# Density theorems for generalized Henig proper efficiency

- 69 Downloads
- 10 Citations

## Abstract

We develop a new, simple technique of proof for density theorems (i.e.,for the sufficient conditions to guarantee that the proper efficient points of a set are dense in the efficient frontier) in an ordered topological vector space. The results are the following: (i) the set of proper efficient points of any compact set*Q* is dense in the set of efficient points with respect to the original topology of the space whenever the ordering cone*K* is weakly closed and admits strictly positive functionals; moreover, if*K* is not weakly closed, then there exists a compact set for which the density statement fails; (ii) if*Q* is weakly compact, then we have only weak density, but if*K* has a closed bounded base, then we can assert the density with respect to the original topology, (iii) there exists a similar possibility to assert the strong density for weakly compact*Q* if additional restrictions are placed on*Q* instead of*K*. These three results are obtained in a unified way as corollaries of the same statement. In this paper, we use the concept of proper efficiency due to Henig. We extend his definition to the setting of a Hausdorff topological vector space.

## Key Words

Vector optimization positive proper efficiency Henig proper efficiency density theorem## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Kuhn, H. W., andTucker, A. W.,
*Nonlinear Programming*, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Edited by J. Neyman, University of California Press, Berkeley, California, pp. 481–492, 1951.Google Scholar - 2.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 - 3.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 - 4.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 - 5.Hartley, R.,
*On Cone Efficiency, Cone Convexity, and Cone Compactness*, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.Google Scholar - 6.Henig, M. I.,
*Proper Efficiency with Respect to Cones*, Journal of Optimization Theory and Applications, Vol. 36, pp. 387–407, 1982.Google Scholar - 7.Gorokhovik, V. V., andRachkovski, N. N.,
*Proper Minimality in Preordered Vector Spaces*, Vestsi Akademii Navuk Belarusi, Seriya Fizika-Matematychnykh Nauk, No. 4, pp. 10–15, 1993 (in Russian).Google Scholar - 8.Arrow, K. J., Barankin, E. W., andBlackwell, D.,
*Admissible Points of Convex Sets*, Contribution to the Theory of Games, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, pp. 87–92, 1953.Google Scholar - 9.Borwein, J. M.,
*On the Existence of Pareto Efficient Points*, Mathematics of Operations Research, Vol. 8, pp. 64–73, 1983.Google Scholar - 10.Borwein, J. M., andZhuang, D.,
*Super Efficiency in Vector Optimization*, Transactions of the American Mathematical Society, Vol. 338, pp. 105–122, 1993.Google Scholar - 11.Zhuang, D.,
*Density Results for Propereffciencies*, SIAM Journal on Control and Optimization, Vol. 32, pp.51–58, 1994.Google Scholar - 12.Dauer, G. P., andGallagher, R. J.,
*Positive Proper Efficient Points and Related Cone Results in Vector Optimization Theory*, SIAM Journal on Control and Optimization, Vol. 28, pp. 158–172, 1990.Google Scholar - 13.Rander, R.,
*A Note on Maximal Points of Convex Sets in l*^{∞}, Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, pp. 351–354, 1951.Google Scholar - 14.Chichilnisky, G., andKalman, 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 - 15.Salz, W.,
*Eine topologische Eigenschaft der Effizienten Punkte konvexer Mengen*, Operations Research Verfahren, Vol. 23, pp. 197–202, 1976.Google Scholar - 16.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 - 17.Petschke, M.,
*On a Theorem of Arrow, Barankin, and Blackwell*, SIAM Journal on Control and Optimization, Vol. 28, pp. 395–401, 1990.Google Scholar - 18.Ferro, F., General Form of the Arrow-Barankin-Blackwell Theorem in Normed Spaces and the l
^{∞}-Case, Journal of Optimization Theory and Applications, Vol. 79, pp. 127–138, 1993.Google Scholar - 19.Gallagher, R. J., andSaleh, O. A.,
*Two Generalizations of a Theorem of Arrow, Barakin, and Blackwell*, SIAM Journal on Control and Optimization, Vol. 31, pp. 247–256, 1993.Google Scholar - 20.Chen, G. Y.,
*Generalized Arrow-Barakin-Blackwell Theorems in Locally Convex Spaces*, Journal of Optimization Theory and Applications, Vol. 84, pp. 93–101, 1995.Google Scholar - 21.
- 22.Borwein, J. M.,
*The Geometry of Pareto Efficiency over Cones*, Mathematische Operationsforschung und Statistik, Serie Optimization, Vol. 11, pp. 235–248, 1980.Google Scholar - 23.
- 24.Bourbaki, N.,
*Espaces Vectoriels Topologiques: Éléments de Mathématique*, Hermann & C^{ie}, Paris, France, Vol. 15,1953.Google Scholar - 25.Jameson, G.,
*Ordered Linear Spaces*, Lecture Notes in Mathematics, Springer Verlag, New York, New York, Vol. 141, 1970.Google Scholar - 26.
- 27.Krasnoselski, M. A.,
*Positive Solutions of Operator Equations*, Fizmatgiz, Moscow, Russia, 1962 (in Russian).Google Scholar - 28.