Summary
In this paper the well-knownLegendre transform-type of non-linear duality is extended to the general vector maximum problem. Duality is studied in terms of the primal and dual objective sets rather than in terms of the underlying feasible solutions. The structure of the objective sets is fully explored. The main result in duality is that under reasonable regularity assumptions there are no duality gaps between the primal and dual objective sets.
Zusammenfassung
In dieser Arbeit wird das bekannte, derLegendre-Transformation nachgebildete Konzept der nichtlinearen Dualität auf das allgemeine Vektor-Maximum-Problem übertragen. Hierbei wird die Dualität der Zielmengen (nicht aber der zulässigen Lösungen) des primalen und dualen Programms untersucht. Die Struktur der Zielmengen wird eingehend studiert. Hauptergebnis ist, daß unter plausiblen Regularitätsbedingungen keine „Lücken” zwischen der primalen und der dualen Zielmenge auftreten.
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This paper was originally written when the author was at the Center for Operations Research and Econometrics, University of Louvain. I wish to thank ProfessorsG. de Ghellinck, J. Drèze, of Louvain, andW. Szwarc, of Wrocław, for stimulating discussions of the subject. I alone am responsible for all remaining errors.
Vorgel. v.:W. Wittmann
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Schönfeld, P. Some duality theorems for the non-linear vector maximum problem. Unternehmensforschung Operations Research 14, 51–63 (1970). https://doi.org/10.1007/BF01918249
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DOI: https://doi.org/10.1007/BF01918249