Abstract
We investigate scalarizations for the determination of Pareto optima of multicriteria optimization problems which deliver properly efficient points in the sense of Geoffrion. Proper efficiency in the sense of Schönfeld is generalized by the simultaneous consideration of several weighted sums of the objective functions. This problem is geometrically interpreted by means of a polyhedral cone. The specialization of the parameters induces an extension of the weighted Chebyshev norm minimization for which we prove conditions for the existence of optimal solutions and statements confirming known more special results.
Zusammenfassung
Wir untersuchen solche skalare Ersatzaufgaben zur Bestimmung der Pareto-Optima mehrkriterieller Optimierungsprobleme, deren Lösungen eigentlich effizient im Sinne von Geoffrion sind. Dabei wird die eigentliche Effizienz im Sinne von Schönfeld durch die Berücksichtigung unterschiedlich gewichteter Summen der Zielfunktionen verallgemeinert und dieses Problem mit Hilfe eines polyedrischen Kegels geometrisch interpretiert. Eine geeignete Spezialisierung der Parameter induziert eine Erweiterung der Minimierung der gewichteten Tschebyscheff-Norm. Hierfür beweisen wir Bedingungen für die Existenz von Optimallösungen und Ergebnisse, die aus der Literatur bekannte Aussagen über speziellere Skalarisierungen beinhalten.
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Weidner, P. The influence of proper efficiency on optimal solutions of scalarizing problems in multicriteria optimization. OR Spektrum 16, 255–260 (1994). https://doi.org/10.1007/BF01720318
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DOI: https://doi.org/10.1007/BF01720318