Abstract
It is proved that the density theorem of Arrow, Barankin, and Blackwell holds in a topological vector space equipped with a weakly closed convex cone to admit strictly positive continuous linear functionals. Moreover, several local versions of the Arrow, Barankin, and Blackwell theorem are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Guerraggio, H., 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.
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. 87–91, 1953.
Hartley, R., On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.
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.
Radner, R., A Note on Maximal Points of Convex Sets in l ∞, Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Edited by L. M. LeCam and J. Neyman, University of California Press, Berkeley, California, Vol. 1, pp. 351–354, 1965.
Majumder, M., Some General Theorems on Efficiency Prices with an Infinite-Dimensional Commodity Space, Journal of Economic Theory, Vol. 5, pp. 1–13, 1972.
Peleg, B., Efficiency Prices for Optimal Consumption Plans, Part 2, Israel Journal of Mathematics, Vol. 9, pp. 222–234, 1971.
Borwein, J. M., The Geometry of Pareto Efficiency over Cones, Mathematische Operationsforshung und Statistik, Series Optimization, Vol. 11, pp. 235–248, 1980.
Jahn, J., A Generalization of a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 26, pp. 999–1005, 1988.
Petschke, M., On a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 28, pp. 395–401, 1990.
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. 247–258, 1993.
Dauer, J. P., and Gallagher, R. J., Positive Efficient Points and Related Cone Results in Vector Optimization Theorem, SIAM Journal on Control and Optimization, Vol. 28, pp. 158–172, 1990.
Fan, K., Minimax Theorems, Proceedings of the National Academy of Sciences of the USA, Vol. 39, pp. 42–47, 1953.
Rudin, W., Functional Analysis, McGraw-Hill, New York, New York, 1973.
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. 127–238, 1993.
Gong, X. H., Connectedness of Efficient Solution Sets for Set-Valued Maps in Normed Spaces, Journal of Optimization Theory and Applications, Vol. 83, pp. 83–96, 1994.
Chen, G. Y., Generalized Arrow-Barankin-Blackwell Theorems in Locally Convex Spaces, Journal of Optimization Theory and Applications, vol. 84, pp. 93–101, 1995.
Gallangher, R. J., The Arrow-Barankin-Blackwell Theorem in a Dual Space Setting, Journal of Optimization Theory and Applications, Vol. 84, pp. 665–674, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zheng, X.Y. Generalizations of a Theorem of Arrow, Barankin, and Blackwell in Topological Vector Spaces. Journal of Optimization Theory and Applications 96, 221–233 (1998). https://doi.org/10.1023/A:1022631621188
Issue Date:
DOI: https://doi.org/10.1023/A:1022631621188