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Exact Separation of k-Projection Polytope Constraints

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Modeling and Optimization: Theory and Applications (MOPTA 2016)

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Abstract

A critical step of any cutting plane algorithm is to find valid inequalities, or more generally valid constraints, that improve the current relaxation of the integer-constrained problem. We consider the k-projection polytope constraints that are a family of constraints based on an inner description of the cut polytope of size k and are applied to k × k principal minors of the matrix variable of a semidefinite optimization relaxation. We propose a bilevel second order cone optimization approach to find the maximally violated k-projection polytope constraint according to a specific depth measure, and reformulate the bilevel problem as a single-level mixed binary second order cone optimization problem. We report computational results using the proposed approach within a cutting plane algorithm on instances of max-cut with 500 and 600 nodes.

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Acknowledgements

We thank an anonymous reviewer for constructive comments that helped us improve this paper.

The second author acknowledges the support of this research by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grant 312125.

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Authors and Affiliations

  1. GERAD & Polytechnique Montreal, Montreal, QC, Canada

    Elspeth Adams & Miguel F. Anjos

Authors
  1. Elspeth Adams
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  2. Miguel F. Anjos
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Correspondence to Miguel F. Anjos .

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Editors and Affiliations

  1. Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

    Martin Takáč

  2. Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

    Tamás Terlaky

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Adams, E., Anjos, M.F. (2017). Exact Separation of k-Projection Polytope Constraints. In: Takáč, M., Terlaky, T. (eds) Modeling and Optimization: Theory and Applications. MOPTA 2016. Springer Proceedings in Mathematics & Statistics, vol 213. Springer, Cham. https://doi.org/10.1007/978-3-319-66616-7_8

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