Absolute Value Variational Inequalities and Dynamical Systems

Main Article Content

Safeera Batool, Muhammad Aslam Noor, Khalida Inayat Noor


In this paper, we consider the absolute value variational inequalities. We propose and analyze the projected dynamical system associated with absolute value variational inequalities by using the projection method. We suggest different iterative algorithms for solving absolute value variational inequalities by discretizing the corresponding projected dynamical system. The convergence of the suggested methods is proved under suitable constraints. Numerical examples are given to illustrate the efficiency and implementation of the methods. Results proved in this paper continue to hold for previously known classes of absolute value variational inequalities.

Article Details


  1. B. H. Ahn, Iterative methods for linear complementarity problems with upper bounds on primary variables, Math. Program. 26(3)(1983), 295-315.
  2. C. Bakxchi and A. Capello, Disequazioni variationalies quasi varionali, applicantioni a problemi di frontiera libera, Vols. I and II, Bologna, Italia. (1978).
  3. B. B. Bin-Mohsin, M. A. Noor, K. I. Noor and R. Latif, Projected dynamical system for variational inequalities, J. Adv. Math. Stud. 11(1)(2018), 1-9.
  4. R. W. Cottle and G. B. Dantzig, Complementarity pivot theory of mathematical programming. Linear Algebra Appl. 1(1968), 103-125.
  5. I. C. Dolcetta and U. Mosco, Implicit complementarity problems and quasi variational inequalities. Var. Ineq. and Compl. Prob. Th. App. (Eds. R.W. Cottle, F. Giannessi and J.L. Lions) John Wiley and Sons, New York, New Jersey, 1980.
  6. J. Dong, D. H. Zhang and A. Nagurney, A projected dynamical systems model of general financial equilibrium with stability analysis. Math. Comput. Model. 24(2)(1996), 35-44.
  7. P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities. Ann. Oper. Res. 44(1)(1993), 7-42.
  8. H. Esmaeili, M. Mirzapour and E. Mahmoodabadi, A fast convergent two-step iterative method to solve the absolute value equation. U.P.B. Sci. Bull., Ser. A. 78(1)(2016), 25-32.
  9. G. Fichera, G. Problemi elastostatistic con vincoli unilaterali il prolema di signorini con ambigue condizone as contorno atti. Acad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nature. Sez. La., 8(7)(1964), 91-140.
  10. T. L. Friesz, D. H. Berstein, N. J. Mehta, R. L. Tobin and S. Ganjliazadeh, Day-to day dynamic network disequilibrium and idealized traveler information systems. Oper. Res. 42(6)(1994), 1120-1136.
  11. T. L. Friesz, D. H. Berstein and R. Stough, Dynamic systems, variational inequalities and control theoretic models for predicting time-varying urban network flows. Transport. Sci. 30(1)(1996), 14-31.
  12. G. Glowinski, J. L.Lions and R. Tremolieres, Numerical Analysis of variational Inequalities. NorthHolland, Amsterdam, 1981.
  13. S. L. Hu and Z. H. Huang, A note on absolute value equations. Optim. Lett. 4(3)(2010), 417-424.
  14. S. L. Hu and Z. H. Huang, A generalized Newton method for absolute value equations associated with second order cones. J. Comput. Appl. Math. 235(5)(2011), 1490-1501.
  15. S. Karamardian, The complementarity problem. Math. Program. 2(1)(1972), 107-109.
  16. Y. F. Ke and C. F. Ma, SOR-like iteration method for solving absolute value equations. Appl. Math. Comput. 311(2017), 195-202.
  17. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia, 1980.
  18. G. M. Korpelevich, An extragradient method for finding saddle points and for other problems. Ekonomika Mat. Metody, 12(4)(1976), 747-756.
  19. C. E. Lemke, Bimatrix equilibrium points and mathematical programming. Manage. Sci. 11(7)(1965), 681-689.
  20. J. L. Lions and G. Stampacchia, Variational inequalities. Commun. Pure Appl. Math. 20(3)(1967), 493-519.
  21. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin. 1971.
  22. C. Q. LV and C. F. Ma, Picard splitting method and Picard CG method for solving the absolute value equation. J. Nonlinear Sci. Appl. 10(2017), 3643-3654.
  23. O. L. Mangasarian, The linear complementarity problem as a separable bilinear program. J. Glob. Optim. 6(2)(1995), 153-161.
  24. O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra Appl. 419(2-3)(2006), 359-367.
  25. O. L. Mangasarian, Absolute value programming. Addison-Wesley Publishing, Boston. 2007.
  26. K. G. Murty, Linear complementarity, linear and nonlinear programming. Heldermann Verlag, Berlin. 1988.
  27. M. A. Noor, On variational inequalities. Ph.D. thesis. Burnel University, U.K. (1975).
  28. M. A. Noor, Mildly Nonlinear variational inequalities. Mathematica. 24(47)(1982), 99-110.
  29. M. A. Noor, Strongly nonlinear variational inequalities. C. R. Math. Rep. Acad. Sci. Canada. 4(4)(1982), 213-218.
  30. M. A. Noor, Iterative Methods for a Class of Complementarity Problems. J. Math. Anal. Appl. 133(2)(1988), 366-382.
  31. M. A. Noor, Some recent advances in variational inequalities. Part I. Basic concepts. New Zealand J. Math. 26(1)(1997), 53-80.
  32. M. A. Noor, Some recent advances in variational inequalties. Part II. Other concepts. New Zealand J. Math. 26(2)(1977), 229-255.
  33. M. A. Noor, Stability of the modified projected dynamical systems. Computer Math. Appl. 44(2002), 1-5.
  34. M. A. Noor, Resolvent dynamical systems for mixed variational inequalities. Korean J. Comput. Appl. Math. 9(1)(2002), 15-26.
  35. M. A. Noor, Some developments in general variational inequalities. Appl. Math. Comput. 152(2004), 199-277.
  36. M. A. Noor, On an implicit method for nonconvex variational inequalities,.J. Optim. Theory Appl. 147(2)(2010), 411-417.
  37. M. A. Noor and K. I. Noor, Iterative methods for variational inequalities and nonlinear programming. Oper. Res. Verf. 31(1979), 455-463.
  38. M. A. Noor, Y. J. Wang and N. Xiu, Some new projection methods for variational inequalities. Appl. Math. Comput. 137(2)(2003), 423-435.
  39. M. A. Noor, K. I. Noor and A. Bnouhachem, On a unified implicit method for variational inequalities. J. Comput. Appl. Math. 249(2013), 69-73.
  40. M. A. Noor, M, K. I. Noor and S. Batool, On generalized absolute value equations. U.P.B. Sci. Bull., Series A, 80(4)(2018), 63-70.
  41. M. A. Noor, J. Iqbal, K. I. Noor and E. Al-Said, Generalized AOR method for solving absolute value complementarity problems J. Appl. Math. 2012(2012), 743861.
  42. O. Prokopyev, O. On equivalent reformulation for absolute value equations. Comput. Optim. App., 44(3)(2009), 363-372.
  43. J. Rohn, A theorem of the alternatives for the equation Ax + B|x| = b. Linear and Multilinear Algebra, 52(6)(2004), 421-426.
  44. J. Rohn, An algorithm for computing all solutions of an absolute value equation. Optim. Lett. 6(5)(2011), 851-856.
  45. G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris. 258(1964), 4413-4416.
  46. Y. S. Xia and J. Wang, On the stability of globally projected dynamical systems. J. Optim. Theory Appl. 106(1)(2000), 129-150.
  47. Y. S. Xia and J. Wang, A recurrent neural network for solving linear projection equation. Neural Networks 13(3)(2000), 337-350.
  48. Y. S. Xia, On convergence conditions of an extended projection neural network. Neural Comput. 17(3)(2005), 515-525.
  49. L. Yong, Particle Swarm Optimization for absolute value equations. J. Comput. Inform. Syst. 6(7)(2010), 2359-2366.