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Mathematical Programming Computation

, Volume 10, Issue 3, pp 423–455 | Cite as

Intersection cuts for single row corner relaxations

  • Ricardo Fukasawa
  • Laurent Poirrier
  • Álinson S. XavierEmail author
Full Length Paper
  • 107 Downloads

Abstract

We consider the problem of generating inequalities that are valid for one-row relaxations of a simplex tableau, with the integrality constraints preserved for one or more non-basic variables. These relaxations are interesting because they can be used to generate cutting planes for general mixed-integer problems. We first consider the case of a single non-basic integer variable. This relaxation is related to a simple knapsack set with two integer variables and two continuous variables. We study its facial structure by rewriting it as a constrained two-row model, and prove that all its facets arise from a finite number of maximal \(\left( \mathbb {Z}\times \mathbb {Z}_+\right) \)-free splits and wedges. The resulting cuts generalize both MIR and 2-step MIR inequalities. Then, we describe an algorithm for enumerating all the maximal \(\left( \mathbb {Z}\times \mathbb {Z}_+\right) \)-free sets corresponding to facet-defining inequalities, and we provide an upper bound on the split rank of those inequalities. Finally, we run computational experiments to compare the strength of wedge cuts against MIR cuts. In our computations, we use the so-called trivial fill-in function to exploit the integrality of more non-basic variables. To that end, we present a practical algorithm for computing the coefficients of this lifting function.

Keywords

Integer programming Lifting Cutting planes 

Mathematics Subject Classification

90C11 90C57 

Notes

Acknowledgements

We would like to thank two anonymous referees for valuable suggestions that led to significant improvements to this manuscript. Fukasawa was supported by NSERC Discovery Grant RGPIN 2014-05623. Poirrier and Xavier were supported by Early Researcher Award Grant ER-11-08-174.

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  • Ricardo Fukasawa
    • 1
  • Laurent Poirrier
    • 1
  • Álinson S. Xavier
    • 1
    Email author
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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