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Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

Author

Listed:
  • Michele Conforti

    (Università degli Studi di Padova)

  • Marianna Santis

    (Sapienza Università di Roma)

  • Marco Summa

    (Università degli Studi di Padova)

  • Francesco Rinaldi

    (Università degli Studi di Padova)

Abstract

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

Suggested Citation

  • Michele Conforti & Marianna Santis & Marco Summa & Francesco Rinaldi, 2021. "Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective," 4OR, Springer, vol. 19(4), pages 531-548, December.
    • Handle: RePEc:spr:aqjoor:v:19:y:2021:i:4:d:10.1007_s10288-020-00459-6
    DOI: 10.1007/s10288-020-00459-6
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    References listed on IDEAS

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    1. V. J. Bowman & G. L. Nemhauser, 1970. "A finiteness proof for modified dantzig cuts in integer programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 17(3), pages 309-313, September.
    2. Jon Lee & Angelika Wiegele, 2017. "Another pedagogy for mixed-integer Gomory," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(4), pages 455-466, December.
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