Skip to main content

Advertisement

SpringerLink
Log in

Two modified inertial projection algorithms for bilevel pseudomonotone variational inequalities with applications to optimal control problems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we investigate two new algorithms for solving bilevel pseudomonotone variational inequality problems in real Hilbert spaces. The advantages of our algorithms are that they only need to calculate one projection on the feasible set in each iteration, and do not require the prior information of the Lipschitz constant of the cost operator. Furthermore, two new algorithms are derived to solve variational inequality problems. We establish the strong convergence of the proposed algorithms under some suitable conditions imposed on parameters. Finally, several numerical results and applications in optimal control problems are reported to illustrate the efficiency and advantages of the proposed algorithms.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J. Global Optim. 52, 627–639 (2012)

    Article  MathSciNet  Google Scholar 

  2. Hieu, D.V., Moudafi, A.: Regularization projection method for solving bilevel variational inequality problem. Optim. Lett. 15, 205–229,. https://doi.org/10.1007/s11590-020-01580-5 (2021)

  3. Thong, D.V., Triet, N.A., Li, X.H., Dong, Q.-L.: Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer. Algorithms 83, 1123–1143 (2020)

    Article  MathSciNet  Google Scholar 

  4. Tan, B., Liu, L., Qin, X.: Self adaptive inertial extragradient algorithms for solving bilevel pseudomonotone variational inequality problems. Jpn. J. Ind. Appl. Math. https://doi.org/10.1007/s13160-020-00450-y (2021)

  5. Cho, S.Y.: A monotone Bregan projection algorithm for fixed point and equilibrium problems in a reflexive Banach space. Filomat 34, 1487–1497 (2020)

    Article  MathSciNet  Google Scholar 

  6. Cho, S.Y.: Implicit extragradient-like method for fixed point problems and variational inclusion problems in a Banach space. Symmetry 12, 998 (2020)

    Article  Google Scholar 

  7. Shehu, Y., Iyiola, O.S., Li, X.H., Dong, Q.-L.: Convergence analysis of projection method for variational inequalities. Comput. Appl. Math. 38, 161 (2019)

    Article  MathSciNet  Google Scholar 

  8. Tan, B., Xu, S., Li, S.: Inertial shrinking projection algorithms for solving hierarchical variational inequality problems. J. Nonlinear Convex Anal. 21, 871–884 (2020)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Dong, Q.-L., Jiang, D., Gibali, A.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algorithms 79, 927–940 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently Parallel Algoritm. Feasibility Optim. Appl. 8, 473–504 (2001)

    Article  MathSciNet  Google Scholar 

  15. Liu, L.: A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings. J. Nonlinear Convex Anal. 20, 471–488 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Slavakis, K., Yamada, I.: Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks. Optimization 67, 1963–2001 (2018)

    Article  MathSciNet  Google Scholar 

  17. Thong, D.V., Hieu, D.V.: A strong convergence of modified subgradient extragradient method for solving bilevel pseudomonotone variational inequality problems. Optimization 69, 1313–1334 (2020)

    Article  MathSciNet  Google Scholar 

  18. Gibali, A., Hieu, D.V.: A new inertial double-projection method for solving variational inequalities. J. Fixed Point Theory Appl. 21, 97 (2019)

    Article  MathSciNet  Google Scholar 

  19. Shehu, Y., Iyiola, O.S.: Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Appl. Numer. Math. 157, 315–337 (2020)

    Article  MathSciNet  Google Scholar 

  20. Shehu, Y., Gibali, A.: New inertial relaxed method for solving split feasibilities. Optim. Lett. https://doi.org/10.1007/s11590-020-01603-1 (2020)

  21. Tan, B., Li, S.: Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems. J. Nonlinear Var. Anal. 4, 337–355 (2020)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. J. Math. 43, 229–238 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Thong, D.V., Hieu, D.V., Rassias, T.M.: Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems. Optim. Lett. 14, 115–144 (2020)

    Article  MathSciNet  Google Scholar 

  27. Thong, D.V., Gibali, A.: Two strong convergence subgradient extragradient methods for solving variational inequalities in Hilbert spaces. Jpn. J. Ind. Appl. Math. 36, 299–321 (2019)

    Article  MathSciNet  Google Scholar 

  28. Gibali, A., Thong, D.V., Tuan, P.A.: Two simple projection-type methods for solving variational inequalities. Anal. Math. Phys. 9, 2203–2225 (2019)

    Article  MathSciNet  Google Scholar 

  29. 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 84, 285–305 (2019)

    Article  MathSciNet  Google Scholar 

  30. Gibali, A., Shehu, Y.: An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces. Optimization 68, 13–32 (2019)

    Article  MathSciNet  Google Scholar 

  31. Beck, A., Guttmann-Beck, N.: FOM—a MATLAB toolbox of first-order methods for solving convex optimization problems. Optim. Methods Softw. 34, 172–193 (2019)

    Article  MathSciNet  Google Scholar 

  32. Hieu, D.V., Cho, Y.J., Xiao, Y.-b., Kumam, P.: Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces. Optimization 69, 2279–2304 (2020)

    Article  MathSciNet  Google Scholar 

  33. Boţ, R.I., Csetnek, E.R., Vuong, P.T.: The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces. Eur. J. Oper. Res. 287, 49–60 (2020)

    Article  MathSciNet  Google Scholar 

  34. Khoroshilova, E.V.: Extragradient-type method for optimal control problem with linear constraints and convex objective function. Optim. Lett. 7, 1193–1214 (2013)

    Article  MathSciNet  Google Scholar 

  35. Preininger, J., Vuong, P.T.: On the convergence of the gradient projection method for convex optimal control problems with bang-bang solutions. Comput. Optim. Appl. 70, 221–238 (2018)

    Article  MathSciNet  Google Scholar 

  36. 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)

    Article  MathSciNet  Google Scholar 

  37. Hieu, D.V., Strodiot, J.J., Muu, L.D.: Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems. J. Comput. Appl. Math. 376, 112844 (2020)

    Article  MathSciNet  Google Scholar 

  38. Bonnans, J.F., Festa, A.: Error estimates for the Euler discretization of an optimal control problem with first-order state constraints. SIAM J. Numer. Anal. 55, 445–471 (2017)

    Article  MathSciNet  Google Scholar 

  39. Pietrus, A., Scarinci, T., Veliov, V.M.: High order discrete approximations to Mayer’s problems for linear systems. SIAM J. Control Optim. 56, 102–119 (2018)

    Article  MathSciNet  Google Scholar 

  40. Bressan, B., Piccoli, B.: Introduction to the Mathematical Theory of Control. Applied Mathematics, vol. 2. American Institute of Mathematical Sciences (AIMS), Springfield (2007)

Download references

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable and constructive comments, which greatly improved the readability and quality of the initial version of the paper.

Author information

Authors and Affiliations

  1. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Bing Tan

  2. Department of Mathematics, Hangzhou Normal University, Zhejiang, China

    Xiaolong Qin

  3. Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, 40447, Taiwan

    Jen-Chih Yao

  4. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Jen-Chih Yao

Authors
  1. Bing Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaolong Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaolong Qin.

Additional information

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

Tan, B., Qin, X. & Yao, JC. Two modified inertial projection algorithms for bilevel pseudomonotone variational inequalities with applications to optimal control problems. Numer Algor 88, 1757–1786 (2021). https://doi.org/10.1007/s11075-021-01093-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01093-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation