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Nonconvex separation theorems and some applications in vector optimization. (English) Zbl 0692.90063

Separation theorems for an arbitrary set and a not necessarily convex set in a linear topological space are proved and applied to vector optimization. Scalarization results for weakly efficient points and properly efficient points are deduced.

MSC:

90C29 Multi-objective and goal programming
46A20 Duality theory for topological vector spaces
90C48 Programming in abstract spaces
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[1] Nehse, R.,A New Concept of Separation, Commentationes Mathematicae Universitatis Carolinae, Vol. 22, pp. 169-179, 1981. · Zbl 0518.46005
[2] Hildenbrandt, R.,Trennung von Mengen und Dualität Nichtkonvexer Optimierungsprobleme, Technische Hochschule Ilmenau, Dissertation A, 1982.
[3] Jahn, J.,Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203-218, 1984. · Zbl 0539.90093 · doi:10.1007/BF02592221
[4] Weidner, P.,Dominanzmengen und Optimalitätsbegriffe in der Vektoroptimierung, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau, Vol. 31, pp. 133-146, 1985. · Zbl 0582.90086
[5] Weidner, P.,Charakterisierung von Mengen Effizienter Elemente in Linearen Räumen auf der Grundlage Allgemeiner Bezugsmengen, Martin-Luther-Universität Halle-Wittenberg, Dissertation A, 1985.
[6] Henig, M. I.,Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 36, pp. 387-407, 1982. · Zbl 0452.90073 · doi:10.1007/BF00934353
[7] Lampe, U.,Dualität und Eigentliche Effizienz in der Vektoroptimierung, Seminarberichte der Sektion Mathematik der Humboldt-Universität zu Berlin, No. 57, pp. 45-54, 1981. · Zbl 0472.90061
[8] Jahn, J.,Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt am Main, Germany, 1986. · Zbl 0578.90048
[9] Kantorowitsch, L. W., andAkilow, G. P.,Funktionalanalysis in Normierten Räumen, Akademie-Verlag, Berlin, Germany, 1978. · Zbl 0359.46017
[10] Gerstewitz, C., andIwanow, E.,Dualität für Nichtkonvexe Vektoroptimierungsprobleme, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau, Vol. 31, pp. 61-81, 1985. · Zbl 0562.90091
[11] Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer, New York, New York, 1975. · Zbl 0336.46001
[12] Gerstewitz, C.,Nichtkonvexe Trennungssätze und Deren Anwendung in der Theorie der Vektoroptimierung, Seminarberichte der Sektion Mathematik der Humboldt-Universität zu Berlin, No. 80, pp. 19-31, 1986. · Zbl 0609.49005
[13] Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319-377, 1974. · Zbl 0268.90057 · doi:10.1007/BF00932614
[14] Weidner, P.,Extensions of the Feasible Range Leaving Invariant the Efficient Point Set, Seminarberichte der Sektion Mathematik der Humboldt-Universität zu Berlin, No. 85, pp. 137-147, 1986. · Zbl 0632.90071
[15] Weidner, P.,On the Characterization of Efficient Points by Means of Monotone Functionals, Optimization, Vol. 19, pp. 53-69, 1988. · Zbl 0659.90079 · doi:10.1080/02331938808843317
[16] Weidner, P.,Monotone Funktionale, Dualkegel und Effiziente Elemente, Seminarberichte der Sektion Mathematik der Humboldt-Universität zu Berlin, No. 80, pp. 110-120, 1986.
[17] Weidner, P.,The Influence of Neglecting Feasible Points and of Varying Preferences on the Efficient Point Set, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau, Vol. 33, pp. 181-188, 1987. · Zbl 0635.90087
[18] Weidner, P.,Complete Efficiency and Interdependencies between Objective Functions in Vector Optimization, Zeitschrift für Operations Research, Vol. 34, pp. 91-115, 1990. · Zbl 0694.90093
[19] Bernau, H.,Interactive Methods for Vector Optimization, Optimization in Mathematical Physics, Edited by B. Brosowski and E. Martensen, Peter Lang, Frankfurt am Main, Germany, pp. 21-36, 1987. · Zbl 0609.90100
[20] Brosowski, B.,A Criterion for Efficiency and Some Applications, Optimization in Mathematical Physics, Edited by B. Brosowski and E. Martensen, Peter Lang, Frankfurt am Main, Germany, pp. 37-59, 1987.
[21] 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. · Zbl 0389.52021 · doi:10.1007/BF00934453
[22] Schönfeld, P.,Some Duality Theorems for the Nonlinear Vector Maximum Problem, Unternehmensforschung, Vol. 14, pp. 51-63, 1970. · Zbl 0194.20401 · doi:10.1007/BF01918249
[23] Elster, K. H., andGöpfert, A.,Recent Results on Duality in Vector Optimization, Recent Advances and Historical Development of Vector Optimization, Edited by J. Jahn and W. Krabs, Springer, Berlin, Germany, pp. 129-136, 1987.
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