Skip to main content
SpringerLink
Log in

Iterative methods for zeros of a monotone variational inclusion in Hilbert spaces

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

In this paper, we introduce implicit and explicit iterative methods for finding a zero of a monotone variational inclusion in Hilbert spaces. As consequence, an improvement modification of an algorithm existing in literature is obtained. A numerical example is given for illustrating our algorithm.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barbu, V.: Nonlinear Semigroup and Differential Equations in Banach Spaces. Academiei Bucuresti, Romania (1976)

    Book  MATH  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  3. Boikanyo, O.A., Morosanu, G.: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635–641 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boikanyo. O.A., Morosanu, G.: A generalization of the regularization proximal point method. Nonlinear Anal. Appl. 2012, Article ID jnaa-00129 (2012). https://doi.org/10.5899/2012/jnaa-00129

  5. Boikanyo, O.A.: The viscosity approximation forward–backward method for zero of the sum of monotone operator. Abstr. Appl. Anal. 2016, Article ID 2371857 (2016)

  6. Brézis, H., Lions, P.L.: Produits infinis de resolvants. Isr. J. Math. 29, 329–345 (1978)

    Article  MATH  Google Scholar 

  7. Bruck, R.E.: A strong convergent iterative solution of \(0 \in U(x)\) for a maximal monotone operator \(U\) in Hilbert space. J. Math. Anal. Appl. 48, 114–126 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buong, N., Duong, L.T.T.: An explicit iterative algorithm for a class of variational inequalities. J. Optim. Theory Appl. 151, 513–528 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buong, N., Phuong, N.T.H.: Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods. J. Optim. Theory Appl 159, 399–411 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buong, N., Ha, N.S., Thuy, N.T.T.: A new explicit iteration method for a class of variational inequalities. Numer. Algorithms 72, 467–481 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buong, N., Quynh, V., Thuy, N.T.T.: A steepest-descent Krasnosel’skii–Mann algorithm for a class of variational inequalities in Banach spaces. J. Fixed Point Theory Appl. 18, 519–532 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buong, N.: Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces. Numer. Algorithms 76(3), 783–798 (2017). https://doi.org/10.1007/s11075-017-0282-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Mann-type steepest-descent and modified hybrid steepest descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29(9–10), 987–1033 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duvaut, D., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  16. Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Plenum, New York (1995)

    Book  MATH  Google Scholar 

  17. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  MATH  Google Scholar 

  18. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271–310 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hlavacek, I., Haslinger, J., Necas, J., Lovicek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1982)

    Google Scholar 

  21. Jiao, H., Wang, F.: On an iterative method for finding a zero to the sum of two maximal monotone operators. J. Appl. Math. 2014, Article ID 414031 (2014)

  22. Kakimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory. 106, 226–240 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khongtham, Y.K.: New iterative method for variational inclusion and fixed point problems. In: Proceeding of the World Congress on Engineering 2014, vol. II WCE 2014, July 2–4, 2014, Lodon UK (2014)

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

    MATH  Google Scholar 

  25. Kluge, R.: Optimal control with minimum problems and variational inequalities. Lect. Notes Comput. Sci. 27, 377–382 (1975)

    Article  MATH  Google Scholar 

  26. Kluge, R.: Approximation methods for nonlinear problems with constrains in forms of variational inequalities. Banach Center Publ. 1, 131–138 (1976)

    Article  Google Scholar 

  27. Lehdili, N., Moudafi, A.: Combining the proximal point algorithm and Tikhonov regularization. Optimization 37, 239–252 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493–519 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liou, Y.C.: Iterative methods for the sum of two monotone operators. J. Appl. Math. 2012, Article ID 638632 (2012)

  30. Martinet, B.: Regularisation d’inéquations variationelles par approximations succivees. Rev. Fr. Inform. Rech. Oper. 4, 154–159 (1970)

    MATH  Google Scholar 

  31. Nevanlinna, O., Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Isr. J. Math. 32, 44–58 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhauser, Boston (1985)

    Book  MATH  Google Scholar 

  33. Passty, G.P.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J. Math. Anal. Appl. 72, 383–390 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Reich, S.: Weak convergence theorem for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  35. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sibony, M.: Sur l’approximation d’equation et inéquation aux derivees partielles nonlineairies de type monotone. J. Math. Anal. Appl. 34, 502–564 (1972)

    Article  Google Scholar 

  38. Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in Hilbert spaces. Math. Program. 87, 189–202 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Takahashi, W., Ueda, Y.: On Reich’s strong convergence theorems for resolvents of accretive operators. J. Math. Anal. Appl. 104, 546–553 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  40. Takahashi, W., Wong, N.C., Yao, J.C.: Two generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications. Taiwan. J. Math. 16(3), 1151–1172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Takahashi, S., Takahashi, W., Toyota, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Takahashi, W., Wong, N.C., Yao, J.C.: Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems. Fixed Point Theory Appl. 2012, 181 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)

    MATH  Google Scholar 

  44. Tian, ChA, Song, Y.: Strong convergence of a regularization method for Rockafellar’s proximal point algorithm. J. Glob. Optim. 55, 831–837 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, S.: On fixed point and variational inclusion problem. Filomat 29(6), 1409–1411 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wu, H.C., Cheng, C.Z.: Iterative methods for equilibrium problems and monotone inclusion problems in Hilbert spaces. J. Appl. Math. 2013, Article ID 280909 (2013)

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, H.K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115–125 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Xu, H.K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. Elsevier, Amsterdam (2001)

    Chapter  Google Scholar 

  51. Yao, Y., Noor, M.A.: On convergence criteria of generalized proximal point algorithm. J. Comput. Appl. Math. 217, 46–55 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zeidler, E.: Nonlinear Functional Analysis and Applications. III: Variational Methods and Optimization. Springer, New York (1985)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors are extremely grateful to the referees for their useful comments, which helped to improve this paper. This work was supported by the Vietnam National Foundation for Science and Technology Development under Grant No. 101.02-2017.305.

Author information

Authors and Affiliations

  1. Duy Tan University, 3 Quang Trung, Da Nang, Viet Nam

    Nguyen Buong

  2. Vietnam Maritime University, Haiphong, Vietnam

    Pham Thi Thu Hoai

Authors
  1. Nguyen Buong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Thi Thu Hoai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Buong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buong, N., Hoai, P.T.T. Iterative methods for zeros of a monotone variational inclusion in Hilbert spaces. Calcolo 55, 7 (2018). https://doi.org/10.1007/s10092-018-0250-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-018-0250-y

Keywords

Mathematics Subject Classification

Search

Navigation