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Duality Gap in Interval Linear Programming

Abstract

This paper deals with the problem of linear programming with inexact data represented by real intervals. We introduce the concept of duality gap to interval linear programming. We give characterizations of strongly and weakly zero duality gap in interval linear programming and its special case where the matrix of coefficients is real. We show computational complexity of testing weakly- and strongly zero duality gap for commonly used types of interval linear programming.

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Notes

  1. 1.

    Preliminary extended abstract of this paper was presented at conference SOR’17 [4]. That 6-pages version contains some of the theorems from Sects. 2 and 3. Few of them are presented without proof. This full version of the paper heavily extends Sect. 2 as well as it contains an additional Sect. 4 with extensions of Rohn’s Theorem.

  2. 2.

    For interval arithmetic and an introduction to interval analysis; see, e.g., books [7, 8].

  3. 3.

    Do not confuse it with the same abbreviation ILP for integer linear programming.

  4. 4.

    We stick with the standard notation in the field where \(\mathsf {\ min\,}\) and \(\mathsf {\ max\,}\) of an infeasible or an unbounded solution is naturally extended to infimum and supremum.

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Acknowledgements

Jana Novotná and Milan Hladík were supported by the Czech Science Foundation Grant P403-18-04735S. The student work was supported by the Grant SVV–2017–260452.

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Correspondence to Jana Novotná.

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Novotná, J., Hladík, M. & Masařík, T. Duality Gap in Interval Linear Programming. J Optim Theory Appl 184, 565–580 (2020). https://doi.org/10.1007/s10957-019-01610-y

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Keywords

  • Interval analysis
  • Linear programming
  • Interval linear programming
  • Duality gap
  • Computational complexity

Mathematics Subject Classification

  • 90C05
  • 90C31
  • 65G40