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Mathematical Programming

, Volume 145, Issue 1–2, pp 199–222 | Cite as

Design and verify: a new scheme for generating cutting-planes

  • Santanu S. Dey
  • Sebastian Pokutta
Full Length Paper Series A

Abstract

A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory–Chvátal procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality \(cx \le d\) where \(c\) and \(d\) are integral, independent of the cutting-plane black-box. In the second step, we verify that the designed inequality is a valid inequality by verifying that the set \(P \cap \{x\in \mathbb R ^n \mid cx \ge d + 1\} \cap \mathbb Z ^n\) is empty using cutting-planes from the black-box. Here \(P\) is the feasible region of the linear-programming relaxation of the IP. We refer to the closure of all cutting-planes that can be verified to be valid using a specific cutting-plane black-box as the verification closure of the considered cutting-plane black-box. This paper undertakes a systematic study of properties of verification closures of various cutting-plane black-box procedures.

Keywords

Verification scheme Cutting planes Integer programming 

Mathematics Subject Classification

90C10 90C57 

Notes

Acknowledgments

The authors are most grateful to the anonymous referees for their detailed remarks that considerably improved the presentation and simplified some of the proofs.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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