. J. Global Optim. 55, 437–457 (2013)

    Article  MathSciNet  Google Scholar 

  7. Cai, G., Gibali, A., Iyiola, O.S., Shehu, Y.: A new double-projection method for solving variational inequalities in Banach spaces. J. Optim. Theory Appl. 178, 219–239 (2018)

    Article  MathSciNet  Google Scholar 

  8. 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)

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Ceng, L.C., Xu, H.K., Yao, J.C.: A hybrid steepest-descent method for variational inequalities in Hilbert spaces. Appl. Anal. 87, 575–589 (2008)

    Article  MathSciNet  Google Scholar 

  11. Ding, X.P.: Algorithm of solutions for mixed implicit quasi-variational inequalities with fuzzy mappings. Comput. Math. Appl. 38, 231–241 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser. 7, 91–140 (1964)

  14. Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963)

  15. Fan, J., Liu, L., Qin, X.: A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities. Optimization 69, 2199–2215 (2020)

    Article  MathSciNet  Google Scholar 

  16. Gibali, A., Thong, D.V., Tuan, P.A.: Two simple projection-type methods for solving variational inequalities. Anal. Math. Phys. 9, 2203–2225 (2019)

    Article  MathSciNet  Google Scholar 

  17. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)

    Article  MathSciNet  Google Scholar 

  18. Hieu, D.V., Cho, Y.J., Xiao, Yi-bin, Kumam, P.: Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces. Optimization 69: 2279–2304 (2020).

  19. Iusem, A.N., Nasri, M.: Korpelevich’s method for variational inequality problems in Banach spaces. J. Global Optim. 50, 59–76 (2011)

    Article  MathSciNet  Google Scholar 

  20. Iiduka, H., Takahashi, W.: Weak convergence of a projection algorithm for variational inequalities in a Banach space. J. Math. Anal. Appl. 339, 668–679 (2008)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. Khanh, P.Q., Thong, D.V., Vinh, N.T.: Versions of the subgradient extragradient method for pseudomonotone variational inequalities. Acta Appl. Math. 170, 319–345 (2020)

    Article  MathSciNet  Google Scholar 

  25. Kien, B.T., Wong, M.M., Wong, N.C., Yao, J.C.: Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis. J. Optim. Theory Appl. 140, 249–263 (2009)

    Article  MathSciNet  Google Scholar 

  26. Li, L., Song, W.: A modified extragradient method for inverse-monotone operators in Banach spaces. J. Global Optim. 44, 609–629 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Marino, G., Muglia, L., Yao, Y.: Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points. Nonlinear Anal. 75, 1787–1789 (2012)

    Article  MathSciNet  Google Scholar 

  29. Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000)

    Article  MathSciNet  Google Scholar 

  30. Noor, M.A.: Some development in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

  32. Shehu, Y.: Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications. J. Global Optim. 52, 57–77 (2012)

    Article  MathSciNet  Google Scholar 

  33. Shehu, Y.: Single projection algorithm for variational inequalities in Banach spaces with application to contact problem. Acta Math. Scientia. 40, 1045–1063 (2020)

    Article  MathSciNet  Google Scholar 

  34. Shehu, Y., Li, X.H., Dong, Q.L.: An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numer. Algor. 84, 365–388 (2020)

    Article  MathSciNet  Google Scholar 

  35. Takahashi, W.: Nonlinear Functional Analysis: Fixed Point Theory and its Applications. Yokohama publishers, Yokohama (2000)

    MATH  Google Scholar 

  36. Thong, D.V., Gibali, A.: Extragradient methods for solving non-Lipschitzian pseudo-monotone variational inequalities. J. Fixed Point Theory Appl. 21, 20 (2019)

    Article  MathSciNet  Google Scholar 

  37. Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algor. 79, 597–610 (2018)

    Article  MathSciNet  Google Scholar 

  38. Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algor. 78, 1045–1060 (2018)

    Article  MathSciNet  Google Scholar 

  39. Thong, D.V., Shehu, Y., Iyiola, O.S.: Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings. Numer. Algor. 84, 795–823 (2020)

    Article  MathSciNet  Google Scholar 

  40. Thong, D.V., Vinh, N.T., Cho, Y.J.: Accelerated subgradient extragradient methods for variational inequality problems. J. Sci. Comput. 80, 1438–1462 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Yao, Y., Liou, Y.C., Yao, J.C.: New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems. Optimization 60, 395–412 (2011)

    Article  MathSciNet  Google Scholar 

  43. Yao, Y., Noor, M.A., Noor, K.I., Liou, Y.C., Yaqoob, H.: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 110, 1211–1224 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the NSF of China (Grant No. 12171062, 11971082), the Natural Science Foundation of Chongqing (Grant No.cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504), and the Program of Chongqing Innovation Research Group Project in University (Grant No. CXQT19018).

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Zhongbing Xie & Gang Cai

  2. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Xiaoxiao Li & Qiao-Li Dong

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  1. Zhongbing Xie
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  2. Gang Cai
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  3. Xiaoxiao Li
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  4. Qiao-Li Dong
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Correspondence to Gang Cai.

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Xie, Z., Cai, G., Li, X. et al. An improved algorithm with Armijo line-search rule for solving pseudomonotone variational inequality problems in Banach spaces. Anal.Math.Phys. 12, 116 (2022). https://doi.org/10.1007/s13324-022-00726-1

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