Abstract
Optimization models often feature disjunctions of polytopes as submodels. Such a disjunctive set is initially (at best) relaxed to its convex hull, which is then refined by branching. To measure the error of the convex relaxation, the (relative) difference between the volume of the convex hull and the volume of the disjunctive set may be used. This requires a method to compute the volume of the disjunctive set.
We propose a revised variant of an old algorithm by Bieri and Nef (Linear Algebra Appl 52:69–97, 1983) for this purpose. The algorithm uses a sweep-plane to incrementally calculate the volume of the disjunctive set as a function of the offset parameter of the sweep-plane.
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References
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Acknowledgements
The authors thank the DFG for support within project A04 in CRC TRR154 and the BMBF Research Campus Modal (fund number 05M14ZAM) and ICT COST Action TD1207 for additional support.
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Anderson, L., Hiller, B. (2019). A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes. In: Fortz, B., Labbé, M. (eds) Operations Research Proceedings 2018. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-18500-8_12
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DOI: https://doi.org/10.1007/978-3-030-18500-8_12
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