Advertisement

Iterative Methods for Variational Inequalities

  • Muhammad Aslam Noor
  • Khalida Inayat Noor
  • Themistocles M. Rassias
Chapter
First Online:
Part of the Springer Optimization and Its Applications book series (SOIA, volume 151)

Abstract

Variational inequalities can be viewed as novel and significant extension of variational principles. A wide class of unrelated problems, which arise in various branches of pure and applied sciences are being investigated in the unified framework of variational inequalities. It is well known that variational inequalities are equivalent to the fixed point problems. This equivalent fixed point formulation has played not only a crucial part in studying the qualitative behavior of complicated problems, but also provide us numerical techniques for finding the approximate solution of these problems. Our main focus is to suggest some new iterative methods for solving variational inequalities and related optimization problems using projection methods, Wiener–Hopf equations and dynamical systems. Convergence analysis of these methods is investigated under suitable conditions. Some open problems are also discussed and highlighted for future research.

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

Notes

Acknowledgements

Authors would like to express their sincere gratitude to Prof. Dr. Abdellah Bnohachem for providing the computational results.

References

  1. 1.
    B. Bin-Mohsin, M.A. Noor, K.I. Noor, R. Latif, Resolvent dynamical systems and mixed variational inequalities. J. Nonl. Sci. Appl. 10, 2925–2933 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Bnouhachem, An additional projection step to He and Liao’s method for solving variational inequalities. J. Comput. Appl. Math. 206, 238–250 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    H. Brezis, Operateurs Maximaux Monotone et Semigroupes de Contractions dans les Espace d’Hilbert (North-Holland, Amsterdam, 1972)Google Scholar
  5. 5.
    P. Daniele, F. Giannessi, A. Maugeri, Equilibrium Problems and Variational Models (Kluwer Academic, London, 2003)zbMATHCrossRefGoogle Scholar
  6. 6.
    J. Dong, D.H. Zhang, A. Nagurney, A projected dynamical systems model of general equilibrium with stability analysis. Math. Comput. Model. 24, 35–44 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    P. Dupuis, A. Nagurney, Dynamical systems and variational inequalities. Anal. Oper. Res. 44, 19–42 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    T.L. Friesz, D.H. Berstein, N.J. Mehta, R.L. Tobin, S. Ganjliazadeh, Day-to day dynamic network disequilibrium and idealized traveler information systems. Oper. Res. 42, 1120–1136 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    T.L. Friesz, D.H. Berstein, R. Stough, Dynamical systems, variational inequalities and control theoretical models for predicting time-varying urban network flows. Trans. Sci. 30, 14–31 (1996)zbMATHCrossRefGoogle Scholar
  10. 10.
    R. Glowinski, J.L. Lions, R. Tremolieres, Numerical Analysis of Variational Inequalities (North Holland, Amsterdam, 1981)zbMATHGoogle Scholar
  11. 11.
    P.T. Harker, J.S. Pang, A Damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetzbMATHGoogle Scholar
  12. 12.
    B.S. He, L.Z. Liao, Improvement of some projection methods for monotone variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    B.S. He, Z.H. Yang, X.M. Yuan, An approximate proximal-extradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300(2), 362–374 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    S. Karamardian, Generalized complementarity problems. J. Opt. Theory Appl. 8, 161–168 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    N. Kikuchi, J.T. Oden, Contact Problems in Elasticity (SAIM Publishing Co., Philadelphia, 1988)zbMATHCrossRefGoogle Scholar
  16. 16.
    G.M. Korpelevich, The extragradient method for finding saddle points and other problems. Ekonomika Mat. Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  17. 17.
    J.L. Lions, G. Stampacchia, Variational inequalities. Commun. Pure Appl. Math. 20, 491–512 (1967)zbMATHCrossRefGoogle Scholar
  18. 18.
    P. Marcotte, J.P. Dussault, A Note on a globally convergent Newton method for solving variational inequalities. Oper. Res. Lett. 6, 35–42 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A. Nagurney, D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications (Kluwer Academic Publishers, Boston, 1996)zbMATHCrossRefGoogle Scholar
  20. 20.
    M.A. Noor, On variational inequalities, PhD thesis, Brunel University, London, 1975Google Scholar
  21. 21.
    M.A. Noor, An iterative schemes for a class of quasi variational inequalities. J. Math. Anal. Appl. 110, 463–468 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    M.A. Noor, General variational inequalities. Appl. Math. Lett. 1(1), 119–121 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M.A. Noor, Wiener–Hopf equations and variational inequalities. J. Optim. Theory Appl. 79, 197–206 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    M.A. Noor, Sensitivity analysis for quasi variational inequalities. J. Optim. Theory Appl. 95, 399–407 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    M.A. Noor, Wiener-Hopf equations technique for variational inequalities. Korean J. Comput. Appl. Math. 7, 581–598 (2000)MathSciNetzbMATHGoogle Scholar
  26. 26.
    M.A. Noor, New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–230 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M.A. Noor, Stability of the modified projected dynamical systems. Comput. Math. Appl. 44, 1–5 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    M.A. Noor, Resolvent dynamical systems for mixed variational inequalities. Korean J. Comput. Appl. Math. 9, 15–26 (2002)MathSciNetzbMATHGoogle Scholar
  29. 29.
    M.A. Noor, Proximal methods for mixed variational inequalities. J. Opt. Theory Appl. 115(2), 447–452 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    M.A. Noor, A Weiner-Hopf dynamical system for variational inequalities. New Zealand J. Math. 31, 173–182 (2002)MathSciNetzbMATHGoogle Scholar
  31. 31.
    M.A. Noor, Fundamentals of mixed quasi variational inequalities. Int. J. Pure Appl. Math. 15(2), 137–250 (2004)MathSciNetzbMATHGoogle Scholar
  32. 32.
    M.A. Noor, Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)MathSciNetzbMATHGoogle Scholar
  33. 33.
    M.A. Noor, Extended general variational inequalities. Appl. Math. Lett. 22(2), 182–185 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    M.A. Noor, On an implicit method for nonconvex variational inequalities. J. Optim. Theory Appl. 147, 411–417 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    M.A. Noor, K.I. Noor, Th.M. Rassias, Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    M.A. Noor, K.I. Noor, H. Yaqoob, On general mixed variational inequalities. Acta. Appl. Math. 110, 227–246 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    M.A. Noor, K.I. Noor, E. Al-Said, On new proximal point methods for solving the variational inequalities. J. Appl. Math. 2012, 7 (2012). Article ID: 412413Google Scholar
  38. 38.
    M.A. Noor, K.I. Noor, A. Bnouhachem, On a unified implicit method for variational inequalities. J. Comput. Appl. Math. 249, 69–73 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    M. Patriksson, Nonlinear Programming and Variational Inequalities: A Unified Approach (Kluwer Academic Publishers, Drodrecht, 1998)zbMATHGoogle Scholar
  40. 40.
    Th.M. Rassias, M.A. Noor, K.I. Noor, Auxiliary principle technique for the general Lax-Milgram lemma. J. Nonl. Funct. Anal. 2018 (2018). Article ID:34Google Scholar
  41. 41.
    S.M. Robinson, Normal maps induced by linear transformations. Math. Oper. Res. 17, 691–714 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    P. Shi, Equivalence of variational inequalities with Wiener-Hopf equations. Proc. Am. Math. Soc. 111, 339–346 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)MathSciNetzbMATHGoogle Scholar
  44. 44.
    K. Taji, M. Fukushima, T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities. Math. Program. 58, 369–383 (1993)zbMATHCrossRefGoogle Scholar
  45. 45.
    Y.S. Xia, On convergence conditions of an extended projection neural network. Neural Comput. 17, 515–525 (2005)zbMATHCrossRefGoogle Scholar
  46. 46.
    Y.S. Xia, J. Wang, On the stability of globally projected dynamical systems. J. Optim. Theory Appl. 106, 129–150 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Y.S. Xia, J. Wang, A recurrent neural network for solving linear projection equation. Neural Comput. 13, 337–350 (2000)Google Scholar
  48. 48.
    H. Yang, M.G.H. Bell, Traffic restraint, road pricing and network equilibrium. Transp. Res. B 31, 303–314 (1997)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Muhammad Aslam Noor
    • 1
  • Khalida Inayat Noor
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.COMSATS University IslamabadIslamabadPakistan
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

Personalised recommendations

Cite chapter

Buy options