On linear vector optimization duality in infinite-dimensional spaces

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2011, 1(3): 407-415. doi: 10.3934/naco.2011.1.407

On linear vector optimization duality in infinite-dimensional spaces

Radu Ioan Boţ ^1, and Sorin-Mihai Grad ^1,

1.	Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany, Germany

Received April 2011 Revised July 2011 Published September 2011

Abstract
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In this paper we extend to infinite-dimensional spaces a vector duality concept recently considered in the literature in connection to the classical vector minimization linear optimization problem in a finite-dimensional framework. Weak, strong and converse duality for the vector dual problem introduced with this respect are proven and we also investigate its connections to some classical vector duals considered in the same framework in the literature.

Keywords: vector optimization., Linear vector duality, cones.

Mathematics Subject Classification: Primary: 90C05; Secondary: 90C25, 90C2.

Citation: Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407

References:

[1]	R. I. Boţ, S. M. Grad and G. Wanka, Classical linear vector optimization duality revisited, Optimization Letters, DOI: 10.1007/s11590-010-0263-1. doi: 10.1007/s11590-010-0263-1.
[2]	R. I. Boţ, S. M. Grad and G. Wanka, "Duality in Vector Optimization," Springer-Verlag, Berlin-Heidelberg, 2009.
[3]	R. I. Boţ and G. Wanka, An analysis of some dual problems in multiobjective optimization (I), Optimization, 53 (2004), 281-300. doi: 10.1080/02331930410001715514.
[4]	A. Guerraggio, E. Molho and A. Zaffaroni, On the notion of proper efficiency in vector optimization, Journal of Optimization Theory and Applications, 82 (1994), 1-21. doi: 10.1007/BF02191776.
[5]	A. H. Hamel, F. Heyde, A. Löhne, C. Tammer and K. Winkler, Closing the duality gap in linear vector optimization, Journal of Convex Analysis, 11 (2004), 163-178.
[6]	J. Jahn, Duality in vector optimization, Mathematical Programming, 25 (1983), 343-353. doi: 10.1007/BF02594784.
[7]	J. Jahn, "Vector Optimization - Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004.
[8]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.
[9]	C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002.
[10]	C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in "Nonsmooth Optimization and Related Topics (Erice 1988), " Plenum, New York, (1988), 437-458.

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References:

[1]	R. I. Boţ, S. M. Grad and G. Wanka, Classical linear vector optimization duality revisited, Optimization Letters, DOI: 10.1007/s11590-010-0263-1. doi: 10.1007/s11590-010-0263-1.
[2]	R. I. Boţ, S. M. Grad and G. Wanka, "Duality in Vector Optimization," Springer-Verlag, Berlin-Heidelberg, 2009.
[3]	R. I. Boţ and G. Wanka, An analysis of some dual problems in multiobjective optimization (I), Optimization, 53 (2004), 281-300. doi: 10.1080/02331930410001715514.
[4]	A. Guerraggio, E. Molho and A. Zaffaroni, On the notion of proper efficiency in vector optimization, Journal of Optimization Theory and Applications, 82 (1994), 1-21. doi: 10.1007/BF02191776.
[5]	A. H. Hamel, F. Heyde, A. Löhne, C. Tammer and K. Winkler, Closing the duality gap in linear vector optimization, Journal of Convex Analysis, 11 (2004), 163-178.
[6]	J. Jahn, Duality in vector optimization, Mathematical Programming, 25 (1983), 343-353. doi: 10.1007/BF02594784.
[7]	J. Jahn, "Vector Optimization - Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004.
[8]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.
[9]	C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002.
[10]	C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in "Nonsmooth Optimization and Related Topics (Erice 1988), " Plenum, New York, (1988), 437-458.

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