Abstract
Using the technique of space theory and set-valued analysis, we establish contractibility results for efficient point sets in a locally convex space and a path connectedness result for a positive proper efficient point set in a reflexive space. We also prove a connectedness result for a positive proper efficient point set in a locally convex space; as an application, we give a connectedness result for an efficient solution set in a locally convex space.
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References
Luc, D. T., Contractibility of Efficient Point Sets in Normed Spaces, Nonlinear Analysis, Vol. 15, pp. 527–535, 1990.
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.
Choo, E. U., and Atkins, D. R., Connectedness in Multiple Linear Fractional Programming, Management Science, Vol. 29, pp. 250–255, 1983.
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.
Luc, D. T., Structure of the Efficient Point Set, Proceedings of the American Mathematical Society, Vol. 95, pp. 433–440, 1985.
Luc, D. T., Connectedness of the Efficient Point Sets in Quasiconcave Maximization, Journal of Mathematical Analysis and Applications, Vol. 122, pp. 346–354, 1987.
Naccacche, P. H., Connectedness of the Set of Nondominated Outcomes in Multicriteria Optimization, Journal of Optimization Theory and Applications, Vol. 25, pp. 459–467, 1978.
Peleg, B., Topological Properties of the Efficient Point Set, Proceedings of the American Mathematical Society, Vol. 35, pp. 531–536, 1972.
Schecter, S., Structure of the Demand Function and Pareto-Optimal Set with Natural Boundary Conditions, Journal of Mathematical Economics, Vol. 5, pp. 1–21, 1978.
Smale, S., Global Analysis and Economics, Part 5: Pareto Theory with Constraints, Journal of Mathematical Economics, Vol. 1, pp. 213–222, 1972.
Warburton, A. R., Quasiconcave Vector Maximization: Connectedness of the Set of Pareto-Optimal and Weak Pareto-Optimal Alternatives, Journal of Optimization Theory and Applications, Vol. 40, pp. 537–557, 1983.
Phelps, R. R., Convex Functions, Monotone Operators, and Differentiability, Lectures Notes in Mathematics, Springer Verlag, New York, NY, Vol. 1364, 1989.
Berge, C., Topological Spaces, Oliver and Boyd, Edinburgh, Scotland, 1963.
Borwein, J. M., and Zhuang, D., Super Efficiency in Vector Optimization, Transactions of the American Mathematical Society, Vol. 338, pp. 105–122, 1993.
Zheng, X. Y., Generalizations of a Theorem of Arrow, Barankin, and Blackwell in Topological Vector Spaces, Journal of Optimization Theory and Applications, Vol. 96, pp. 221–233, 1998.
Aubin, J. P., and Ekeland, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, NY, 1984.
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Zheng, X.Y. Contractibility and Connectedness of Efficient Point Sets. Journal of Optimization Theory and Applications 104, 717–737 (2000). https://doi.org/10.1023/A:1004649928081
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DOI: https://doi.org/10.1023/A:1004649928081