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Polyhedral approximation in mixed-integer convex optimization

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Abstract

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro.

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Notes

  1. In Lemma 3 we provide an explicit counterexample.

  2. The results reported here are based on Pajarito version 0.1. The latest release, version 0.4, has been almost completely rewritten with significant algorithmic advances, which will be discussed in upcoming work with Chris Coey.

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Acknowledgements

We thank Chris Coey for proofreading and Madeleine Udell and the anonymous reviewers for their comments. M. Lubin was supported by the DOE Computational Science Graduate Fellowship, which is provided under grant number DE-FG02-97ER25308. The work at LANL was funded by the Center for Nonlinear Studies (CNLS) and was carried out under the auspices of the NNSA of the U.S. DOE at LANL under Contract No. DE-AC52-06NA25396. J.P. Vielma was funded by NSF grant CMMI-1351619.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Miles Lubin & Juan Pablo Vielma

  2. Los Alamos National Laboratory, Los Alamos, NM, USA

    Emre Yamangil & Russell Bent

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  1. Miles Lubin
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Correspondence to Miles Lubin.

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Lubin, M., Yamangil, E., Bent, R. et al. Polyhedral approximation in mixed-integer convex optimization. Math. Program. 172, 139–168 (2018). https://doi.org/10.1007/s10107-017-1191-y

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