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On the exact separation of mixed integer knapsack cuts

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Abstract

During the last decades, much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets, which we call knapsack cuts. Bixby et al. (The sharpest cut: the impact of Manfred Padberg and his work. MPS/SIAM Series on Optimization, pp. 309–326, 2004) empirically observe that, within the context of branch-and-cut algorithms to solve mixed integer programming problems, the most important inequalities are knapsack cuts derived by the mixed integer rounding (MIR) procedure. In this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set. The main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships. The exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them. Using these benchmarks on MIPLIB 3.0 and MIPLIB 2003 instances we analyze the performance of MIR inequalities. Our computations, which are performed in exact arithmetic, are surprising: In the vast majority of the instances in which knapsack cuts yield bound improvements, MIR cuts alone achieve over 87% of the observed gain.

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Authors and Affiliations

  1. Mathematical Sciences Department, IBM Research, Yorktown Heights, NY, USA

    Ricardo Fukasawa

  2. School of Business, Universidad Adolfo Ibañez, Santiago, Chile

    Marcos Goycoolea

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  1. Ricardo Fukasawa
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  2. Marcos Goycoolea
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Correspondence to Ricardo Fukasawa.

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Fukasawa, R., Goycoolea, M. On the exact separation of mixed integer knapsack cuts. Math. Program. 128, 19–41 (2011). https://doi.org/10.1007/s10107-009-0284-7

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  • DOI: https://doi.org/10.1007/s10107-009-0284-7

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