Article ID: MTJPAM-D-20-00052

Title: Nonconvex Bifunction General Variational Inequalities


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00052; Volume 4 / Issue 3 / Year 2022 (Special Issue), Pages 1-8

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: 23 December 2020, Accepted: 18 February 2021, Published: 8 May 2021.

Corresponding Author: Muhammad Aslam Noor (Email address: noormaslam@gmail.com)

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Abstract

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction general variational inequality. Using the auxiliary principle technique, we suggest and analyze some iterative methods for solving the nonconvex bifunction general variational inequalities. We prove that the convergence of these methods either requires only pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. The ideas and techniques of this paper may stimulate further research in this field.

Keywords: Convex programming, Nonlinear programming, Bifunction variational inequalities, Nonconvex functions, Auxiliary principle technique, Convergence

References:
  1. E. A. Al-Said, M. A. Noor and Th. M. Rassias, Numerical solutions of third-order obstacle problems, Internat. J. Comput. Math. 69 (1-2), 75-84, 1998.
  2. M. Bounkhel, L. Tadj and A. Hamdi, Iterative schemes to solve nonconvex variational problems, J. Inequal. Pure Appl. Math. 4, 1-14, 2003.
  3. F. H. Clarke, Y. S. Ledyaev and Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, Berlin, 1998.
  4. G. P. Crepsi, J. Ginchev and M. Rocca, Minty variational inequalities, increase along rays property and optimization, J. Optim. Theory Appl. 123, 479-496, 2004.
  5. G. P. Crepsi, J. Ginchev and M. Rocca, Existence of solutions and star-shapedness in Minty variational inequalities, J. Global Optim. 32, 485-493, 2005.
  6. G. P. Crepsi, J. Ginchev and M. Rocca, Increasing along rays property for vector functions, J. Nonconvex Anal. 7, 39-50, 2006.
  7. G. P. Crepsi, J. Ginchev and M. Rocca, Some remarks on the Minty vector variational principle, J. Math. Aanl. Appl. 345, 165-175, 2008.
  8. Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined by bifunction, Comput. Math. Appl. 53, 1306-1316, 2007.
  9. R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981.
  10. C. S. Lalitha and M. Mehra, Vector variational inequalities with cone-pseudomonotone bifunction, Optim. 54, 327-338,2005.
  11. M. A. Noor, General variational inequalities, Appl. Math Lett. 1 (2), 119-121, 1988.
  12. M. A. Noor, Quasi variational inequalities, Appl. Math Lett. 1 (4), 367-370, 1988.
  13. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251, 217-229, 2000.
  14. M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152, 199-277, 2004.
  15. M. A. Noor, Projection methods for nonconvex variational inequalities, Optim. Letters 3, 411-418, 2009.
  16. M. A. Noor, Implicit Iterative methods for nonconvex variational inequalities, J. Optim. Theory Appl. 143, 619-624, 2009.
  17. M. A. Noor, Iterative methods for general nonconvex variational inequalities, Albanian J. Math. 3, 117-127, 2009.
  18. M. A. Noor, Some iterative methods for general nonconvex variational inequalities, Comput. Math. Modelling, 21, 97-108, 2010.
  19. M. A. Noor, An extragradient algorithm for solving general nonconvex variational inequalities, Appl. Math. 23, 917-921, 2010.
  20. M. A. Noor, On an implicit method for nonconvex variational inequalities, J. Optim. Theory Appl. 147, 411-417, 2010.
  21. M. A. Noor and Th. M. Rassias, A class of projection methods for general variational inequalities, J. Math. Anal. Appl. 268 (1), 334-343, 2002.
  22. M. A. Noor and Th. M. Rassias, On general hemiequilibrium problems, J. Math. Anal. Appl. 324, 1417-1428, 2004.
  23. M. A. Noor, K. I. Noor and M. Th. Rassias, New trends in general variational inequalities, Acta Appl. Mathematica, 170 (1), 981-1046, 2020.
  24. M. A. Noor, K. I. Noor and Th. M. Rassias, Some aspects of avriational inequalities, J. Comput. Appl. Math. 47, 285-312, 1993.
  25. M. A. Noor, K. I. Noor and Th. M. Rassias, Invitation to variational inequalities, in Analysis, Geometry and Groups: A Riemann Legacy Volume H. M. Srivastava and Th. M. Rassias, Eds. , pp. 373, Hadronic Press, Nonantum, MA, 1993.
  26. M. A. Noor, K. I. Noor and Th. M. Rassias, Set-valued resolvent equations and mixed variational inequalities, J. Math. Anal. Appl. 220, 741-759, 1998.
  27. G. A. Poliquin, R. T. Rockafellar and J. L. Thibautl, Local differentiability of distance functions, Trans. Amer. Math. Soc. 352, 5231-5249, 2000.
  28. G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, Contesi Rendi de’s Academie Sciences de Paris, 258, 4413-4416, 1964.