Recent Developments in General Quasi Variational Inequalities

Main Article Content

A. A. AlShejari, M. A. Noor, K. I. Noor


In this paper, we present a number of new and known numerical techniques for solving general quasi variational inequalities, introduced by Noor [34] in 1988, using various techniques including projection, Wiener-Hopf equations, auxiliary principle, dynamical systems coupled with finite difference approach and sensitivity analysis. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis is also investigated. Some special cases are discussed as applications of the main results. Several open problems are suggested for future research.

Article Details


  1. F. Alvarez, Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space, SIAM J. Optim. 14 (2004), 773–782.
  2. W.F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, New York, 1992.
  3. Ashish, M. Rani, R. Chugh, Julia Sets and Mandelbrot Sets in Noor Orbit, Appl. Math. Comput. 228 (2014), 615–631.
  4. Ashish, R. Chugh, M. Rani, Fractals and Chaos in Noor Orbit: A Four-Step Feedback Approach, Lap Lambert Academic Publishing, Saarbrucken, Germany, (2021).
  5. A. Barbagallo, S. Guarino Lo Bianco, A Random Elastic Traffic Equilibrium Problem via Stochastic Quasi-Variational Inequalities, Commun. Nonlinear Sci. Numer. Simul. 131 (2024), 107798.
  6. A. Bensoussan, J.L. Lions, Application des Inequalities Variationnelles en Control Stochastique, Bordas (Dunod), Paris, 1978.
  7. S.Y. Cho, A.A. Shahid, W. Nazeer, S.M. Kang, Fixed Point Results for Fractal Generation in Noor Orbit and s-Convexity, SpringerPlus. 5 (2016), 1843.
  8. R.W. Cottle, J.S. Pang, R.E. Stone, The Linear Complementarity Problem, SIAM, 2009.
  9. G. Cristescu, L. Lupsa, Non-Connected Convexities and Applications, Kluwer Academic Publishers, Dordrecht, Holland, 2002.
  10. G. Cristescu, M. Gianu, Detecting the Non-Convex Sets With Youness and Noor Types Convexities, Bul. Stiint. Univ. Politeh. Timis. Ser. Mat.-Fiz. 55 (2010), 20–27.
  11. G. Cristescu, G. Mihail, Shape Properties of Noor’s g-Convex Sets, in: Proceeding of the 12th Symposium of Mathematics and Its Applications, Timisoara, Romania, (2009), 1–13.
  12. S. Dafermos, Sensitivity Analysis in Variational Inequalities, Math. Oper. Res. 13 (1988), 421–434.
  13. S. Dey, S. Reich, A Dynamical System for Solving Inverse Quasi-Variational Inequalities, Optimization. (2023).
  14. P. Dupuis, A. Nagurney, Dynamical Systems and Variational Inequalities, Ann. Oper. Res. 44 (1993), 7–42.
  15. R. Glowinski, J.L. Lions, R. Tremolieres, Numerical Analysis of Variational Inequalities, North Holland, Amsterdam, 1981.
  16. R. Glowinski, P. Le Tallec, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989.
  17. S. Jabeen, B. B. Mohsen, M. Aslam Noor, K. Inayat Noor, Inertial Projection Methods for Solving General QuasiVariational Inequalities, AIMS Math. 6 (2021), 1075–1086.
  18. S. Jabeen, M.A. Noor, K.I. Noor, Some Inertial Methods for Solving of System of Quasi Variational Inequalities, J. Adv. Math. Stud. 15 (2022), 01–10.
  19. M.A. Noor, K.I. Noor, A.G. Khan, Dynamical Systems for Quasi Variational Inequalities, Ann. Funct. Anal. 6 (2015), 193–209.
  20. D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, SIAM, Philadelphia, 2000.
  21. G.M. Korpelevich, The Extragradient Method for Finding Saddle Points and Other Problems, Ekon. Mat. Metody. 12 (1976), 747–756.
  22. Y.C. Kwun, A.A. Shahid, W. Nazeer, S.I. Butt, M. Abbas, S.M. Kang, Tricorns and Multicorns in Noor Orbit With s-Convexity, IEEE Access. 7 (2019), 95297–95304.
  23. J.L. Lions, G. Stampacchia, Variational Inequalities, Commun. Pure Appl. Math. 20 (1967), 493–519.
  24. C.E. Lemke, Bimatrix Equilibrium Points and Mathematical Programming, Manage. Sci. 11 (1965), 681–689.
  25. N. Mijajlovic, M. Jacimovic, M.A. Noor, Gradient-Type Projection Methods for Quasi-Variational Inequalities, Optim. Lett. 13 (2018), 1885–1896.
  26. A. Nagurney, D. Zhang, Projected Dynamical Systems and Variational Inequalities With Applications, Kluwer Academic Publishers, Boston, 1996.
  27. C.P. Niculescu, L.E. Persson, Convex Functions and Their Applications, Springer, New York, 2018.
  28. M.A. Noor, Riesz-Frachet Theorem and Monotonicity, MS Thesis, Queen’s University, Kingston, Ontario, 1971.
  29. M.A. Noor, On Variational Inequalities, PhD Thesis, Brunel University, London, 1975.
  30. M.A. Noor, An Iterative Scheme for a Class of Quasi Variational Inequalities, J. Math. Anal. Appl. 110 (1985), 463–468.
  31. M.A. Noor, Generalized Quasi Complementarity Problems, J. Math. Anal. Appl. 120 (1986), 321–327.
  32. M.A. Noor, The Quasi-Complementarity Problem, J. Math. Anal. Appl. 130 (1988), 344–353.
  33. M.A. Noor, General Variational Inequalities, Appl. Math. Lett. 1 (1988), 119–122.
  34. M.A. Noor, Quasi Variational Inequalities, Appl. Math. Lett. 1 (1988), 367–370.
  35. M.A. Noor, Fixed Point Approach for Complementarity Problems, J. Math. Anal. Appl. 133 (1988), 437–448.
  36. M.A. Noor, General Algorithm for Variational Inequalities, J. Optim. Theory Appl. 73 (1992), 409–413.
  37. M.A. Noor, Sensitivity Analysis for Quasi Variational Inequalities, J. Optim. Theory Appl. 95 (1997), 399–407.
  38. M.A. Noor, New Approximation Schemes for General Variational Inequalities, J. Math. Anal. Appl. 251 (2000), 217–229.
  39. M.A. Noor, Variational Inequalities for Fuzzy Mappings (III), Fuzzy Sets Syst. 110 (2000), 101–108.
  40. M.A. Noor, A Wiener-Hopf Dynamical System and Variational Inequalities, N. Z. J. Math. 31 (2002), 173–182.
  41. M.A. Noor, Some Developments in General Variational Inequalities, Appl. Math. Comput. 152 (2004), 199–277.
  42. M.A. Noor, Stability of the Modified Projected Dynamical Systems, Comput. Math. Appl. 44 (2002), 1–5.
  43. M.A. Noor, Merit Functions for Quasi Variational Inequalities, J. Math. Inequal. 1 (2007), 259–269.
  44. M.A. Noor, Extended General Variational Inequalities, Appl. Math. Lett. 22 (2009), 182–186.
  45. M.A. Noor, Principles of Variational Inequalities, Lap-Lambert Academic Publishing, Saarbrucken, Germany, 2009.
  46. M.A. Noor, On General Quasi Variational Inequalities, J. King Saud Univ. Sci. 24 (2012), 81–88.
  47. M.A. Noor, E.A. Al-Said, Change of Variable Method for Generalized Complementarity Problems, J. Optim. Theory Appl. 100 (1999), 389–395.
  48. M.A. Noor, K.I. Noor, Sensitivity Analysis for Quasi-Variational Inclusions, J. Math. Anal. Appl. 236 (1999), 290–299.
  49. M.A. Noor, K.I. Noor, Some Novel Aspects of Quasi Variational Inequalities, Earthline J. Math. Sci. 10 (2022), 1–66.
  50. M.A. Noor, K.I. Noor, Dynamical System Technique for Solving Quasi Variational Inequalities, U.P.B. Sci. Bull., Ser. A. 84 (2022), 55–66.
  51. M.A. Noor, K.I. Noor, A.G. Khan, Some Tterative Schemes for Solving Extended General Quasi Variational Inequalities, Appl. Math. Inf. Sci. 7 (2013), 917–925.
  52. M.A. Noor, K.I. Noor, B.B. Mohsen, Some New Classes of General Quasi Variational Inequalities, AIMS Math. 6 (2021), 6406–6421.
  53. M.A. Noor, K.I. Noor, A. Bnouhachem, Some New Iterative Methods for Variational Inequalities, Canad. J. Appl. Math. 3 (2021), 1–17.
  54. M.A. Noor, K.I. Noor, M.Th. Rassias, New Trends in General Variational Inequalities, Acta Appl. Math. 170 (2020), 981–1064.
  55. M.A. Noor, K.I. Noor, T.M. Rassias, Some Aspects of Variational Inequalities, J. Comput. Appl. Math. 47 (1993), 285–312.
  56. M.A. Noor, K.I. Noor, Th.M. Rassias, Parametric General Quasi Variational Inequalities, Math. Commun. 15 (2010), 205–212.
  57. M.A. Noor, W. Oettli, On General Nonlinear Complementarity Problems and Quasi Equilibria, Le Math. 49 (1994), 313–331.
  58. M. Patriksson, Nonlinear Programming and Variational Inequalities: A Unified Approach, Kluwer Academic Publishers, Dordrecht, 1998.
  59. S.M. Robinson, Normal Maps Induced by Linear Transformations, Math. Oper. Res. 17 (1992), 691–714.
  60. K. Sanaullah, S. Ullah, N.M. Aloraini, A Self Adaptive Three-Step Numerical Scheme for Variational Inequalities, Axioms. 13 (2024), 57.
  61. P. Shi, Equivalence of Variational Inequalities With Wiener-Hopf Equations, Proc. Amer. Math. Soc. 111 (1991), 339–346.
  62. Y. Shehu, A. Gibali, S. Sagratella, Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces, J. Optim. Theory Appl. 184 (2020), 877–894.
  63. G. Stampacchia, Formes Bilineaires Coercitives sur les Ensembles Convexes, C. R. Acad. Sci. Paris. 258 (1964), 4413–4416.
  64. S. Suantai, M.A. Noor, K. Kankam, P. Cholamjiak, Novel Forward?backward Algorithms for Optimization and Applications to Compressive Sensing and Image Inpainting, Adv. Differ. Equ. 2021 (2021), 265.
  65. P. Tseng, A Modified Forward-Backward Splitting Method for Maximal Monotone Mappings, SIAM J. Control Optim. 38 (2000), 431–446.
  66. Y. Xia, J. Wang, A Recurrent Neural Network for Solving Linear Projection Equations, Neural Networks 13 (2000), 337?350.
  67. Y.S. Xia, J. Wang, On the Stability of Globally Projected Dynamical Systems, J. Optim. Theory Appl. 106 (2000), 129–150.
  68. E.A. Youness, E-Convex Sets, E-Convex Functions, and E-Convex Programming, J. Optim. Theory Appl. 102 (1999), 439–450.