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Journal of Systems Science and Complexity

, Volume 20, Issue 3, pp 344–349 | Cite as

Role of α-Pseudo-Univex Functions in Vector Variational-Like Inequality Problems

  • S. K. MishraEmail author
  • Shouyang Wang
  • K. K. Lai
Article

Abstract

In this paper, we introduce a new class of generalized convex function, namely, α-pseudo-univex function, by combining the concepts of pseudo-univex and α-invex functions. Further, we establish some relationships between vector variational-like inequality problems and vector optimization problems under the assumptions of α-pseudo-univex functions. Results obtained in this paper present a refinement and improvement of previously known results.

Keywords

α-pseudo-univex functions vector optimization vector variational-like inequality problems 

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References

  1. 1.
    M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 1981, 80: 545–550.CrossRefGoogle Scholar
  2. 2.
    G. Mastroeni, Some remarks on the role of generalized convexity in the theory of variational inequalities, in Generalized Convexity and Optimization for Economic and Financial Decisions (ed. by G. Giorgi and F. Rossi), Pitagora Editrice, Bologna, Italy, 1999, 271–281.Google Scholar
  3. 3.
    F. Gianessi, Theorems of alternative, quadratic programs and complementarity problems, in Variational Inequality and Complementarity Problems (ed. by R. W. Cottle, F. Giannessi, and J. L. Lions), John Wiley and Sons, New York, 1980, 151–186.Google Scholar
  4. 4.
    F. Giannessi, Vector variational inequalities and vector equilibria: Mathematical Theories, Nonconvex Optimization and its Applications, 2000, 38: 423–432.Google Scholar
  5. 5.
    G. Ruiz-Garzon, R. Osuna-Gomez, and Rufian-Lizan, Relationships between vector variational-like inequality and optimization problems, European Journal of Operational Research, 2004, 157: 113–119.CrossRefGoogle Scholar
  6. 6.
    X. Q. Yang, Generalized convex functions and vector variational inequalities, Journal of Optimization Theory and Applications, 1993, 79: 563–580.CrossRefGoogle Scholar
  7. 7.
    X. Q. Yang and G. Y. Chen, A class of nonconvex functions and prevariational inequalities, Journal of Mathematical Analysis and Applications, 1992, 169: 359–373.CrossRefGoogle Scholar
  8. 8.
    X. Q. Yang and C. J. Goh, On vector variational inequalities: Application to vector equilibria, Journal of Optimization Theory and Applications, 1997, 95: 431–443.CrossRefGoogle Scholar
  9. 9.
    C. R. Bector, S. Chandra, S. Gupta, and S. K. Suneja, Univex sets, functions and univex nonlinear programming, in Lecture Notes in Economics and Mathematical Systems 405, Springer Verlag, Berlin, 1994, 1–18.Google Scholar
  10. 10.
    S. K. Mishra, On multiple objective optimization with generalized univexity, Journal of Mathematical Analysis and Applications, 1998, 224: 131–148.CrossRefGoogle Scholar
  11. 11.
    S. K. Mishra and G. Giorgi, Optimality and duality with generalized semi-univexity, Opsearch, 2000, 37: 340–350.Google Scholar
  12. 12.
    S. K. Mishra, S. Y. Wang, and K. K. Lai, Nondifferentiable multiobjective programming under generalized d-univexity, European Journal of Operational Research, 2005, 160: 218–226.CrossRefGoogle Scholar
  13. 13.
    N. G. Rueda, M. A. Hanson, and C. Singh, Optimality and duality with generalized convexity, Journal of Optimization Theory and Applications , 1995, 86: 491–500.CrossRefGoogle Scholar
  14. 14.
    M. A. Noor, On generalized preinvex functions and monotonicities, Journal of Inequalities in Pure and Applied Mathematics, 2004, 5(4): Article 110.Google Scholar
  15. 15.
    T. Weir and B. Mond, Preinvex functions in multiobjective optimization, Journal of Mathematical Analysis and Applications, 1988, 136: 29–38.CrossRefGoogle Scholar
  16. 16.
    R. Osuna-Gomez, A. Rufian-Lizana, and P. Ruiz-Canales, Invex functions and generalized convexity in multiobjective programming, Journal of Optimization Theory and Applications, 1998, 98: 651–661.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics and Computer Science, College of Basic Sciences and HumanitiesG. B. Pant University of Agriculture and TechnologyPantnagarIndia
  2. 2.Institute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.Department of Management SciencesCity University of Hong KongKowloon, Hong KongChina

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