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Contractibility of the efficient frontier of three-dimensional simply-shaded sets. (English) Zbl 1054.90065
Summary: The aim of this paper is to study the geometrical and topological structure of the efficient frontier of simply-shaded sets in a three-dimensional Euclidean space with respect to the usual positive cone. Our main result concerns the contractibility of the efficient frontier and refines a recent result of A. Danilidis, N. Hadjisavvas, and S. Schaible [J. Optimization Theory Appl. 93, No. 3, 517–524 (1997; Zbl 0879.90159)] regarding the connectedness of the efficient outcome set for three-criteria optimization problems involving continuous semistrictly quasiconcave objective functions.

90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization
Full Text: DOI
[1] Daniilidis, A., Hadjisavvas, N., and Schaible, S., Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems, Journal of Optimization Theory and Applications, Vol. 93, pp. 517–524, 1997. · Zbl 0879.90159
[2] Luc, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 319, 1989.
[3] Benoist, J., Connectedness of the Efficient Set for Strictly Quasiconcave Sets, Journal of Optimization Theory and Applications, Vol. 96, pp. 627–654, 1998. · Zbl 0907.90227
[4] Debreu, G., Theory of Value, John Wiley, New York, NY, 1959. · Zbl 0193.20205
[5] Stoer, J., and Witzgall, C., Convexity and Optimization in Finite Dimensions, I, Springer Verlag, Berlin, Germany, 1970. · Zbl 0203.52203
[6] Bredon, E. G., Topology and Geometry, Graduate Texts in Mathematics, Springer Verlag, New York, NY, Vol. 139, 1993.
[7] Bonnisseau, J. M., and Cornet, B., Existence of Equilibria When Firms Follow Bounded Losses Pricing Rule, Journal of Mathematical Economics, Vol. 17, pp. 119–147, 1988. · Zbl 0657.90014
[8] Benoist, J., and Popovici, N., The Structure of the Efficient Frontier of Finite-Dimensional Completely-Shaded Sets, Journal of Mathematical Analysis and Applications, Vol. 250, pp. 98–117, 2000. · Zbl 1009.90104
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