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Light on the infinite group relaxation I: foundations and taxonomy

Author

Listed:
  • Amitabh Basu

    (The Johns Hopkins University)

  • Robert Hildebrand

    (Institute for Operations Research, ETH Zürich)

  • Matthias Köppe

    (University of California, Davis)

Abstract

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled Some continuous functions related to corner polyhedra I, II (Math Program 3:23–85, 359–389, 1972a, b). The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general $$k\ge 1$$ k ≥ 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.

Suggested Citation

  • Amitabh Basu & Robert Hildebrand & Matthias Köppe, 2016. "Light on the infinite group relaxation I: foundations and taxonomy," 4OR, Springer, vol. 14(1), pages 1-40, March.
    • Handle: RePEc:spr:aqjoor:v:14:y:2016:i:1:d:10.1007_s10288-015-0292-9
    DOI: 10.1007/s10288-015-0292-9
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    References listed on IDEAS

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    Cited by:

    1. Amitabh Basu & Michele Conforti & Marco Di Summa & Giacomo Zambelli, 2019. "Optimal Cutting Planes from the Group Relaxations," Management Science, INFORMS, vol. 44(4), pages 1208-1220, November.
    2. Amitabh Basu & Michele Conforti & Marco Di Summa & Joseph Paat, 2019. "The Structure of the Infinite Models in Integer ProgrammingAbstract: The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces ," Management Science, INFORMS, vol. 44(4), pages 1412-1430, November.

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