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The Attainment of the Solution of the Dual Program in Vertices for Vectorial Linear Programs

In: Multiobjective Programming and Goal Programming

Author

Listed:
  • Frank Heyde
  • Andreas Löhne
  • Christiane Tammer

    (Martin-Luther-University Halle Halle-Wittenberg)

Abstract

This article is a continuation of Löhne A, Tammer C (2007: A new approach to duality in vector optimization. Optimization 56(1–2):221–239) [14]. We developed in [14] a duality theory for convex vector optimization problems, which is different from other approaches in the literature. The main idea is to embed the image space Rq of the objective function into an appropriate complete lattice, which is a subset of the power set of Rq. This leads to a duality theory which is very analogous to that of scalar convex problems. We applied these results to linear problems and showed duality assertions. However, in [14] we could not answer the question, whether the supremum of the dual linear program is attained in vertices of the dual feasible set. We show in this paper that this is, in general, not true but, it is true under additional assumptions.

Suggested Citation

  • Frank Heyde & Andreas Löhne & Christiane Tammer, 2009. "The Attainment of the Solution of the Dual Program in Vertices for Vectorial Linear Programs," Lecture Notes in Economics and Mathematical Systems, in: Vincent Barichard & Matthias Ehrgott & Xavier Gandibleux & Vincent T'Kindt (ed.), Multiobjective Programming and Goal Programming, pages 13-24, Springer.
  • Handle: RePEc:spr:lnechp:978-3-540-85646-7_2
    DOI: 10.1007/978-3-540-85646-7_2
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