Abstract
In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X →Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Choo, E. U., andAtkins, D. R.,Bicriteria Linear Fractional Programming, Journal of Optimization Theory and Applications, Vol. 36, pp. 203–220, 1982.
Choo, E. U., andAtkins, D. R.,Connectedness in Multiple Linear Fractional Programming, Management Sciences, Vol. 29, pp. 250–255, 1983.
Warburton, A. R.,Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives, Journal of Optimization Theory and Applications, Vol. 40, pp. 537–557, 1983.
Jameson, G.,Ordered Linear Spaces, Springer-Verlag, Berlin, Germany, 1970.
Vogel, W.,Vektoroptimierung in Produkträumen, Anton Hain, Meisenheim am Glan, Germany, 1977.
Jahn, J.,Mathematical Vector Optimization in Partially-Ordered Linear Spaces, Peter Lang, Frankfurt/Main, Germany, 1986.
Luc, D. T.,Connectedness of the Efficient Point Set in Quasiconcave Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 133, pp. 346–354, 1987.
Bitran, G. R., andMagnanti, T. L.,The Structure of Admissible Points with Respect to Cone Dominance, Journal of Optimization Theory and Applications, Vol. 29, pp. 573–614, 1979.
Naccache, P. H.,Connectedness of the Set of Nondominated Outcomes in Multicriteria Optimization, Journal of Optimization Theory and Applications, Vol. 25, pp. 459–467, 1978.
Papageorgiou, N. S.,Pareto Efficiency in Locally Convex Spaces, I, Numerical Functional Analysis and Optimization, Vol. 8, pp. 83–116, 1985.
Wernsdorff, R.,On the Connectedness of the Set of Efficient Points in Convex Optimization Problems with Multiple or Random Objectives, Mathematische Operationsforschung und Statistik, Vol. 15, pp. 379–387, 1984.
Luc, D. T.,Structure of the Efficient Point Set, Proceedings of the American Mathematical Society, Vol. 95, pp. 433–440, 1985.
Holmes, R. B.,Geometrical Functional Analysis and Its Applications, Springer, New York, New York, 1975.
Brosowski, B., andConci, A.,On Vector Optimization and Parametric Programming, Segundas Jornadas Latino-Americanas de Matematica Applicada, Vol. 2, pp. 483–495, 1983.
Helbig, S.,Parametric Optimization with a Bottleneck Objective and Vector Optimization, Recent Advances and Historical Development of Vector Optimization, Edited by J. Jahn and W. Krabs, Springer-Verlag, Berlin, Germany, pp. 146–159, 1987.
Helbig, S.,A Scalarization for Vector Optimization Problems in Locally Convex Spaces, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 69, pp. T315-T316, 1989.
Da Silva, A. R.,Infinite Parametrische Optimierung, Doctoral Thesis, University of Frankfurt, Frankfurt, Germany, 1986.
Jahn, J., andSachs, E.,Generalized Quasiconvex Mappings and Vector Optimization, SIAM Journal of Control and Optimization, Vol. 24, pp. 306–322, 1986.
Brosowski, B.,Parametric Semi-Infinite Optimization, Peter Lang, Frankfurt/Main, Germany, 1982.
Robertson, A. P., andRobertson, W.,Topological Vector Spaces, University Press, Cambridge, England, 1964.
Guddat, J.,Parametrische Optimierung und Vektoroptimierung, Anwendungen der Linearen Parametrischen Optimierung, Edited by K. Lommatzsch, Akademie-Verlag, Berlin, Germany, pp. 54–75, 1979.
Krabs, W.,Einführung in die Kontrolltheorie, Wissenschaftliche Buchgesellschaft, Darmstadt, Germany, 1978.
Jahn, J.,Parametric Approximation Problems Arising In Vector Optimization, Journal of Optimization Theory and Applications, Vol. 54, pp. 503–516, 1987.
Author information
Authors and Affiliations
Additional information
Communicated by P. L. Yu
The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.
Rights and permissions
About this article
Cite this article
Helbig, S. On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces. J Optim Theory Appl 65, 257–270 (1990). https://doi.org/10.1007/BF01102345
Issue Date:
DOI: https://doi.org/10.1007/BF01102345