Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013, Article ID 237428, 7 pages
http://dx.doi.org/10.1155/2013/237428
Research Article

Nonsmooth Multiobjective Fractional Programming with Local Lipschitz Exponential -Invexity

Chung-Jen Junior College of Nursing, Health Sciences and Management, Chia-Yi 62241, Taiwan

Received 3 June 2013; Accepted 4 September 2013

Academic Editor: Gue Lee

Copyright © 2013 Shun-Chin Ho. 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 Scopus
  2. I. Ahmad, R. P. Agarwal, S. K. Gupta, and N. Kailey, “Generalized second-order mixed symmetric duality in nondifferentiable mathematical programming,” Abstract and Applied Analysis, vol. 2011, Article ID 103597, 75 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. T. Antczak, “A class of B-p,r-invex functions and mathematical programming,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 187–206, 2003. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Antczak, “Generalized fractional minimax programming with B-p,r-invexity,” Computers and Mathematics with Applications, vol. 56, no. 6, pp. 1505–1525, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. T. Antczak, “Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity,” Journal of Global Optimization, vol. 45, no. 2, pp. 319–334, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. B. D. Craven, “Invex functions and constrained local minima,” Bulletin of the Australian Mathematical Society, vol. 24, no. 3, pp. 357–366, 1981. View at Publisher · View at Google Scholar
  7. J. Lee and S. Ho, “Optimality and duality for multiobjective fractional problems with r-invexity,” Taiwanese Journal of Mathematics, vol. 12, no. 3, pp. 719–740, 2008. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. C. R. Bector, S. Chandra, and I. Husain, “Optimality conditions and duality in subdifferentiable multiobjective fractional programming,” Journal of Optimization Theory and Applications, vol. 79, no. 1, pp. 105–125, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. S. C. Ho and H. C. Lai, “Optimality and duality for nonsmooth minimax fractional programming problem with exponential p,r-invexity,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 3, pp. 433–447, 2012. View at Google Scholar
  10. M. H. Kim and G. M. Lee, “On duality theorems for nonsmooth Lipschitz optimization problems,” Journal of Optimization Theory and Applications, vol. 110, no. 3, pp. 669–675, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. D. S. Kim and S. Schaible, “Optimality and duality for invex nonsmooth multiobjective programming problems,” Optimization, vol. 53, no. 2, pp. 165–176, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. H. Kuk, G. M. Lee, and D. S. Kim, “Nonsmooth multiobjecttve programs with V-ρ-invexity,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 4, pp. 405–412, 1998. View at Google Scholar · View at Scopus
  13. H. C. Lai and S. C. Ho, “Optimality and duality for nonsmooth multiobjective fractional programming problems involving exponential V-r-invexity,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 6, pp. 3157–3166, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. H. C. Lai and S. C. Ho, “Duality for a system of multiobjective problems with exponential type invexity functions,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 1, pp. 97–110, 2012. View at Google Scholar
  15. G. M. Lee, “Nonsmooth invexity in multiobjective programming,” Journal of Information & Optimization Sciences, vol. 15, no. 1, pp. 127–136, 1994. View at Google Scholar
  16. S. K. Mishra and R. N. Mukherjee, “On generalised convex multi-objective nonsmooth programming,” Journal of the Australian Mathematical Society Series B, vol. 38, no. 1, pp. 140–148, 1996. View at Publisher · View at Google Scholar · View at Scopus
  17. T. W. Reiland, “Nonsmooth invexity,” Bulletin of the Australian Mathematical Society, vol. 42, pp. 437–446, 1990. View at Google Scholar
  18. F. H. Clarke, Optimization and Non-Smooth Analysis, Wiley-Interscience, New York, NY, USA, 1983.
  19. J. B. Hiriart-Urruty, “On optimality conditions in nondifferentiable programming,” Mathematical Programming, vol. 14, no. 1, pp. 73–86, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. J. C. Liu, “Optimality and duality for multiobjective fractional programming involving nonsmooth (F, ρ)-convex functions,” Optimization, vol. 36, no. 4, pp. 333–346, 1996. View at Publisher · View at Google Scholar · View at Scopus