A Novel Inertial Projection and Contraction Method for Solving Pseudomonotone Variational Inequality Problems

Abstract

In this paper, we introduce a new algorithm which combines the inertial contraction projection method and the Mann-type method (Mann in Proc. Am. Math. Soc. 4:506–510, 1953) for solving monotone variational inequality problems in real Hilbert spaces. The strong convergence of our proposed algorithm is proved under some standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm and the main result.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Alber, Ya., Iusem, A.N.: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Anal. 9, 315–335 (2001)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14, 773–782 (2004)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    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)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differ. Equ. 179, 278–310 (2002)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2, 1–34 (2000)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)

    Google Scholar 

  7. 7.

    Bot, R.I., Csetnek, E.R.: A hybrid proximal-extragradient algorithm with inertial effects. Numer. Funct. Anal. Optim. 36, 951–963 (2015)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    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)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bot, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71, 519–540 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Bot, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1, 29–49 (2016)

    MathSciNet  MATH  Google Scholar 

  11. 11.

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

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bot, R.I., Csetnek, E.R., Laszlo, S.C.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO J. Comput. Optim. 4, 3–25 (2016)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Cai, X., Gu, G., He, B.: On the \(O(\frac{1}{t})\) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339–363 (2014)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012)

    Google Scholar 

  15. 15.

    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)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    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)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    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 

  18. 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. 19.

    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)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Chen, C., Ma, S., Yang, J.: A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J. Optim. 25, 2120–2142 (2015)

    MathSciNet  MATH  Google Scholar 

  21. 21.

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

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Dong, Q.L., Cho, Y.J., Rassias, T.M.: The projection and contraction methods for finding common solutions to variational inequality problems. Optim. Lett. 12, 1871–1896 (2018)

    MathSciNet  MATH  Google Scholar 

  23. 23.

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

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Dong, Q.L., Gibali, A., Jiang, D., Ke, S.H.: Convergence of projection and contraction algorithms with outer perturbations and their applications to sparse signals recovery. J. Fixed Point Theory Appl. 20, 16 (2018). https://doi.org/10.1007/s11784-018-0501-1

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Dong, Q.L., Jiang, D., Gibali, A.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algorithms 9, 927–940 (2018)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    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)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    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)

    MathSciNet  MATH  Google Scholar 

  28. 28.

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

    MathSciNet  MATH  Google Scholar 

  29. 29.

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

    Google Scholar 

  30. 30.

    Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. In: Allgower, G., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations. Lectures in Applied Mathematics, vol. 26, pp. 265–284. AMS, Providence (1990)

    Google Scholar 

  31. 31.

    He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)

    MathSciNet  MATH  Google Scholar 

  32. 32.

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

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Kopecká, E., Reich, S.: A note on alternating projections in Hilbert space. J. Fixed Point Theory Appl. 12, 41–47 (2012)

    MathSciNet  MATH  Google Scholar 

  34. 34.

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

    MathSciNet  MATH  Google Scholar 

  35. 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. 36.

    Maingé, P.E.: Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set-Valued Anal. 15, 67–79 (2007)

    MathSciNet  MATH  Google Scholar 

  37. 37.

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

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 34, 876–887 (2008)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Maingé, P.E.: Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure. Comput. Math. Appl. 3, 720–728 (2016)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Maingé, P.E., Gobinddass, M.L.: Convergence of one step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Maingé, P.E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)

    MathSciNet  MATH  Google Scholar 

  42. 42.

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

    MathSciNet  MATH  Google Scholar 

  43. 43.

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

    MathSciNet  MATH  Google Scholar 

  44. 44.

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

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Moudafi, A., Elisabeth, E.: An approximate inertial proximal method using enlargement of a maximal monotone operator. Int. J. Pure Appl. Math. 5, 283–299 (2003)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)

    Google Scholar 

  49. 49.

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

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Shehu, Y., Cholamjiak, P.: Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo 56, 4 (2019)

    MathSciNet  MATH  Google Scholar 

  51. 51.

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

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91, 123–140 (1996)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    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

    MATH  Article  Google Scholar 

  54. 54.

    Thong, D.V., Vinh, N.T., Cho, Y.J.: Accelerated subgradient extragradient methods for variational inequality problems. J. Sci. Comput. (2019). https://doi.org/10.1007/s10915-019-00984-5

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    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

    MATH  Article  Google Scholar 

  56. 56.

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

    MathSciNet  MATH  Google Scholar 

  57. 57.

    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)

    MathSciNet  MATH  Google Scholar 

  58. 58.

    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)

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Xiao, Y.B., Huang, N.J., Wong, M.M.: Well-posedness of hemivariational inequalities and inclusion problems. Taiwan. J. Math. 15, 1261–1276 (2011)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of generalized evolution variational inequalities in Banach spaces. Appl. Math. Lett. 25, 914–920 (2012)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Xiu, N.H., Zhang, J.Z.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–587 (2003)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank three anonymous reviewers 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. P. Cholamjiak was supported by Thailand Research Fund and University of Phayao under the project RSA6180084 and UOE62001. This work was partially supported by Thailand Science Research and Innovation under the project IRN62W0007.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Duong Viet Thong.

Additional information

Dedicated to Professor Pham Ky Anh on the occasion of his 70th birthday

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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 (2020). https://doi.org/10.1007/s10440-019-00297-7

Download citation

Keywords

  • Inertial contraction projection method
  • Mann-type method
  • Pseudomonotone mapping
  • Pseudomonotone variational inequality problem

Mathematics Subject Classification (2010)

  • 65Y05
  • 65K15
  • 68W10
  • 47H05
  • 47H10