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

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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.

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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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Authors and Affiliations

  1. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Santanu S. Dey & Sebastian Pokutta

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  1. Santanu S. Dey
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  2. Sebastian Pokutta
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Correspondence to Santanu S. Dey.

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Santanu S. Dey was supported by NSF Grant CMMI1149400.

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Dey, S.S., Pokutta, S. Design and verify: a new scheme for generating cutting-planes. Math. Program. 145, 199–222 (2014). https://doi.org/10.1007/s10107-013-0645-0

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