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Inertial Viscosity Iterative Method for Solving Pseudo-monotone Variational Inequality Problems and Fixed Point Problems

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Abstract

In this paper, we investigate a new inertial viscosity extragradient algorithm for solving variational inequality problems for pseudo-monotone and Lipschitz continuous operator and fixed point problems for quasi-nonexpansive mappings in real Hilbert spaces. Strong convergence theorems are obtained under some appropriate conditions on the parameters. Finally, we give some numerical experiments to show the advantages of our proposed algorithms. The results obtained in this paper extend and improve some recent works in the literature.

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References

  1. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, Wiley, New York, 1984

    MATH  Google Scholar 

  2. Bot, R. I., Csetnek, E. R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization, 61(1), 35–65 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bot, R. I., Csetnek, E. R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators. SIAM J. Optim., 23(4), 2011–2036 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bot, R. I., Hendrich, C.: A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim., 23(4), 2541–2565 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bot, R. I., Csetnek, E. R., Heinrich, A., et al.: On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems. Math. Program., 150, 251–279 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bot, R. I., Csetnek, E. R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion. Appl. Math. Comput., 256, 472–487 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Bot, R. I., Csetnek, E. R.: An inertial Tseng’s type proximal algorithm for nonsmooth and nonconvex optimization problems. J. Optim. Theory Appl., 171, 600–616 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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  MATH  Google Scholar 

  9. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms, 56, 301–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chidume, C. E.: Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics, Vol. 1965, Springer-Verlag London, London, 2009

    Book  MATH  Google Scholar 

  13. Cholamjiak, P., Thong, D. V., Cho, Y. J.: A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta. Appl. Math., 169, 217–245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cottle, R. W., Yao, J. C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl., 75, 281–295 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dong, Q. L., Cho, Y. J., Zhong, L. L., et al.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim., 70, 687–704 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Facchinei, F., Pang, J. S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, Vol. I. Springer, New York, 2003

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984

    MATH  Google Scholar 

  20. Goebel, K., Kirk, W. A.: On Metric Fixed Point Theory, Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  21. Harker, P. T., Pang, J. S.: A damped-Newton method for the linear complementarity problem, In: Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., Vol. 26, American Mathematical Society, Providence, RI, 1990, 265–284

    Google Scholar 

  22. 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  MATH  Google Scholar 

  23. Hu, X., Wang, J.: Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network. IEEE Trans. Neural Netw., 17, 1487–1499 (2006)

    Article  Google Scholar 

  24. Iusem, A. N., Nasri, M.: Korpelevichs method for variational inequality problems in Banach spaces. J. Glob. Optim., 50, 59–76 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in refelexive Banach spaces. SIAM J. Optim., 21, 1319–1344 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Shehu, Y.: Iterative methods for split feasibility problems in certain Banach spaces. J. Nonlinear Convex Anal., 16, 2315–2364 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Shehu, Y., Iyiola, O. S., Enyi, C. D.: An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces. Numer. Algorithms, 72, 835–864 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shehu, Y., Iyiola, O. S.: Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method. J. Fixed Point Theory Appl., 19, 2483–2510 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shehu, Y., Dong, Q. L., Jiang, D.: Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization, 68, 385–409 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Solodov, M. V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim., 34, 1814–1830 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag. Optim., 13, 1–21 (2018)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Thong, D. V., Hieu, D. V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization, 67, 83–102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Thong, D. V., Hieu, D. V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math., 341, 80–98 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Thong, D. V., Hieu, D. V.: Modified Tseng’s extragradient algorithms for variational inequality problems. J. Fixed Point Theory Appl., 20, 152 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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  MATH  Google Scholar 

  46. Thong, D. V., Triet, N. A., Li, X., et al.: Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer. Algorithms, 83, 1123–1143 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  47. Thong, D. V., Hieu, D. V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algorithms, 80, 1283–1307 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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)

    Article  MathSciNet  MATH  Google Scholar 

  49. Thong, D. V., Triet, N. A., Li, X. H., et al.: Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer. Algorithms, 83, 1123–1143 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim., 38, 431–446 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematics Science, Chongqing Normal University, Chongqing, 401331, P. R. China

    Gang Cai

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

    Qiao Li Dong & Yu Peng

Authors
  1. Gang Cai
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  2. Qiao Li Dong
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  3. Yu Peng
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Corresponding author

Correspondence to Gang Cai.

Additional information

Supported by the NSF of China (Grant Nos. 11771063, 11971082 and 12171062), 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), Open Fund of Tianjin Key Lab for Advanced Signal Processing (Grant No. 2019ASP-TJ03)

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Cai, G., Dong, Q.L. & Peng, Y. Inertial Viscosity Iterative Method for Solving Pseudo-monotone Variational Inequality Problems and Fixed Point Problems. Acta. Math. Sin.-English Ser. 38, 937–952 (2022). https://doi.org/10.1007/s10114-022-0243-2

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  • DOI: https://doi.org/10.1007/s10114-022-0243-2

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