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The linearization problem of a binary quadratic problem and its applications

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  • Hao Hu

    (University of Waterloo)

  • Renata Sotirov

    (Tilburg University)

Abstract

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore–Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. We also present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.

Suggested Citation

  • Hao Hu & Renata Sotirov, 2021. "The linearization problem of a binary quadratic problem and its applications," Annals of Operations Research, Springer, vol. 307(1), pages 229-249, December.
    • Handle: RePEc:spr:annopr:v:307:y:2021:i:1:d:10.1007_s10479-021-04310-x
    DOI: 10.1007/s10479-021-04310-x
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