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Journal of Applied Mathematics and Computation

ISSN Online: 2576-0653 ISSN Print: 2576-0645 CODEN: JAMCEZ
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ArticleOpen Access http://dx.doi.org/10.26855/jamc.2023.06.006

Some Classes of Exponentially General Variational-like Inequalities

Muhammad Aslam Noor*, Khalida Inayat Noor

Department of Mathematics, COMSATS University Islamabad, Park Road, Islamabad, Pakistan.

*Corresponding author: Muhammad Aslam Noor

Published: July 30, 2023

Abstract

In this paper, we define and introduce some new classes of exponentially variational-like inequalities, which are called exponentially general variational-like inequalities. It is shown that the optimality conditions of differentiable exponentially general preinvex functions can be characterized by exponentially general variational-like inequalities. These exponentially inequalities have applications in mathematical programming, information science, net work, data analysis and machine leaarning. We apply the auxiliary principle technique to suggest and analyse some new inertal iterative methods for finding the approximate solutions of these exponentially variational iequalities. Convergence criteria is discussed under some mild conditions. Our methods of anlyzing the convergence criteria is very simple as compared with other teachniques. Several special casess, which can be obtained, are discussed as applications. We have only considered the theoretical aspects of these methods. Numerical implementation and comparison with other echnqiues is an open problem. Results obtained in this paper can be viewed as significant improvement of previously known results. The ideas and technqiues of this paper stimluate further research in different areas of pure and applied sciences.

Keyword

Convex functions, minimum, exponentially convex functions, variational-like inequalities, iterative methods

References

[1] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80(1981), 545-550.

[2] A. Ben-Isreal and B. Mond, What is invexity? J. Austral. Math. Soc., Ser. B, 28(1), (1986), 1-9.

[3] S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189(1995), 901-908.

[4] M. A. Noor, Variational-like inequalities, Optimization, 30(1994), 323-330

[5] M. A. Noor and K. I. Noor, Higher order strongly exponentially preinvex functions, J. Appl. Math. Inform. 39(3-4) (2021), 469-485.

[6] G. Ruiz-Garzion, R. Osuna-Gomez and A. Rufian-Lizan, Generalized invex monotonicity, European J. Oper. Research, 144(2003), 501-512.

[7] T. Weir and B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl., 136(1988), 29-38.

[8] X. M. Yang, Q. Yang and K. L. Teo, Criteria for generalized invex monotonicities, European J. Oper. Research, 164(1), (2005), 115-119..

[9] J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20(1967), 493-512.

[10] R. Glowinski, J. J. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

[11] D. L. Zhu and P. Marcotte, Cocoercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim. 6(1996), 714-726.

[12] M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic Publishers, Dordrecht, 1998.

[13] G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Paris, 258(1964), 4413-4416.

[14] G. Alirezaei and R. Mazhar, On exponentially concave functions and their impact in information theory, J. Inform. Theory Appl. 9(5), (2018), 265-274.

[15] T. Antczak, On (p,r)-invex sets and functions, J. Math. Anal. Appl. 263(2001), 355-379.

[16] M. Avriel, r-Convex functions. Math. Program., 2(1972), 309-323.

[17] N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52(1929), 1-66.

[18] J. Pecaric and J. Jaksetic, On exponential convexity, Euler-Radau expansions and stolarsky means, Rad Hrvat. Matematicke Znanosti, 17(515), (2013), 81-94.

[19] S. Pal and T. K. Wong, On exponentially concave functions and a new information geometry, Annals. Prob. 46(2), (2018), 1070-1113.

[20] Y. X. Zhao, S. Y. Wang and L. Coladas Uria, Characterizations of r-Convex Functions, J. Optim. Theory Appl. 145(2010), 186–195.

[21] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992.

[22] M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi equilibria, Le Matemat. 49(1994), 313-331.

[23] G. Cristescu and L. Lupsa, Non Connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, 2002.

Copyright

© 2023 by the author(s).
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How to cite this paper

Some Classes of Exponentially General Variational-like Inequalities

How to cite this paper: Muhammad Aslam Noor, Khalida Inayat Noor. (2023) Some Classes of Exponentially General Variational-like Inequalities. Journal of Applied Mathematics and Computation7(2), 249-256.

DOI: http://dx.doi.org/10.26855/jamc.2023.06.006