Using a set-valued dual cost function we give a new approach to duality theory for linear vector optimization problems. We develop the theory very close to the scalar case. Especially, in contrast to known results, we avoid the appearance of a duality gap in case of b = 0. Examples are given.

Contact details are reproduced from the original publication and may be historical.

Andreas H. Hamel

Martin-Luther-University Halle-Wittenberg, Dept. of Mathematics and Computer Science,
Theodor-Lieser-Str. 5, 06099 Halle, Germany

hamel@mathematik.uni-halle.de

Frank Heyde

Martin-Luther-University Halle-Wittenberg, Dept. of Mathematics and Computer Science,
Theodor-Lieser-Str. 5, 06099 Halle, Germany

Andreas Löhne

Martin-Luther-University Halle-Wittenberg, Dept. of Mathematics and Computer Science,
Theodor-Lieser-Str. 5, 06099 Halle, Germany

Christiane Tammer

Martin-Luther-University Halle-Wittenberg, Dept. of Mathematics and Computer Science,
Theodor-Lieser-Str. 5, 06099 Halle, Germany

Kristin Winkler

Martin-Luther-University Halle-Wittenberg, Dept. of Mathematics and Computer Science,
Theodor-Lieser-Str. 5, 06099 Halle, Germany

  1. S. Brumelle: Duality for multiple objective convex programs, Math. Oper. Res. 6(2) (1981) 159–172.https://doi.org/10.1287/moor.6.2.159Google Scholar
  2. A. Balbas, A. Heras, P. Jiminez: Duality theory and slackness conditions in multiobjective linear programming, Comp. Math. Appl. 37 (1999) 101–109.https://doi.org/10.1016/S0898-1221(99)00062-0Google Scholar
  3. H. W. Corley: Existence and Lagrangian duality for maximization of set-valued functions, J. Optimization Theory Appl. 54(3) (1987) 489–501.https://doi.org/10.1007/BF00940198Google Scholar
  4. H. W. Corley: Duality theory for the matrix linear programming problem, J. Math. Anal. Appl. 104 (1984) 47–52.https://doi.org/10.1016/0022-247X(84)90028-3Google Scholar
  5. G. B. Dantzig: Linear Programming and Extensions, A RAND Corporation Research Study, Princeton University Press. XVI, Princeton, N.J. (1963).https://doi.org/10.1515/9781400884179Google Scholar
  6. M. Ehrgott: Multicriteria Optimization, Lect. Notes in Econ. Math. Systems 491, Springer, Berlin (2000).https://doi.org/10.1007/978-3-662-22199-0Google Scholar
  7. J. Focke: Vektormaximumproblem und parametrische Optimierung, Math. Operationsforsch. Statistik 4(5) (1973) 365–369.https://doi.org/10.1080/02331887308801134Google Scholar
  8. A. Göpfert, R. Nehse: Vektoroptimierung, BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1990).Google Scholar
  9. A. Göpfert, H. Riahi, C. Tammer, C. Zalinescu: Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics 17, Springer, New York (2003).Google Scholar
  10. H. Isermann: On some relations between a dual pair of multiple objective linear programs, Z. Oper. Res., Ser. A 22 (1978) 33–41.https://doi.org/10.1007/BF01917642Google Scholar
  11. H. Isermann: Duality in multiple objective linear programming, in: Multiple Criteria Probl. Solving, Proc. Conf. Buffalo 1977, Lect. Notes in Econ. Math. Systems 155 (1978) 274–285.https://doi.org/10.1007/978-3-642-46368-6_12Google Scholar
  12. E. H. Ivanov, R. Nehse: Some results on dual vector optimization problems, Optimization 16(4) (1985) 507–517.https://doi.org/10.1080/02331938508843043Google Scholar
  13. J. S. H. Kornbluth: Duality, indifference and sensitivity analysis in multiple objective linear programming, Oper. Res. Quart. 25 (1974) 599–614.https://doi.org/10.1057/jors.1974.108Google Scholar
  14. D. T. Luc: Theory of Vector Optimization, Lect. Notes in Econ. Math. Systems 319, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-50280-4Google Scholar
  15. J. Jahn: Mathematical Vector Optimization in Partially Ordered Linear Spaces, Methoden und Verfahren der Mathematischen Physik 31, Verlag Peter Lang, Frankfurt a. M. (1986).Google Scholar
  16. J. Jahn: Set-valued optimization. A survey, Report No. 264 of Institute of Applied Mathematics, University Erlangen-Nürnberg, Germany (2000).Google Scholar
  17. C. R. Rao, S. K. Mitra: Generalized Inverse of Matrices and its Applications, Wiley & Sons, New York (1971).Google Scholar
  18. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton, N.J. (1970).Google Scholar
  19. W. Rödder: A generalized saddlepoint theory. Its application to duality theory for linear vector optimum problems, Eur. J. Oper. Res. 1 (1977) 55–59.https://doi.org/10.1016/S0377-2217(77)81008-4Google Scholar
  20. W. Vogel: Lineares Optimieren, Akademische Verlagsgesellschaft, Leipzig (1970).Google Scholar

A. H. Hamel, F. Heyde, A. Löhne, Ch. Tammer, K. Winkler. “Closing the Duality Gap in Linear Vector Optimization.” Journal of Convex Analysis 11 (2004), No. 1, 163–178. https://doi.org/10.68381/jca11011