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, 56:4 | Cite as

Iterative method with inertial for variational inequalities in Hilbert spaces

  • Yekini ShehuEmail author
  • Prasit Cholamjiak
Article
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Abstract

Strong convergence property for Halpern-type iterative method with inertial terms for solving variational inequalities in real Hilbert spaces is investigated under mild assumptions in this paper. Our proposed method requires only one projection onto the feasible set per iteration, the underline operator is monotone and uniformly continuous which is more applicable than most existing methods for which strong convergence is achieved and our method includes the inertial extrapolation step which is believed to increase the rate of convergence. Numerical comparisons of our proposed method with some other related methods in the literature are given.

Keywords

Variational inequalities Monotone operator Inertial terms Strong convergence Hilbert spaces 

Mathematics Subject Classification

47H05 47J20 47J25 65K15 90C25 

Notes

References

  1. 1.
    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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2(1), 1–34 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differ. Equ. 179(1), 278–310 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward–backward algorithm for convex minimization. SIAM J. Optim. 24, 232–256 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Apostol, R.Y., Grynenko, A.A., Semenov, V.V.: Iterative algorithms for monotone bilevel variational inequalities. J. Comp. Appl. Math. 107, 3–14 (2012)Google Scholar
  7. 7.
    Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  8. 8.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)zbMATHGoogle Scholar
  9. 9.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)CrossRefGoogle Scholar
  10. 10.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion. Appl. Math. Comput. 256, 472–487 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bot, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1, 29–49 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bot, R.I., Csetnek, E.R.: An inertial forward–backward–forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algoritm. 71, 519–540 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012)zbMATHGoogle 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. Global Optim. 46, 635–646 (2010)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Chen, C., Chan, R.H., Ma, S., Yang, J.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8, 2239–2267 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Denisov, S., Semenov, V., Chabak, L.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar
  21. 21.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  22. 22.
    Halpern, B.: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 73, 957–961 (1967)CrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185, 166–173 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    He, B.-S., Yang, Z.-H., Yuan, X.-M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Iusem, A.N., Gárciga Otero, R.: Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer. Funct. Anal. Optim. 22, 609–640 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Iusem, A.N., Nasri, M.: Korpelevich’s method for variational inequality problems in Banach spaces. J. Global Optim. 50, 59–76 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    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)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Khobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)CrossRefGoogle Scholar
  32. 32.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  33. 33.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)CrossRefGoogle Scholar
  34. 34.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ékon. Mat. Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Lorenz, D.A., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Maingé, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344, 876–887 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Maingé, P.-E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Global Optim. 61, 193–202 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Marcotte, P.: Applications of Khobotov’s algorithm to variational and network equlibrium problems. Inf. Syst. Oper. Res. 29, 258–270 (1991)zbMATHGoogle Scholar
  43. 43.
    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)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ochs, P., Brox, T., Pock, T.: iPiasco: inertial proximal algorithm for strongly convex optimization. J. Math. Imaging Vis. 53, 171–181 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Polyak, B.T.: Some methods of speeding up the convergence of iterarive methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)Google Scholar
  46. 46.
    Shehu, Y., Iyiola, O.S.: Strong convergence result for monotone variational inequalities. Numer. Algoritm. 76, 259–282 (2017)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Shehu, Y., Iyiola, O.S.: Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators. Numer. Algoritm. 79, 529–553 (2018)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
  50. 50.
    Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Xu, H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66(2), 1–17 (2002)MathSciNetGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of NigeriaNsukkaNigeria
  2. 2.Institute of MathematicsUniversity of WürzburgWürzburgGermany
  3. 3.School of ScienceUniversity of PhayaoPhayaoThailand

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