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Journal of Optimization Theory and Applications

, Volume 111, Issue 1, pp 173–194 | Cite as

Existence and Density Results for Proper Efficiency in Cone Compact Sets

  • X. D. H. Truong
Article

Abstract

Existence and density results are established for positive proper efficient points, Henig proper efficient points, and superefficient points in cone compact sets.

Cones proper efficiency cone compact sets existence and density theorems 

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References

  1. 1.
    Kuhn, H. W., and Tucker, A. W., Nonlinear Programming, Proceeding of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Edited by J. Neyman, University of California Press, Berkeley, California, pp. 481–492, 1952.Google Scholar
  2. 2.
    Benson, H. P., An Improved Definition of Proper Efficiency for Vector Maximization with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 232–241, 1979.Google Scholar
  3. 3.
    Borwein, J. M., Proper Efficient Points for Maximization with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.Google Scholar
  4. 4.
    Borwein, J. M., and Zhuang, D., Super Efficiency in Vector Optimization, Transactions of the American Mathematical Society, Vol. 338, pp. 105–122, 1993.Google Scholar
  5. 5.
    Geoffrion, A. M., Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 616–630, 1968.Google Scholar
  6. 6.
    Guerraggio, A., Molho, E., and Zaffaroni, A., On the Notion of Proper Efficiency in Vector Optimization, Journal of Optimization Theory and Applications, Vol. 82, pp. 1–21, 1994.Google Scholar
  7. 7.
    Hartley, R., On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.Google Scholar
  8. 8.
    Henig, M. I., Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 36, pp. 387–407, 1982.Google Scholar
  9. 9.
    Savaragi, Y., Nakayama, H., and Tanino, T., Theory of Multiobjective Optimization, Academic Press, Orlando, Florida, 1985.Google Scholar
  10. 10.
    Zheng, X. Y., Proper Efficiency in Locally Convex Topological Vector Spaces, Journal of Optimization Theory and Applications, Vol. 94, pp. 469–486, 1997.Google Scholar
  11. 11.
    Arrow, K. J., Barankin, E. W., and Blackwell, D., Admissible Points of Convex Sets, Contributions to the Theory of Games, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, Vol. 2, pp. 87–92, 1953.Google Scholar
  12. 12.
    Bitran, G. R., and Magnanti, T. L., The Structure of Admissible Points with Respect to Cone Dominance, Journal of Optimization Theory and Applications, Vol. 29, pp. 573–614, 1979.Google Scholar
  13. 13.
    Borwein, J. M., On the Existence of Pareto Efficient Points, Mathematics of Operation Research, Vol. 8, pp. 64–73, 1980.Google Scholar
  14. 14.
    Chen, G. Y., Generalized ArrowBarankinBlackwell Theorem, in Locally Convex Spaces, Journal of Optimization Theory and Applications, Vol. 84, pp. 93–101, 1995.Google Scholar
  15. 15.
    Chichilnisky, G., and Kalman, P. J., Applications of Functional Analysis to Models of Efficient Allocation of Economic Resources, Journal of Optimization Theory and Applications, Vol. 30, pp. 19–32, 1980.Google Scholar
  16. 16.
    Dauer, J. P., and Gallagher, R. J., Positive Proper Efficient Points and Related Cones Results in Vector Optimization Theory, SIAM Journal on Control and Optimization, Vol. 28, pp. 158–172, 1990.Google Scholar
  17. 17.
    Ferro, F., General Form of the ArrowBarankinBlackwell Theorem in Normed Spaces and the l -Case, Journal of Optimization Theory and Applications, Vol. 79, pp. 127–138, 1993.Google Scholar
  18. 18.
    Gallagher, R. J., The ArrowBarankinBlackwell Theorem in a Dual Space Setting, Journal of Optimization Theory and Applications, Vol. 84, pp. 665–674, 1995.Google Scholar
  19. 19.
    Gallagher, R. J., and Saleh, O. A., Two Generalizations of a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 31, pp. 247–256, 1993.Google Scholar
  20. 20.
    Jahn, J., A Generalization of a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 26, pp. 999–1005, 1988.Google Scholar
  21. 21.
    Peleg, B., Efficiency Prices for Optimal Consumption Plans, Part 2, Israel Journal of Mathematics, Vol. 9, pp. 222–234, 1971.Google Scholar
  22. 22.
    Petschke, M., On a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 28, pp. 395–401, 1990.Google Scholar
  23. 23.
    Radner, R., A Note on Maximal Points of Convex Sets in lS, Proceeding of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Edited by L. M. Le Cam and J. Neyman, University of California Press, Berkeley, California, pp. 351–354, 1967.Google Scholar
  24. 24.
    Song, W., Generalization of the ArrowBarankinBlackwell Theorem in a Dual Space Setting, Journal of Optimization Theory and Applications, Vol. 95, pp. 225–230, 1997.Google Scholar
  25. 25.
    Fu, W. T., On the Density of Proper Efficient Points, Proceedings of the American Mathematical Society, Vol. 124, pp. 1213–1217, 1996.Google Scholar
  26. 26.
    Makarov, E. K., and Rachkovski, N. N., Density Theorem for Generalized Henig's Proper Efficiency, Journal of Optimization Theory and Applications, Vol. 91, pp. 419–437, 1996.Google Scholar
  27. 27.
    Bednarczuk, E. M., and Song, W., Some More Density Results for Proper Efficiency, Journal of Mathematical Analysis and Applications, Vol. 231, pp. 345–354, 1999.Google Scholar
  28. 28.
    Zheng, X. Y., The Domination Property for Efficiency in Locally Convex Topological Vector Spaces, Journal of Mathematical Analysis and Applications, Vol. 213, pp. 455–467, 1997.Google Scholar
  29. 29.
    Zhuang, D., Density Results for Proper Efficiency, SIAM Journal on Control and Optimization, Vol. 32, pp. 51–58, 1994.Google Scholar
  30. 30.
    Cambini, A., and Martein, L., On the Existence of Efficient Points, Optimization, Vol. 28, pp. 283–290, 1994.Google Scholar
  31. 31.
    Corley, H. W., An Existence Result for Maximization with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 31, pp. 277–281, 1980.Google Scholar
  32. 32.
    Luc, D. T., Theory of Vector Optimization, Springer Verlag, Berlin, Germany, 1989.Google Scholar
  33. 33.
    Dunford, N., and Schwartz, J. T., Linear Operators, Part 1, Wiley Interscience, New York, NY, 1958.Google Scholar
  34. 34.
    Krasnoselskii, M. A., Positive Solutions of Operator Equations, Noordhoff, Groningen, Holland, 1964.Google Scholar
  35. 35.
    Truong, X. D. H., On the Existence of Efficient Points in Locally Convex Spaces, Journal of Global Optimization, Vol. 4, pp. 265–278, 1994.Google Scholar
  36. 36.
    Schaefer, H. H., Topological Vector Spaces, Springer Verlag, New York, NY, 1971.Google Scholar
  37. 37.
    Isac, G., Pareto Optimization in Infinite-Dimensional Spaces: The Importance of Nuclear Cones, 4th SIAM Conference on Optimization, Chicago, Illinois, 1992.Google Scholar
  38. 38.
    Bakhtin, I. A., Cones in Banach Spaces, Voronezh State Pedagogical Institute, Voronezh, Russia, 1977 (in Russian).Google Scholar
  39. 39.
    Sterna-Karwat, A., On the Existence of Cone-Maximal Points in Real Topological Linear Spaces, Israel Journal of Mathematics, Vol. 54, pp. 34–41, 1986.Google Scholar
  40. 40.
    Truong, X. D. H., A Note on a Class of Cones Ensuring the Existence of Efficient Points in Bounded Complete Sets, Optimization, Vol. 31, pp. 141–152, 1994.Google Scholar
  41. 41.
    Guerraggio, A., Unpublished Manuscript, 1999.Google Scholar
  42. 42.
    Boissard, N., Superefficient Solutions and Superinfima in Vector Optimization, Journal of Mathematical Analysis and Applications, Vol. 228, pp. 37–50, 1998.Google Scholar
  43. 43.
    Isac, G., Sur les Points a Support Conique dans les Espaces Localement Convexes, Annals of the Faculty of Sciences of Kinshasa, Zair University, Vol. 3, pp. 281–291, 1977.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • X. D. H. Truong
    • 1
  1. 1.Hanoi Institute of MathematicsBoho, HanoiVietnam

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