Skip to main content
Log in

On the notion of proper efficiency in vector optimization

  • Survey Paper
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider the main definitions of proper efficiency for a vector optimization problem in topological linear spaces. The implications among these definitions generalize the inclusion structure holding in Euclidean spaces with componentwise ordering.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Stadler, W.,A Survey of Multicriteria Optimization or the Vector Maximal Problem, Part 1: 1776–1960, Journal of Optimization Theory and Applications, Vol. 29, pp. 1–52, 1979.

    Google Scholar 

  2. Dauer, J. P., andStadler, W.,A Survey of Vector Optimization in Infinite-Dimensional Spaces, Part 2, Journal of Optimization Theory and Applications, Vol. 51, pp. 205–251, 1986.

    Google Scholar 

  3. Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, pp. 481–492, 1951.

  4. Zaffaroni, A.,Proper Efficiency and Nonconical Orderings, Studio Matematico No. 11, Università Bocconi, Milano, Italy, 1991 (in Italian).

    Google Scholar 

  5. Borwein, J. M., andZhuang, D.,Superefficiency in Vector Optimization, Transactions of the American Mathematical Society, Vol. 338, pp. 105–122, 1993.

    Google Scholar 

  6. Guerraggio, A., Molho, E., andZaffaroni, A.,La Nozione di Punto Propriamente Pareto-Efficiente, Technical Paper No. 56, Istituto di Matematica Finanziaria, Università di Torino, Torino, Italy, 1990.

    Google Scholar 

  7. 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 

  8. Jahn, J.,A Characterization of Properly Minimal Elements of a Set, SIAM Journal on Control and Optimization, Vol. 23, pp. 649–656, 1985.

    Google Scholar 

  9. Giorgi, G., andGuerraggio, A.,Approssimazioni Coniche Locali: Proprietà Algebriche e Topologiche, Studio Matematico No. 14, Università Bocconi, Milano, Italy, 1992.

    Google Scholar 

  10. Giorgi, G., andGuerraggio, A.,On the Notion of Tangent Cone, Optimization, Vol. 25, pp 11–23, 1992.

    Google Scholar 

  11. Jahn, J.,Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt am Main, Germany, 1986.

    Google Scholar 

  12. Hurwicz, L.,Programming in Linear Spaces, Studies in Linear and Nonlinear Programming, Edited by K. J. Arrow, L. Hurwicz, and H. Uzawa, Stanford University Press, Stanford, California, 1958.

    Google Scholar 

  13. Dauer, J. P., andGallagher, R. J.,Positive Proper Efficiency and Related Cone Results in Vector Optimization Theory, SIAM Journal on Control and Optimization, Vol. 28, pp. 158–172, 1990.

    Google Scholar 

  14. Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618–630, 1968.

    Google Scholar 

  15. Hartley, R.,On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.

    Google Scholar 

  16. Benson, H. P.,An Improved Definition of Proper Efficiency for Vector Minimization with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 232–241, 1978.

    Google Scholar 

  17. Borwein, J. M.,The Geometry of Pareto Efficiency over Cones, Mathematische Operationsforschung and Statistik, Serie Optimization, Vol. 11, pp. 235–248, 1980.

    Google Scholar 

  18. Henig, M. I.,Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 36, pp. 387–407, 1982.

    Google Scholar 

  19. Hazen, G. B., andMorin, T. L.,Optimality Conditions in Nonconical Multiple-Objective Programming, Journal of Optimization Theory and Applications, Vol. 40, pp. 25–60, 1983.

    Google Scholar 

  20. Klinger, A.,Improper Solutions of the Vector Maximum Problem, Operations Research, Vol. 15, pp. 570–572, 1967.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by F. Giannessi

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guerraggio, A., Molho, E. & Zaffaroni, A. On the notion of proper efficiency in vector optimization. J Optim Theory Appl 82, 1–21 (1994). https://doi.org/10.1007/BF02191776

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02191776

Key Words

Navigation