A new descent alternating direction method with LQP regularization for the structured variational inequalities

Original Paper
First Online:
Received:
Accepted:
  • 19 Downloads

Abstract

In this paper, we suggest and analyze a new logarithmic–quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The main contribution of this paper, the predictor is obtained via solving LQP system approximately under significantly relaxed accuracy criterion and the new iterate is obtained by using a new direction with a new step size \(\alpha _k\). Global convergence of the proposed method is proved under certain assumptions. We also reported some numerical results to illustrate the efficiency of the proposed method.

Keywords

Variational inequalities Monotone operator Logarithmic–quadratic proximal method Projection method Alternating direction method 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The first author is grateful to Prof. Hafida Bouloiz for her constant help.

References

  1. 1.
    Bnouhachem, A., Benazza, H., Khalfaoui, M.: An inexact alternating direction method for solving a class of structured variational inequalities. Appl. Math. Comput. 219, 7837–7846 (2013)MathSciNetMATHGoogle Scholar
  2. 2.
    Bnouhachem, A.: On LQP alternating direction method for solving variational. J. Inequal. Appl. 2014(80), 1–15 (2014)MATHGoogle Scholar
  3. 3.
    Bnouhachem, A., Xu, M.H.: An inexact LQP alternating direction method for solving a class of structured variational inequalities. Comput. Math. Appl. 67, 671–680 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bnouhachem, A., Ansari, Q.H.: A descent LQP alternating direction method for solving variational inequality problems with separable structure. Appl. Math. Comput. 246, 519–532 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Bnouhachem, A., Hamdi, A.: Parallel LQP alternating direction method for solving variational inequality problems with separable structure. J. Inequal. Appl. 2014(392), 1–14 (2014)MathSciNetMATHGoogle Scholar
  6. 6.
    Bnouhachem, A., Hamdi, A.: A hybrid LQP alternating direction method for solving variational inequality problems with separable structure. Appl. Math. Inf. Sci. 9(3), 1259–1264 (2015)MathSciNetGoogle Scholar
  7. 7.
    Bnouhachem, A., Al-Homidan, S., Ansari, Q.H.: New descent LQP alternating direction methods for solving a class of structured variational inequalities. Fixed Point Theory Appl. 2015(137), 1–11 (2015)MathSciNetMATHGoogle Scholar
  8. 8.
    Bnouhachem, A., Latif, A., Ansari, Q.H.: On the \(O(1/t)\) convergence rate of the alternating direction method with LQP regularization for solving structured variational inequality problems. J. Inequal. Appl. 2016(297), 1–14 (2016)MathSciNetMATHGoogle Scholar
  9. 9.
    Bnouhachem, A., Bensi, F., Hamdi, A.: On alternating direction method for solving variational inequality problems with separable structure. J. Nonlinear Sci. Appl. 10(1), 175–185 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bnouhachem, A., Ansari, Q.H., Yao, J.C.: SQP alternating direction for structured vriational inequality. J. Nonlinear Convex Anal. 19(3), 461–476 (2018)Google Scholar
  11. 11.
    Bnouhachem A., Rassias T.M.: SQP alternating direction method with a new optimal step size for solving variational inequality problems with separable structure. Appl. Anal. Discrete Math. (in Press) Google Scholar
  12. 12.
    Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64, 81–101 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Eckstein, J.: Some saddle-function splitting methods for convex programming. Optim. Methods Softw. 4, 75–83 (1994)CrossRefGoogle Scholar
  14. 14.
    Gabay, D.B., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput. Math. Appl. 2, 17–40 (1976)CrossRefMATHGoogle Scholar
  15. 15.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrange Methods: Applications to the Solution of Boundary-valued Problems. Studies in Mathematics and Its Applications, vol. 15, pp. 299–331. North Holland, Amsterdam, The Netherlands (1983)CrossRefGoogle Scholar
  16. 16.
    He, B.S., Liao, L.Z., Han, D.R., Yang, H.: A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92, 103–118 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    He, B.S.: Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities. Comput. Optim. Appl. 42, 195–212 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    He, B.S., Tao, M., Yuan, X.M.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22, 313–340 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hou, L.S.: On the \(O(1/t)\) convergence rate of the parallel descent-like method and parallel splitting augmented Lagrangian method for solving a class of variational inequalities. Appl. Math. Comput. 219, 5862–5869 (2013)MathSciNetMATHGoogle Scholar
  20. 20.
    Jiang, Z.K., Bnouhachem, A.: A projection-based prediction–correction method for structured monotone variational inequalities. Appl. Math. Comput. 202, 747–759 (2008)MathSciNetMATHGoogle Scholar
  21. 21.
    Jiang, Z.K., Yuan, X.M.: New parallel descent-like method for sloving a class of variational inequalities. J. Optim. Theory Appl. 145, 311–323 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998)MathSciNetMATHGoogle Scholar
  23. 23.
    Li, M.: A hybrid LQP-based method for structured variational inequalities. Int. J. Comput. Math. 89(10), 1412–1425 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Tao, M., Yuan, X.M.: On the \(O(1/t)\) convergence rate of alternating direction method with logarithmic–quadratic proximal regularization. SIAM J. Optim. 22(4), 1431–1448 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim. 7, 951–965 (1997)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wang, K., Xu, L.L., Han, D.R.: A new parallel splitting descent method for structured variational inequalities. J. Ind. Man. Optim. 10(2), 461–476 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Yuan, X.M., Li, M.: An LQP-based decomposition method for solving a class of variational inequalities. SIAM J. Optim. 21(4), 1309–1318 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zhang, W., Han, D., Jiang, S.: A modified alternating projection based prediction–correction method for structured variational inequalities. Appl. Numer. Math. 83, 12–21 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Abdellah Bnouhachem
    • 1
    • 2
  • Themistocles M. Rassias
    • 3
  1. 1.School of Management and EngineeringNanjing UniversityNanjingPeople’s Republic of China
  2. 2.ENSAIbn Zohr UniversityAgadirMorocco
  3. 3.Department of MathematicsNational Technical University of AthensAthensGreece

Personalised recommendations