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Disjunctive cuts in Mixed-Integer Conic Optimization

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Abstract

This paper studies disjunctive cutting planes in Mixed-Integer Conic Programming. Building on conic duality, we formulate a cut-generating conic program for separating disjunctive cuts, and investigate the impact of the normalization condition on its resolution. In particular, we show that a careful selection of normalization guarantees its solvability and conic strong duality. Then, we highlight the shortcomings of separating conic-infeasible points in an outer-approximation context, and propose conic extensions to the classical lifting and monoidal strengthening procedures. Finally, we assess the computational behavior of various normalization conditions in terms of gap closed, computing time and cut sparsity. In the process, we show that our approach is competitive with the internal lift-and-project cuts of a state-of-the-art solver.

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Notes

  1. Personal communication with Gurobi and Mosek developers.

  2. Our code is released at https://github.com/mtanneau/CLaP.

  3. Code available at https://github.com/mtanneau/CLaP.

  4. Personal communication with CPLEX developers.

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Acknowledgements

The second author was supported by an FRQNT excellence doctoral scholarship, and a Mitacs Globalink research award. We thank Pierre Bonami, Andrea Tramontani and Sven Wiese for several helpful discussions on the topic, as well as the anonymous referees for their comments and suggestions that helped improve the paper.

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Authors and Affiliations

  1. CERC, Polytechnique Montréal - Canada Excellence Research Chair, C.P. 6079, Succ. Centre-ville, Montréal, H3C 3A7, QC, Canada

    Andrea Lodi & Mathieu Tanneau

  2. MIT Sloan School of Management, Cambridge, MA, United States

    Juan-Pablo Vielma

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  1. Andrea Lodi
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  2. Mathieu Tanneau
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Correspondence to Mathieu Tanneau.

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Lodi, A., Tanneau, M. & Vielma, JP. Disjunctive cuts in Mixed-Integer Conic Optimization. Math. Program. 199, 671–719 (2023). https://doi.org/10.1007/s10107-022-01844-1

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