Strengthening cuts for mixed integer programs

Recently, it has been shown that minimal inequalities for a continuous relaxation of mixed integer linear programs are associated with maximal lattice-free convex sets. In this paper, we show how to lift these inequalities for integral nonbasic variables by considering maximal lattice-free convex sets in a higher-dimensional space. We apply this approach to several examples. In particular, we identify cases where the lifting is unique.

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“…This lemma is closely related to a result of Balas and Jeroslow [3]. It implies the following property.…”

Section: Lemma 6 Let ψ Be a Minimal Valid Function And π Be A Minimsupporting

confidence: 66%

“…Note that if ψ is extreme for (3), then ψ is minimal. (3) and (ψ, π ) is valid for (1), then (ψ, π ) is extreme for (1). (3), and π 1 = π 2 = π since π 1 ≥ π and π 2 ≥ π .…”

Section: Introductionmentioning

confidence: 99%

A Geometric Perspective on Lifting

Conforti

Cornuéjols

Zambelli

2011

Operations Research

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“…The function φ is called the 'fill-in function' [25,26] or alternatively the above procedure is called the 'monoidal strengthening' technique [7]. The validity of the fill-in function rests on the fact that integer variables corresponding to columns r and r + u where u ∈ Z 2 must obtain the same coefficient in any facet-defining inequality for (1).…”

Section: Coefficients Of Integer Variables In (1)mentioning

confidence: 99%

“…What would be preferable would be valid inequalities for (1) taking into account the integrality of the integer non-basic variables, even if these inequalities cut off parts of the set (2). One way to keep the strong coefficients of the continuous variables and to improve further the coefficients of the integer non-basic variables is to use lifting, based on the use of so-called 'fill-in functions' [25,26] or the 'monoidal strengthening' technique [7].…”

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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“…Once a violated cut has been found as a solution of (5), the cut can be easily strengthened a posteriori through the Balas and Jeroslow [10] procedure. Such a strengthening can be seen as finding the best split disjunction for a given set of Farkas multipliers.…”

Section: Separating Disjunctive Cuts In Milpmentioning

confidence: 99%

Disjunctive Inequalities: Applications And Extensions

Belotti

Liberti

Lodi

et al. 2011

Wiley Encyclopedia of Operations Research and Management Science

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We survey some applications and extensions of disjunctive programming with special emphasis on recent developments. Specifically, after recalling the basic ingredients of disjunctive inequalities we report on recent results in the context of mixed integer linear programming. We then consider the application of disjunctive constraints both as modeling tool and cutting planes in mixed integer nonlinear programming. Finally, we discuss the application of disjunctions as branching conditions in enumerative algorithms, as opposed to the cutting approach.

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Strengthening cuts for mixed integer programs

Cited by 96 publications

References 4 publications

A Geometric Perspective on Lifting

A Geometric Perspective on Lifting

On the Practical Strength of Two-Row Tableau Cuts

Disjunctive Inequalities: Applications And Extensions

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