A Geometric Perspective on Lifting

Conforti, Michele; Cornuéjols, Gérard; Zambelli, Giacomo

doi:10.1287/opre.1110.0916

Cited by 48 publications

(56 citation statements)

References 16 publications

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“…Recently, this construction has been studied in [19], [13], [9]. We next demonstrate that inequality (12) is a crooked cross cut for T .…”

Section: 'Monoidal' Strengtheningmentioning

confidence: 89%

“…Subsequently, Cornuéjols and Margot [16] gave an exact characterization of the split cuts and intersection cuts based on maximal lattice-free triangles and maximal lattice-free quadrilaterals that yield facet-defining inequalities for this set. Many authors have extended these results to semi-infinite version of the canonical k-row set and to higher values of k [12] [32], to sets with more structure, such as bounds on nonbasic variables [2], the integrality of non-basic variables [20,22], the nonnegativity of basic integer variables [10,11], [21], [25], and both the integrality of non-basic variables and non-negativity of basic integer variables [9] [13]. See [18] for a recent survey on the topic.…”

Section: Introductionmentioning

confidence: 99%

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Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

2011

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In this paper, we study the relationship between 2D lattice-free cuts, the family of cuts obtained by taking two-row relaxations of a mixed-integer program (MIP) and applying intersection cuts based on maximal lattice-free sets in R 2 , and various types of disjunctions. Recently, Li and Richard (2007) studied disjunctive cuts obtained from t-branch split disjunctions of mixed-integer sets (these cuts generalize split cuts). Balas (2009) initiated the study of cuts for the two-row continuous group relaxation obtained from 2-branch split disjunctions. We study these cuts (and call them cross cuts) for the two-row continuous group relaxation, and for general MIPs. We also consider cuts obtained from asymmetric 2-branch disjunctions which we call crooked cross cuts. For the two-row continuous group relaxation, we show that unimodular cross cuts (the coefficients of the two split inequalities form a unimodular matrix) are equivalent to the cuts obtained from maximal lattice-free sets other than type 3 triangles. We also prove that all 2D lattice-free cuts and their S-free extensions are crooked cross cuts. For general mixed integer sets, we show that crooked cross cuts can be generated from a structured three-row relaxation. Finally, we show that for the corner relaxation of an MIP, every crooked cross cut is a 2D lattice-free cut.

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“…Recently, this construction has been studied in [19], [13], [9]. We next demonstrate that inequality (12) is a crooked cross cut for T .…”

Section: 'Monoidal' Strengtheningmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

2011

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“…On the other hand, Dey and Wolsey [25,20] and Conforti, Cornuéjols and Zambelli [16] considered the problem of exploiting any integrality constraint on non-basic variables. They propose a way to, given a valid intersection cut for P I , strengthen the coefficients of the nonbasic integer variables by solving a so-called lifting problem.…”

Section: Overviewmentioning

confidence: 99%

The strength of multi-row models

2014

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We develop a method for computing facet-defining valid inequalities for any mixed-integer set P J . Our practical implementation does not return only facetdefining inequalities, but it is able to find a separating cut whenever one exists. The separator is not comparable in speed with the specific cutting-plane generators used in branch-and-cut solvers, but it is general-purpose. We can thus use it to compute cuts derived from any reasonably small relaxation P J of a general mixedinteger problem, even when there exists no specific implementation for computing cuts with P J . Exploiting this, we evaluate, from a computational perspective, the usefulness of cuts derived from several types of multi-row relaxations. In particular, we present results with four different strengthenings of the two-row intersection cut model, and multi-row models with up to fifteen rows. We conclude that only fully-strengthened two-row cuts seem to offer a significant advantage over two-row intersection cuts. Our results also indicate that the improvement obtained by going from models with very few rows to models with up to fifteen rows may not be worth the increased computing cost. OverviewIn the last years, there has been a renewed interest in the MIP research community for finding new ways to compute general-purpose cutting-planes. Specifically, one of the subjects of attention was the generation of so-called multi-row cutting-planes, i.e. inequalities that are valid for relaxations with multiple rows of the problem to be solved.In this paper, we aim at evaluating accurately the strength of the multi-row relaxations that are used to generate these cutting planes. To that end, we develop techniques to

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“…Several variants of (2) have been studied; see for example [2,3,10,11,14,15,17,20,21,29] and [19] for a recent survey on the topic. This relaxation (2) has at least two appealing features : i) it is possible to obtain a complete characterization of all facet-defining inequalities using intersections cuts derived from the so-called maximal lattice-free convex sets and ii) the cutting planes derived from these facet-defining inequalities have the strongest possible coefficients for the continuous variables.…”

Section: Introductionmentioning

confidence: 99%

On the Practical Strength of Two-Row Tableau Cuts

Dey

Lodi

Tramontani³

et al. 2014

INFORMS Journal on Computing

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Following the flurry of recent theoretical work on cutting planes from two-row mixed integer group relaxations of an LP tableau, we report on computational tests to evaluate the strength of two-row cuts based on lattice-free triangles having more than one integer point on one side. A heuristic procedure to generate such triangles (referred to in the literature as "type 2" triangles) is presented, and then the coefficients of the integer variables are tightened by lifting.To test the effectiveness of triangle cuts, we compare the gap closed using Gomory mixed integer cuts for one round, the gap closed in one round using all the triangle cuts generated by our heuristic and the gap closed by a small number of two-row split cuts. Our tests are carried out on randomly generated instances designed to represent different problem features by varying the number of integer non-basic variables, bounds, non-negativity constraints and density, as well as on the classical MIPLIB instances.The outcome of this computational analysis is some insight into key characteristics of MIP instances whose presence makes two-row triangle cuts computationally effective. In particular it appears to be necessary that the tableau row pairs are dense, and more subjectively that the non-basic continuous variables are "important". Unfortunately these characteristics seem rarely to be present among real life instances, and more specifically the tableau rows of the MIPLIB instances are far from dense. A preliminary version of this work has been published in [18].

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A Geometric Perspective on Lifting

Cited by 48 publications

References 16 publications

Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

The strength of multi-row models

On the Practical Strength of Two-Row Tableau Cuts

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