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Integral simplex using decomposition with primal cutting planes

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Abstract

This paper concentrates on the addition of cutting planes to the integral simplex using decomposition (ISUD) of Zaghrouti et al. (Oper Res 62(2):435–449, 2014). This method solves the set partitioning problem by iteratively improving an existing feasible solution. We present the algorithm in a primal language and relate it to existing augmenting methods. The resulting theoretical properties, stronger than the ones already known, simplify termination proofs and deepen the geometrical insights on ISUD in particular. We show that primal cuts, that is, cutting planes that are tight at the current feasible integer solution, can be used to improve the performance of the algorithm, and further that such cutting planes are enough to solve each augmentation problem. We propose efficient separation procedures for well-known polyhedral inequalities, namely primal clique and odd-cycle cuts. Numerical results demonstrate the effectiveness of primal cutting planes; tests are performed on small and large-scale set partitioning problems from aircrew and bus-driver scheduling instances up to 1600 constraints and 570,000 variables.

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Notes

  1. A formulation of SPP as an integer linear programming problem is given in Sect. 2.2.

  2. The above definition includes local search algorithms, whose analysis is, however, outside of the scope of the present paper that is concerned with simplex-type algorithms that are able to prove optimality of the solution.

  3. More details on this feature and its implications are given in Sect. 2.2.

  4. For the determination of the clique of maximal weight in \(\mathcal {G}_l\), we use the cliquer open source library, available at http://users.aalto.fi/~pat/cliquer.html, based on the algorithm described in [19].

  5. http://people.brunel.ac.uk/~mastjjb/jeb/info.html.

  6. CPLEX is freely available for academic and research purposes under the IBM academic initiative: http://www-03.ibm.com/ibm/university/academic. When referring to CPLEX, we always refer to the version 12.4 of this software, with single-thread settings (all other settings being default).

  7. GENCOL is a commercial software developed at the GERAD research center and now owned by the AD OPT company, a division of KRONOS.

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Acknowledgements

This work was supported by a Collaborative Research and Development Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Kronos Inc. Samuel Rosat benefitted of a grant from the International Internship Program of the Fonds de Recherche du Québec Nature et Technologies (FRQNT).

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Authors and Affiliations

  1. GERAD and École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montreal, QC, H3C 3A7, Canada

    Samuel Rosat, Issmail Elhallaoui & François Soumis

  2. CERC and École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montreal, QC, H3C 3A7, Canada

    Andrea Lodi

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  1. Samuel Rosat
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Correspondence to Andrea Lodi.

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Rosat, S., Elhallaoui, I., Soumis, F. et al. Integral simplex using decomposition with primal cutting planes. Math. Program. 166, 327–367 (2017). https://doi.org/10.1007/s10107-017-1123-x

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