## A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes

Please always quote using this URN: urn:nbn:de:0297-zib-69489

- 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. Naively, this can be done via inclusion/exclusion and leveraging the existing code for the volume of polytopes. However, this is often inefficient. We propose a revised variant of an old algorithm by Bieri and Nef (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.

Author: | Lovis AndersonORCiD, Benjamin HillerORCiD |
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Document Type: | ZIB-Report |

Tag: | Disjunctive Programming; Union of Polytopes; Volume Algorithm |

MSC-Classification: | 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] |

CCS-Classification: | G. Mathematics of Computing |

PACS-Classification: | 00.00.00 GENERAL |

Publishing Institution: | Zuse Institute Berlin (ZIB) |

Date of first Publication: | 2018/07/17 |

Series (Serial Number): | ZIB-Report (18-37) |

ISSN: | 1438-0064 |