Abstract
In connection with the remarkable and rapid development of linear and nonlinear vector optimization, an increasing number of papers were devoted to the construction and investigation of dual problems. Several concepts of duality were treated, especially Lagrange duality and Fenchel duality (cf. Elster /1/, Elster/Iwanow /3/, Jahn /8/). Apart of linear problems we can find extensive investigations of convex problems and more and more of nonconvex problems. In the last case some difficulties can arise concerning p.e. separation theorems which are sufficiently general. In the present paper a direct duality theorem is derived for rather general (nonconvex) vector optimization problems. For the proof a separation theorem is needed where the separating functional is nonlinear (more precisely: convex). The results are formulated in finite dimensional spaces, but an extension to infinite dimensions is possible.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
ELSTER, K.-H.: Neuere Entwicklungen der Vektoroptimierung. X. Internat. Kongreß über Anwendungen der Math. i. den Ingenieurwissenschaften, Weimar 1984. Berichte Heft 4, S. 63–66.
ELSTER, K.-H./ELSTER, R.: Relationships between posynomial programming and vector optimization (To appear).
ELSTER, K.-H./IWANOW, E.: Über Fortschritte auf dem Gebiet der mehrkriteriellen Entscheidungsprobleme. Wiss. Zeitschr. TH Ilmenau 31 (1985) 2, 15–36.
ESTER, J./SCHWARTZ, B.: Ein erweitertes Effizienztheorem. Math. Operationsforschung u. Statist., ser. optimization 14 (1983), 331–342.
GERSTEWITZ, C./IWANOW, E.: Dualität für nichtkonvexe Vektor-Optimierungsprobleme. Wiss. Zeitschr. TH Ilmenau 31 (1985) 2, 61–81.
GERTH, C./WEIDNER, P.: Nonconvex separation theorems and some applications to the vector optimization (To appear in JOTA).
GÜPFERT, A.: Multicriteria duality, examples and advances: Lecture Notes in Econ. and Math. Systems Nr. 273, Springer Verlag 1986, 52–53.
JAHN, J.: Mathematical Vector Optimization in Partially Ordered Linear Spaces. Meth. u. Verf. der math. Physik Bd. 31. Vlg. Peter Lang, Frankfurt/M. (1986).
KLÜTZLER, R.: Dualität bei diskreten Steuerproblemen. Math. Operationsforsch. u. Statist., ser. optimization 12 (1931) 3, 411–420.
WEIDNER, P.: Charakterisierung von Mengen effizienter Elemente in linearen Räumen auf der Grundlage allgemeiner Bedingungen. Diss. A, MLU Halle-Wittenberg (1995).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Elster, KH., Göpfert, A. (1987). Recent Results on Duality in Vector Optimization. In: Jahn, J., Krabs, W. (eds) Recent Advances and Historical Development of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46618-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-46618-2_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-18215-3
Online ISBN: 978-3-642-46618-2
eBook Packages: Springer Book Archive