Table of Contents
ISRN Applied Mathematics
Volume 2014, Article ID 904640, 11 pages
http://dx.doi.org/10.1155/2014/904640
Research Article

Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

1Centre for Mathematical Sciences, Banasthali University, Rajasthan 304022, India
2Department of Applied Sciences and Humanities, ITM University, Gurgaon 122017, India
3Department of Mathematics, South Asian University, New Delhi 110021, India

Received 18 February 2014; Accepted 25 March 2014; Published 22 May 2014

Academic Editors: P.-y. Nie and S. Utyuzhnikov

Copyright © 2014 Pallavi Kharbanda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J.-P. Vial, “Strong and weak convexity of sets and functions,” Mathematics of Operations Research, vol. 8, no. 2, pp. 231–259, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. A. Hanson and B. Mond, “Necessary and sufficient conditions in constrained optimization,” Mathematical Programming, vol. 37, no. 1, pp. 51–58, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. V. Jeyakumar and B. Mond, “On generalised convex mathematical programming,” Australian Mathematical Society B, vol. 34, no. 1, pp. 43–53, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. A. Hanson, R. Pini, and C. Singh, “Multiobjective programming under generalized type I invexity,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 562–577, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. A. Liang, H. X. Huang, and P. M. Pardalos, “Optimality conditions and duality for a class of nonlinear fractional programming problems,” Journal of Optimization Theory and Applications, vol. 110, no. 3, pp. 611–619, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. T. R. Gulati, I. Ahmad, and D. Agarwal, “Sufficiency and duality in multiobjective programming under generalized type I functions,” Journal of Optimization Theory and Applications, vol. 135, no. 3, pp. 411–427, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. L. Ye, “d-invexity and optimality conditions,” Journal of Mathematical Analysis and Applications, vol. 162, no. 1, pp. 242–249, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. Antczak, “Multiobjective programming under d-invexity,” European Journal of Operational Research, vol. 137, no. 1, pp. 28–36, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. K. Mishra and M. A. Noor, “Some nondifferentiable multiobjective programming problems,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 472–482, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. Nahak and R. N. Mohapatra, “d-ρ-η-θ-invexity in multiobjective optimization,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 6, pp. 2288–2296, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H. Slimani and M. S. Radjef, “Nondifferentiable multiobjective programming under generalized dI-invexity,” European Journal of Operational Research, vol. 202, no. 1, pp. 32–41, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. C. R. Bector, S. K. Suneja, and S. Gupta, “Univex functions and univex nonlinear programming,” in Proceeding of the Administrative Sciences Association of Canada, pp. 115–124, 1992.
  14. N. G. Rueda, M. A. Hanson, and C. Singh, “Optimality and duality with generalized convexity,” Journal of Optimization Theory and Applications, vol. 86, no. 2, pp. 491–500, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. K. Mishra, “On multiple-objective optimization with generalized univexity,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 131–148, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. K. Mishra, S. Y. Wang, and K. K. Lai, “Nondifferentiable multiobjective programming under generalized d-univexity,” European Journal of Operational Research, vol. 160, no. 1, pp. 218–226, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. I. Ahmad, “Efficiency and duality in nondifferentiable multiobjective programming involving directional derivative,” Applied Mathematics, vol. 2, no. 4, pp. 452–460, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  18. P. Kharbanda, D. Agarwal, and D. Sinha, “Nonsmooth multiobjective optimization involving generalized univex functions,” Opsearch, 2013. View at Publisher · View at Google Scholar
  19. S. K. Suneja and M. K. Srivastava, “Optimality and duality in nondifferentiable multiobjective optimization involving d-type I and related functions,” Journal of Mathematical Analysis and Applications, vol. 206, no. 2, pp. 465–479, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. T. Weir and B. Mond, “Pre-invex functions in multiple objective optimization,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 29–38, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. T. Weir and V. Jeyakumar, “A class of nonconvex functions and mathematical programming,” Bulletin of the Australian Mathematical Society, vol. 38, no. 2, pp. 177–189, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Mititelu, “Invex sets,” Studii și Cercetări Matematice, vol. 46, no. 5, pp. 529–532, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, NY, USA, 1983. View at MathSciNet