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Proper efficiency with respect to cones

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Abstract

Strict separation by a cone is used here to redefine proper efficiency. Two versions of the properness, which unify and generalize known definitions, are presented. Necessary and sufficient conditions for the existence of the set of properly efficient decisions and characterization of this set in terms of the supports of the decision set are given.

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References

  1. Cochrane, J. L., andZeleny, M., Editors,Multiple Criteria Decision-Making, University of South Carolina Press, Columbia, South Carolina, 1973.

    Google Scholar 

  2. Bitran, G., andMagnanti, T.,The Structure of Admissible Points with Respect to Cone Dominance, Journal of Optimization Theory and Applications, Vol. 29, pp. 573–614, 1979.

    Google Scholar 

  3. Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Second Berkeley Symposium on Mathematical Statistics and Probability, Edited by J. Neyman, University of California Press, Berkeley, California, 1951.

    Google Scholar 

  4. Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 613–630, 1968.

    Google Scholar 

  5. Borwein, J.,Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.

    Google Scholar 

  6. Benson, B.,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 

  7. Benson, H., andMorin, T.,The Vector Maximization Problems: Proper Efficiency and Stability, SIAM Journal on Applied Mathematics, Vol. 32, pp. 64–72, 1977.

    Google Scholar 

  8. Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multi-Objectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.

    Google Scholar 

  9. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  10. Henig, M. I.,A Cone Separation Theorem, Journal of Optimization Theory and Applications, Vol. 36, pp. 451–455, 1982.

    Google Scholar 

  11. Arrow, K. J., Barankin, E. W., andBlackwell, D.,Admissible Points of Convex Sets, Contribution to the Theory of Games, Edited by H. W. Kulan and A. W. Tucker, Princeton University Press, Princeton, New Jersey, 1953.

    Google Scholar 

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

    Google Scholar 

  13. Stoer, J., andWitzgall, L.,Convexity and Optimization in Finite Dimensions, Vol. 1, Springer-Verlag, Berlin, Germany, 1970.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Management, Tel Aviv University, Tel Aviv, Israel

    M. I. Henig (Lecturer)

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  1. M. I. Henig
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Additional information

Communicated by P. L. Yu

The research was done while the author was a visiting professor at the University of British Columbia, Vancouver, British Columbia, Canada.

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Henig, M.I. Proper efficiency with respect to cones. J Optim Theory Appl 36, 387–407 (1982). https://doi.org/10.1007/BF00934353

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  • DOI: https://doi.org/10.1007/BF00934353

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