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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 26808, 16 pages
http://dx.doi.org/10.1155/IJMMS/2006/26808

Some relations between variational-like inequalities and efficient solutions of certain nonsmooth optimization problems

1Departamento de Matemática, Universidade Federal do Paraná, CP 19081 CEP, Curitiba, Paraná 81531-990, Brazil
2Departamento de Matemática Aplicada, Universidade Estadual de Campinas, CP 6065, Campinas, São Paulo 13083-859, Brazil
3Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, Sevilla 41012, Spain

Received 28 November 2005; Revised 4 June 2006; Accepted 21 June 2006

Copyright © 2006 Hindawi Publishing Corporation. 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. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1984. View at Zentralblatt MATH · View at MathSciNet
  2. A. Ben-Israel and B. Mond, “What is invexity?” Journal of the Australian Mathematical Society. Series B, vol. 28, no. 1, pp. 1–9, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. J. V. Brandão, M. A. Rojas-Medar, and G. N. Silva, “Invex nonsmooth alternative theorem and applications,” Optimization, vol. 48, no. 2, pp. 239–253, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. Y. Chen and B. D. Craven, “Existence and continuity of solutions for vector optimization,” Journal of Optimization Theory and Applications, vol. 81, no. 3, pp. 459–468, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. Y. Chen and S. J. Li, “Existence of solutions for a generalized vector quasivariational inequality,” Journal of Optimization Theory and Applications, vol. 90, no. 2, pp. 321–334, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1983. View at Zentralblatt MATH · View at MathSciNet
  7. B. D. Craven and B. M. Glover, “Invex functions and duality,” Journal of the Australian Mathematical Society. Series A, vol. 39, no. 1, pp. 1–20, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. G. Duvaut and J.-L. Lions, Les inéquations en Mécanique et en Physique, Dunod, Paris, 1972. View at Zentralblatt MATH · View at MathSciNet
  9. A. M. Geoffrion, “Proper efficiency and the theory of vector maximization,” Journal of Mathematical Analysis and Applications, vol. 22, no. 3, pp. 618–630, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in Variational Inequalities and Complementarity Problems (Proc. Internat. School, Erice, 1978), R. W. Cottle, F. Giannessi, and J.-L. Lions, Eds., pp. 151–186, Wiley, Chichester, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. Giannessi, “On Minty variational principle,” in New Trends in Mathematical Programming, vol. 13 of Appl. Optim., pp. 93–99, Kluwer Academic, Massachusetts, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Giorgi and A. Guerraggio, “Various types of nonsmooth invex functions,” Journal of Information & Optimization Sciences, vol. 17, no. 1, pp. 137–150, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1976.
  14. 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
  15. V. Jeyakumar, “A generalization of a minimax theorem of Fan via a theorem of the alternative,” Journal of Optimization Theory and Applications, vol. 48, no. 3, pp. 525–533, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. I. V. Konnov and J. C. Yao, “On the generalized vector variational inequality problem,” Journal of Mathematical Analysis and Applications, vol. 206, no. 1, pp. 42–58, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. G. M. Lee, “On relations between vector variational inequality and vector optimization problem,” in Progress in Optimization (Perth, 1998), X. Q. Yang, A. I. Mees, M. E. Fisher, and L. S. Jennings, Eds., vol. 39 of Appl. Optim., pp. 167–179, Kluwer Academic, Dordrecht, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. M. Lee, D. S. Kim, and H. Kuk, “Existence of solutions for vector optimization problems,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 90–98, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. M. Lee, D. S. Kim, B. S. Lee, and N. D. Yen, “Vector variational inequality as a tool for studying vector optimization problems,” Nonlinear Analysis, vol. 34, no. 5, pp. 745–765, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. M. Lee and S. H. Kum, “On implicit vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 104, no. 2, pp. 409–425, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Park, “Some coincidence theorems on acyclic multifunctions and applications to KKM theory,” in Fixed Point Theory and Applications (Halifax, NS, 1991), pp. 248–277, World Scientific, New Jersey, 1992. View at Google Scholar · View at MathSciNet
  22. T. D. Phuong, P. H. Sach, and N. D. Yen, “Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function,” Journal of Optimization Theory and Applications, vol. 87, no. 3, pp. 579–594, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. T. W. Reiland, “Nonsmooth invexity,” Bulletin of the Australian Mathematical Society, vol. 42, no. 3, pp. 437–446, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, no. 28, Princeton University Press, New Jersey, 1970. View at Zentralblatt MATH · View at MathSciNet
  25. X. Q. Yang, “Generalized convex functions and vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, no. 3, pp. 563–580, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet