American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[2]

R. I. Boţ, S. M. Grad and G. Wanka, "Duality in Vector Optimization," Springer-Verlag, Berlin-Heidelberg, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Jahn, Duality in vector optimization, Mathematical Programming, 25 (1983), 343-353. doi: 10.1007/BF02594784.  Google Scholar

[7]

J. Jahn, "Vector Optimization - Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004.  Google Scholar

[8]

R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.  Google Scholar

[9]

C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

R. I. Boţ, S. M. Grad and G. Wanka, "Duality in Vector Optimization," Springer-Verlag, Berlin-Heidelberg, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Jahn, Duality in vector optimization, Mathematical Programming, 25 (1983), 343-353. doi: 10.1007/BF02594784.  Google Scholar

[7]

J. Jahn, "Vector Optimization - Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004.  Google Scholar

[8]

R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.  Google Scholar

[9]

C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002.  Google Scholar

[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.  Google Scholar

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