Intersection Cuts and Corner Polyhedra

Graduate Texts in Mathematics - Integer Programming ◽

10.1007/978-3-319-11008-0_6 ◽

2014 ◽

pp. 235-280

Author(s):

Michele Conforti ◽

Gérard Cornuéjols ◽

Giacomo Zambelli

Keyword(s):

Intersection Cuts ◽

Corner Polyhedra

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Approximation of Corner Polyhedra with Families of Intersection Cuts

Integer Programming and Combinatorial Optimization - Lecture Notes in Computer Science ◽

10.1007/978-3-319-59250-3_5 ◽

2017 ◽

pp. 51-62 ◽

Cited By ~ 1

Author(s):

Gennadiy Averkov ◽

Amitabh Basu ◽

Joseph Paat

Keyword(s):

Intersection Cuts ◽

Corner Polyhedra

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Approximation of Corner Polyhedra with Families of Intersection Cuts

SIAM Journal on Optimization ◽

10.1137/17m1128939 ◽

2018 ◽

Vol 28 (1) ◽

pp. 904-929 ◽

Cited By ~ 1

Author(s):

Gennadiy Averkov ◽

Amitabh Basu ◽

Joseph Paat

Keyword(s):

Intersection Cuts ◽

Corner Polyhedra

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Multirow Intersection Cuts Based on the Infinity Norm

INFORMS Journal on Computing ◽

10.1287/ijoc.2020.1027 ◽

2021 ◽

Author(s):

Álinson S. Xavier ◽

Ricardo Fukasawa ◽

Laurent Poirrier

Keyword(s):

Optimization Problems ◽

Valid Inequalities ◽

Linear Optimization ◽

Computational Cost ◽

General Purpose ◽

Mixed Integer ◽

Infinity Norm ◽

Intersection Cuts ◽

Linear Optimization Problems ◽

Mixed Integer Linear Optimization

When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

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