Skip to main content
Springer Nature Link
Log in

Auxiliary Principle Technique for Hierarchical Equilibrium Problems

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

Abstract

In this paper, building upon auxiliary principle technique and using proximal operator, we introduce a new explicit algorithm for solving monotone hierarchical equilibrium problems. The considered problem is a monotone equilibrium problem, where the constraint is the solution set of a set-valued variational inequality problem. The strong convergence of the proposed algorithm is studied under strongly monotone and Lipschitz-type assumptions of the bifunction. By combining with parallel techniques, the convergence result is also established for the equilibrium problem involving a finite system of demicontractive mappings. Several fundamental experiments are provided to illustrate the numerical behavior of the proposed algorithm and comparison with other known algorithms is studied.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

References

  1. Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  4. Brézis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll. Unione Mat. Ital. 6, 293–300 (1972)

    MathSciNet  MATH  Google Scholar 

  5. Al-Homidan, S., AlShahrani, M., Ansari, Q.H.: Weak sharp solutions for equilibrium problems in metric spaces. J. Nonlinear Convex Anal. 16(7), 1185–1193 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Amini-Harandi, A., Ansari, Q.H., Farajzadeh, A.P.: Existence of equilibria in complete metric spaces. Taiwan. J. Math. 16, 777–785 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ansari, Q.H.: Metric Spaces Including Fixed Point Theory and Set-Valued Maps. Narosa Publishing House, New Delhi (2010)

    MATH  Google Scholar 

  8. Ansari, Q.H.: Ekeland’s variational principle and its extensions with applications. In: Almezel, S., Ansari, Q.H., Khamsi, M.A. (eds.) Fixed Point Theory and Applications, pp. 65–99. Springer, Cham (2014)

    Google Scholar 

  9. Ansari, Q.H., Balooee, J.: Auxiliary principle technique for solving regularized nonconvex mixed equilibrium problems. Fixed Point Theory 20(2), 431–450 (2019)

    Article  MATH  Google Scholar 

  10. Ansari, Q.H., Balooee, J., Dogan, K.: Iterative schemes for solving regularized nonconvex mixed equilibrium problems. J. Nonlinear Convex Anal. 18(4), 607–622 (2017)

    MathSciNet  Google Scholar 

  11. Ansari, Q.H., Balooee, J., Petrusel, A.: Some remarks on regularized multivalued nonconvex equilibrium problems. Miskolc Math. Notes 18(2), 573–593 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ansari, Q.H., Köbis, E., Yao, J.-C.: Vector Variational Inequalities and Vector Optimization—Theory and Applications. Springer, Berlin (2018)

    Book  MATH  Google Scholar 

  13. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bianchi, M., Schaible, S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Glob. Optim. 30, 121–134 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Castellani, M., Pappalardo, M., Passacantando, M.: Existence results for nonconvex equilibrium problems. Optim. Methods. Softw. 325(1), 49–58 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225–240 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds.) Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht (2003)

    Google Scholar 

  18. Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cohen, G.: Auxiliary principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Anh, P.N., Le Thi, H.A.: An Armijo-type method for pseudomonotone equilibrium problems and its applications. J. Glob. Optim. 57(3), 803–820 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Anh, P.N., Hien, N.D., Tuan, P.M.: Computational errors of the extragradient method for equilibrium problems. Bull. Malays. Math. Soc. 42, 2835–2858 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64(2), 429–451 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Combettes, P.L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process. 51, 1771–1782 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Iiduka, H.: Fixed point optimization algorithmand its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Marcotte, P.: Network design problem with congestion effects: a case of bilevel programming. Math. Program. 34(2), 142–162 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria. Springer Nature Switzerland, Cham (2019)

    Book  MATH  Google Scholar 

  29. Chbani, Z., Riahi, H.: Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities. Optimization 64, 1285–1303 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Bento, G.C., Cruz Neto, J.X., Lopes, J.O., Soares Jr., P.A., Soubeyran, A.: Generalized proximal distances for bilevel equilibrium problems. SIAM J. Optim. 26, 810–830 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Maingé, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 3, 529–538 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Xu, H.K.: Viscosity method for hierarchical fixed point approach to variational inequalities. Taiwan. J. Math. 14(2), 463–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Xu, M.H., Li, M., Yang, C.C.: Neural networks for a class of bilevel variational inequalities. J. Glob. Optim. 44, 535–552 (2009)

    Article  MATH  Google Scholar 

  35. Anh, P.N., Muu, L.D.: Lagrangian duality algorithms for finding a global optimal solution to mathematical programs with affine equilibrium constraints. Nonlinear Dyn. Syst. Theory 6(3), 225–244 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Yao, Y., Marino, G., Muglia, L.: A modified Korpelevichś method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559–569 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62, 271–283 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Solodov, M.: An explicit descent method for bilevel convex optimization. J. Convex Anal. 14, 227–237 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Slavakis, K., Yamada, I.: Robust wideband beamforming by the hybrid steepest descentmethod. IEEE Trans. Signal Process. 55, 4511–4522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, Hoboken (1984)

    MATH  Google Scholar 

  42. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)

    MATH  Google Scholar 

  43. Anh, P.N., Le Thi, H.A.: New subgradient extragradient methods for solving monotone bilevel equilibrium problems. Optimization 68(11), 2097–2122 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their helpful and constructive comments that helped us very much in improving the paper. In this research, the first author was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2019.303.

Author information

Authors and Affiliations

  1. Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

    Pham Ngoc Anh

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari

  3. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Qamrul Hasan Ansari

Authors
  1. Pham Ngoc Anh
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Qamrul Hasan Ansari
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Qamrul Hasan Ansari.

Additional information

Communicated by Jen-Chih Yao.

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

Anh, P.N., Ansari, Q.H. Auxiliary Principle Technique for Hierarchical Equilibrium Problems. J Optim Theory Appl 188, 882–912 (2021). https://doi.org/10.1007/s10957-021-01814-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01814-1

Keywords

Mathematics Subject Classification