Abstract
With this note we bring again into attention a vector dual problem neglected by the contributions who have recently announced the successful healing of the trouble encountered by the classical duals to the classical linear vector optimization problem. This vector dual problem has, different to the mentioned works which are of set-valued nature, a vector objective function. Weak, strong and converse duality for this “new-old” vector dual problem are proven and we also investigate its connections to other vector duals considered in the same framework in the literature. We also show that the efficient solutions of the classical linear vector optimization problem coincide with its properly efficient solutions (in any sense) when the image space is partially ordered by a nontrivial pointed closed convex cone, too.
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References
Boţ R.I., Grad S.-M., Wanka G.: Duality in Vector optimization. Springer, Berlin (2009)
Boţ R.I., Wanka G.: An analysis of some dual problems in multiobjective optimization (I). Optimization 53(3), 281–300 (2004)
Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154(1), 29–50 (2007)
Chinchuluun A., Pardalos P.M., Migdalas A., Pitsoulis L.: Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)
Focke J.: Vektormaximumproblem und parametrische Optimierung. Mathematische Operationsforschung und Statistik 4(5), 365–369 (1973)
Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Springer, Berlin (2002)
Hamel A.H., Heyde F., Löhne A., Tammer C., Winkler K.: Closing the duality gap in linear vector optimization. J. Convex Analy. 11(1), 163–178 (2004)
Heyde F., Löhne A., Tammer C.: Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Math. Methods Oper. Res. 69(1), 159–179 (2009)
Heyde F., Löhne A., Tammer C.: The attainment of the solution of the dual program in vertices for vectorial linear programs. In: Barichard, V., Ehrgott, M., Gandibleux, X., T’Kindt, V. (eds) Multiobjective Programming and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol. 618, pp. 13–24. Springer, Berlin (2009)
Isermann H.: Duality in multiple objective linear programming. In: Zionts, S. (ed.) Multiple Criteria Problem Solving. Lecture Notes in Economics and Mathematical Systems, vol. 155, pp. 274–285. Springer, Berlin (1978)
Isermann H.: On some relations between a dual pair of multiple objective linear programs. Zeitschrift für Operations Research 22(1), A33–A41 (1978)
Isermann H.: Proper efficiency and the linear vector maximum problem. Oper. Res. 22(1), 189–191 (1974)
Ivanov E.H., Nehse R.: Some results on dual vector optimization problems. Optimization 16(4), 505–517 (1985)
Jahn J.: Duality in vector optimization. Mathematical Programming 25(3), 343–353 (1983)
Jahn J.: Vector Optimization—Theory, Applications, and Extensions. Springer, Berlin (2004)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Wanka G., Boţ R.I.: A new duality approach for multiobjective convex optimization problems. J. Nonlinear Convex Anal. 3(1), 41–57 (2002)
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Research partially supported by DFG (German Research Foundation), project WA 922/1-3.
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Boţ, R.I., Grad, SM. & Wanka, G. Classical linear vector optimization duality revisited. Optim Lett 6, 199–210 (2012). https://doi.org/10.1007/s11590-010-0263-1
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DOI: https://doi.org/10.1007/s11590-010-0263-1