Title: Iterative algorithms for solving nonlinear quasi-variational inequalities
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-21-00048; Volume 4 / Issue 1 / Year 2022, Pages 44-58
Document Type: Research Paper
Author(s): Muhammad Aslam Noor a , Khalida Inayat Noor b
aDepartment of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
bDepartment of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
Received: 18 July 2021, Accepted: 7 September 2021, Published: 5 October 2021.
Corresponding Author: Muhammad Aslam Noor (Email address: email@example.com)
Full Text: PDF
In this paper, we consider the quasi-variational inequalities. It is shown that quasi-variational inequalities are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inequalities and related optimization problems are suggested by using projection methods, Wiener-Hopf equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results.
Keywords: Variational inequalities, projection method, Wiener-Hopf equations, dynamical system, convergence, applicationsReferences:
- 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, 773–782, 2003.
- A. S. Antipin, M. Jacimovic and N. Mijajlovic, Extra gradient method for solving quasi variational inequalities, Optimization 67, 103–112, 2018.
- A. Bensoussan and J. L. Lions, Application des inéqualities variationnelles en contrôl stochastique, Paris, Bordas(Dunod), 1978.
- D. Chan and J. Pang, The generalized quasi-variational inequality problem, Math. Oper. Res. 7, 211–22, 1982.
- R. W. Cottle, Nonlinear programs with positively bounded Jacobians, J. Soc. Indust. Appl. Math. 14, 147–158, 1966.
- R. W. Cottle, J,-S. Pang and R. E. Stone, The linear complementarity problem,, SIAM Publ. 2009.
- P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Annals Oper. Research 44, 7–42, 1993.
- R. Glowinski, J. L. Lions and R. Tremolieres, Numerical analysis of variational inequalities, North Holland, Amsterdam, 1981.
- S. Jabeen, B. B.Mohsin, M. A. Noor and K. I. Noor, Inertial projection methods for solving general quasi-variational inequalities, AIMS Math. 6 (2), 1075–1086, 2021.
- S. Karamardian, Generalized complementarity problems, J. Opt. Theory Appl. 8, 161–168, 1971.
- E. Lemke, Bimatrix equilibrium points, and mathematical programming, Management Sci. 11, 681–689, 1965.
- D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, SIAM, Philadelphia, 2000.
- G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika Mat. Metody 12, 747–756, 1976.
- O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett. 3, 101–108, 2009.
- N. Mijajlovic, J. Milojica and M. A. Noor, Gradient-type projection methods for quasi variational inequalities, Optim. Lett. 13, 1885–1896, 2019.
- H. Moosaei, S. Ketabchi, M.A. Noor, J. Iqbal and V. Hooshyarbakhsh, Some techniques for solving absolute value equations, Appl. Math. Comput. 268, 696–705, 2015.
- A. Nagurney and D. Zhang, Projected dynamical systems and variational inequalities with applications, Kluwer Academic Publishers, Boston, Dordrecht, London 1996.
- M. A. Noor, On Variational Inequalities, PhD Thesis, Brunel University, London, U. K., 1975.
- M. A. Noor, An iterative scheme for class of quasi variational inequalities, J. Math. Anal. Appl. 110, 463–468, 1985.
- M. A. Noor, Generalized quasi mildly nonlinear variational inequalities: in Variational Methods in Engineering, (Edit.C. Brebbia), 3.3-3.11. Springer-Verlag, Berlin, 1985.
- M. A. Noor, The quasi-complementarity problem, J. Math. Anal. Appl. 130, 344–353, 1988.
- M. A. Noor, Fixed point approach for complementaritty problems, J. Math. Anal. Appl. 133, 437–448, 1988.
- M. A. Noor, An iterative algorithm for variational inequalities, J. Math. Anal. Appl. 158, 448-455, 1991.
- M. A. Noor, Sensitivity analysis for quasi variational inequalities, J. Optim. Theory Appl. 95 (2), 399–407, 1997.
- M. A. Noor, A Wiener-Hopf dynamical system and variational inequalities, New Zealand J. Math. 31, 173–182, 2002.
- M. A. Noor, Some dvelopments in general variational inequalites, Appl. Math. Comput. 152, 199–277, 2004.
- M. A. Noor, Hemivariational inequalities, J. Appl. Math. Computing 17 (1-2), 59–72, 2005.
- M. A. Noor and K. I. Noor, From representaion theorems to variational inequalities, in: Computational Mathematics and Variational Analysis (Edits: N. J. Daras, T. M. Rassias), Springer Optimization and Its Applications, 159 (2020), 261–277. https:doi.org/10.1007/978-3-030-44625-3-15261
- M. A. Noor and K. I. Noor, Some new classes of strongly generalized preinvex functions, TWMS J. Pure Appl. Math. 12 (2), 2021.
- M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi equilibria, Le Mathematiche 49, 313–331, 1994.
- M. A. Noor, K. I. Noor and B. B. Mohsen, Some new classes of general quasi variational inequalities, AIMS Math. 6 (6), 6406–6421, 2021.
- M. A. Noor and S. Zarae, Linear quasi complementarity problems, Utilitas. Math. 27, 249–260, 1985.
- M. A. Noor, K. I. Noor and A. Bnouhachem, Some new iterative methods for variational inequalities, Canad. J. Appl. Math. 3 (1), 1–17, 2021.
- M. A. Noor, K. I. Noor, and M. T. Rassias, New trends in general variational inequalities, Acta Appl. Math. 170 (1), 981–1046, 2020.
- M. A. Noor, K. I. Noor and Th. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47, 285–312, 1993.
- M. A. Noor, K. I. Noor and S. Batool, On generalized absolute value equations, U.P.B. Sci. Bull., Series A. 80 (4), 63–70, 2018.
- M. A. Noor, J. Iqbal, K. I. Noor and E. Al-Said, On an iterative method for solving absolute value equations, Optim. Lett. 6, 1027–1033, 2012.
- M. A. Noor, J. Iqbal, K.I. Noor and E. Al-Said, Generalized AOR method for solving absolute complementarity problems, J. Appl. Math. 2012, 2012, doi:10.1155/ 2012/743-861.
- M. A. Noor, K. I. Noor, A. Hamdi and E. H. El-Shemas, On difference of two monotone operators, Optim. Letters 3, 329–335, 2009.
- P. D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationary principles, Acta Mech. 42, 160–183, 1983.
- P. D. Panagiotopoulos, Hemivariational inequalities, applications to mechanics and engineering, Springer Verlag, Berlin, 1993.
- M. Patriksson, Nonlinear programming and variational inequalities: A unified approach, Kluwer Acadamic publishers, Drodrecht, 1998.
- S. M. Robinson, Normal maps induced by linear transformations, Math. Oper. Res. 17, 691–714, 1992.
- P. Shi, Equivalence of variational inequalities with Wiener-Hopf equations, Proc. Amer. Math. Soc. 111, 339–346, 1991.
- Y. Shehu, A. Gibali and S. Sagratella, Inertial projection-type method for solving quasi variational inequalities in real Hilbert space, J. Optim. Theory Appl. 184, 877–894, 2020.
- G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258, 4413–4416, 1964.
- H. J. Wang, D. X. Cao, H. Liu and L. Qiu, Numerical validation for sytems of absolute value equations, Calcol 54, 669–683, 2017.
- J. -J. Zhang, The relaxed nonlinear PHSS-like iteration method for absolute value equations, Appl. Math. Comput. 265, 266–274, 2015.