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Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems

  • Duong Viet ThongEmail author
  • Dang Van Hieu
  • Themistocles M. Rassias
Original Paper
  • 47 Downloads

Abstract

In this paper, two new algorithms are introduced for solving a pseudomontone variational inequality problem with a Lipschitz condition in a Hilbert space. The algorithms are constructed around three methods: the subgradient extragradient method, the inertial method and the viscosity method. With a new stepsize rule is incorporated, the algorithms work without any information of Lipschitz constant of operator. The weak convergence of the first algorithm is established, while the second one is strongly convergent which comes from the viscosity method. In order to show the computational effectiveness of our algorithms, some numerical results are provided.

Keywords

Subgradient extragradient method Inertial method Variational inequality Pseudomonotone mapping Lipschitz continuity 

Notes

Acknowledgements

The authors would like to thank anonymous reviewrs for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper. This work is supported by Vietnam (National Foundation for Science and Technology Development (NAFOSTED)) under the project: 101.01-2019.320.

References

  1. 1.
    Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Matematicheskie Metody 12, 1164–1173 (1976)Google Scholar
  3. 3.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  4. 4.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities; Applications to Free Boundary Problems. Wiley, New York (1984)zbMATHGoogle Scholar
  5. 5.
    Cai, G., Gibali, A., Iyiola, O.S., Shehu, Y.: A new double-projection method for solving variational inequalities in Banach space. J. Optim. Theory Appl. 178, 219–239 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cui, S., Shanbhag, U.V.: On the optimality of single projection variants of extragradient schemes for monotone stochastic variational inequality problems. arXiv:1904.11076v1
  11. 11.
    Cui, S., Shanbhag, U.V.: On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems. In: 55th IEEE Conference on Decision and Control, CDC 2016, Las Vegas, NV, USA, 12–14 December 2016, IEEE, pp. 4510–4515 (2016)Google Scholar
  12. 12.
    Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dong, Q.L., Gibali, A., Jiang, D., Tang, Y.: Bounded perturbation resilience of extragradient-type methods and their applications. J. Inequal. Appl. 2017, 280 (2017).  https://doi.org/10.1186/s13660-017-1555-0 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dong, Q.L., Lu, Y.Y., Yang, J.F.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65, 2217–2226 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, T.M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Volume I. Springer Series in Operations Research. Springer, New York (2003)zbMATHGoogle Scholar
  19. 19.
    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)MathSciNetzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  23. 23.
    Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  29. 29.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)zbMATHCrossRefGoogle Scholar
  31. 31.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Moudafi, A.: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Shehu, Y., Dong, Q.L., Jiang, D.: Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68, 385–409 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for inequality variational problems. Numer. Algorithms 79, 597–610 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    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. (2019).  https://doi.org/10.1007/s11590-019-01391-3 CrossRefGoogle Scholar
  42. 42.
    Thong, D.V., Vinh, N.T., Cho, Y.J.: Accelerated subgradient extragradient methods for variational inequality problems. J. Sci. Comput. 80, 1438–1462 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Thong, D.V., Vinh, N.T., Cho, Y.J.: New strong convergence theorem of the inertial projection and contraction method for variational inequality problems. Numer. Algorithms (2019).  https://doi.org/10.1007/s11075-019-00755-1 CrossRefGoogle Scholar
  44. 44.
    Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J. Optim. Theory Appl. 176, 399–409 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algorithms 81, 269–291 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of generalized evolution variational inequalities in Banach space. Appl. Math. Lett. 25, 914–920 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Wang, Y.M., Xiao, Y.B., Wang, X., Cho, Y.J.: Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems. J. Nonlinear Sci. Appl. 9, 1178–1192 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559–569 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer. Algorithms 80, 741–752 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Yang, J., Liu, H., Liu, Z.: Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization 76, 2247–2258 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Yang, J., Liu, H.: A modified projected gradient method for monotone variational inequalities. J. Optim. Theory Appl. 179, 197–211 (2018)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Duong Viet Thong
    • 1
    Email author
  • Dang Van Hieu
    • 2
  • Themistocles M. Rassias
    • 3
  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of MathematicsCollege of Air ForceNha TrangVietnam
  3. 3.Department of MathematicsNational Technical University of AthensAthensGreece

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