Skip to main content
SpringerLink
Log in

Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature.

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. Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno (English translation: “On Signorini’s elastostatic problem with ambiguous boundary conditions”). Atti Accad. Naz. Lincei VIII. Ser. Rend. Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963)

    MATH  Google Scholar 

  2. Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno (English translation: “Elastostatic problems with unilateral constraints: the Signorini’s problem with ambiguous boundary conditions”). Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat., Sez. I VIII. Ser. 7, 91–140 (1964)

  3. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. Acad. Sci Paris 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Antipin, A.S., Jaćimović, M., Mijajlovi, N.: Extragradient method for solving quasivariational inequalities. Optimization 67, 103–112 (2018)

    Article  MathSciNet  Google Scholar 

  6. Mosco, U.: Implicit variational problems and quasi variational inequalities. Lecture Notes in Mathathematics, vol. 543. Springer, Berlin (1976)

    MATH  Google Scholar 

  7. Noor, M.A.: An iterative scheme for a class of quasi variational inequalities. J. Math. Anal. Appl. 110, 463–468 (1985)

    Article  MathSciNet  Google Scholar 

  8. Noor, M.A.: Quasi variational inequalities. Appl. Math. Lett. 1, 367–370 (1988)

    Article  MathSciNet  Google Scholar 

  9. Noor, M.A., Noor, K.I., Khan, A.G.: Some iterative schemes for solving extended general quasi variational inequalities. Appl. Math. Inf. Sci. 7, 917–925 (2013)

    Article  MathSciNet  Google Scholar 

  10. Mijajlović, N., Jaćimović, M., Noor, M.A.: Gradient-type projection methods for quasi-variational inequalities. Optim. Lett. 13, 1885–1896 (2019)

    Article  MathSciNet  Google Scholar 

  11. Aussel, D., Sagratella, S.: Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality. Math. Methods Oper. Res. 85, 3–18 (2017)

    Article  MathSciNet  Google Scholar 

  12. Facchinei, F., Kanzow, C., Karl, S., Sagratella, S.: The semismooth Newton method for the solution of quasi-variational inequalities. Comput. Optim. Appl. 62, 85–109 (2015)

    Article  MathSciNet  Google Scholar 

  13. Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Program. 144, 369–412 (2014)

    Article  MathSciNet  Google Scholar 

  14. Latorre, V., Sagratella, S.: A canonical duality approach for the solution of affine quasi-variational inequalities. J. Global Optim. 64, 433–449 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. 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 (2003)

    Article  MathSciNet  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)

    Article  MathSciNet  Google Scholar 

  20. Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344, 876–887 (2008)

    Article  MathSciNet  Google Scholar 

  21. Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 791–803 (1964)

    MathSciNet  Google Scholar 

  22. 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)

    Article  MathSciNet  Google Scholar 

  23. Lorenz, D.A., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)

    Article  MathSciNet  Google Scholar 

  24. Ochs, P., Brox, T., Pock, T.: iPiasco: inertial proximal algorithm for strongly convex optimization. J. Math. Imaging Vis. 53, 171–181 (2015)

    Article  MathSciNet  Google Scholar 

  25. Boţ, 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 

  26. Shehu, Y.: Convergence rate analysis of inertial Krasnoselskii–Mann-type iteration with applications. Numer. Funct. Anal. Optim. 39, 1077–1091 (2018)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. Boţ, 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)

    Article  MathSciNet  Google Scholar 

  30. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Non-Expansive Mappings. Marcel Dekker Inc, New York (1984)

    MATH  Google Scholar 

  31. Noor, M.A., Oettli, W.: On general nonlinear complementarity problems and quasi equilibria. Matematiche (Catania) 49, 313–331 (1994)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  33. Ryazantseva, I.P.: First-order methods for certain quasi-variational inequalities in a Hilbert space. Comput. Math. Math. Phys. 47, 183–190 (2007)

    Article  MathSciNet  Google Scholar 

  34. Antipin, A.S., Jaćimović, M., Mijajlovi, N.: A second-order iterative method for solving quasi-variational inequalities. Comput. Math. Math. Phys. 53, 258–264 (2013)

    Article  MathSciNet  Google Scholar 

  35. Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. Discrete Contin. Dyn. Syst. 31, 1383–1396 (2011)

    Article  MathSciNet  Google Scholar 

  36. Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov’s accelerated forward-backward method is actually faster than \(1/k^2\). SIAM J. Optim. 26, 1824–1834 (2016)

    Article  MathSciNet  Google Scholar 

  37. Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: a library of quasi-variational inequality test problems. Pac. J. Optim. 9, 225–250 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous referees and editor whose insightful comments helped to considerably improve an earlier version of this paper. The research of the first author is supported by an ERC Grant from the Institute of Science and Technology (IST).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Yekini Shehu

  2. Institute of Science and Technology (IST), Am Campus 1, 3400, Klosterneuburg, Vienna, Austria

    Yekini Shehu

  3. Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  4. The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838, Haifa, Israel

    Aviv Gibali

  5. Department of Computer, Control, and Management, Engineering, Antonio Ruberti, Sapienza University of Rome, Rome, Italy

    Simone Sagratella

Authors
  1. Yekini Shehu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aviv Gibali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Simone Sagratella
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aviv Gibali.

Additional information

Radu Ioan Boţ.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shehu, Y., Gibali, A. & Sagratella, S. Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces. J Optim Theory Appl 184, 877–894 (2020). https://doi.org/10.1007/s10957-019-01616-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01616-6

Keywords

Mathematics Subject Classification

Search

Navigation