infinite group relaxation
Light on the infinite group relaxation. I: Foundations and taxonomy. 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 {it 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 $kge 1$ 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 The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes.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.
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References in zbMATH (referenced in 22 articles )
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Sorted by year (- Di Summa, Marco: Piecewise smooth extreme functions are piecewise linear (2020)
- Basu, Amitabh; Conforti, Michele; Di Summa, Marco; Paat, Joseph: The structure of the infinite models in integer programming (2019)
- Basu, Amitabh; Conforti, Michele; Di Summa, Marco; Zambelli, Giacomo: Optimal cutting planes from the group relaxations (2019)
- Basu, Amitabh; Dey, Santanu S.; Paat, Joseph: Nonunique lifting of integer variables in minimal inequalities (2019)
- Basu, Amitabh; Sankaranarayanan, Sriram: Can cut-generating functions be good and efficient? (2019)
- Basu, Amitabh; Conforti, Michele; Di Summa, Marco; Paat, Joseph: Extreme functions with an arbitrary number of slopes (2018)
- Basu, Amitabh; Hildebrand, Robert; Molinaro, Marco: Minimal cut-generating functions are nearly extreme (2018)
- Dey, Santanu S.; Iroume, Andres; Wang, Guanyi: The strength of multi-row aggregation cuts for sign-pattern integer programs (2018)
- Dey, Santanu S.; Molinaro, Marco: Theoretical challenges towards cutting-plane selection (2018)
- Hong, Chun Yu; Köppe, Matthias; Zhou, Yuan: Equivariant perturbation in Gomory and Johnson’s infinite group problem. V: Software for the continuous and discontinuous 1-row case (2018)
- Köppe, Matthias; Zhou, Yuan: Equivariant perturbation in Gomory and Johnson’s infinite group problem. VI: The curious case of two-sided discontinuous minimal valid functions (2018)
- Lebair, Teresa M.; Basu, Amitabh: Approximation of minimal functions by extreme functions (2018)
- Köppe, Matthias; Wang, Jiawei: Structure and interpretation of dual-feasible functions (2017)
- Köppe, Matthias; Zhou, Yuan: New computer-based search strategies for extreme functions of the Gomory-Johnson infinite group problem (2017)
- Santana, Asteroide; Dey, Santanu S.: Some cut-generating functions for second-order conic sets (2017)
- Basu, Amitabh; Conforti, Michele; Di Summa, Marco; Paat, Joseph: Extreme functions with an arbitrary number of slopes (2016)
- Basu, Amitabh; Hildebrand, Robert; Köppe, Matthias: Light on the infinite group relaxation. I: Foundations and taxonomy (2016)
- Basu, Amitabh; Hildebrand, Robert; Köppe, Matthias: Light on the infinite group relaxation. II: Sufficient conditions for extremality, sequences, and algorithms (2016)
- Basu, Amitabh; Hildebrand, Robert; Molinaro, Marco: Minimal cut-generating functions are nearly extreme (2016)
- Hong, Chun Yu; Köppe, Matthias; Zhou, Yuan: Software for cut-generating functions in the Gomory-Johnson model and beyond (2016)