Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$k=2$$ k = 2

Basu, Amitabh; Hildebrand, Robert; Köppe, Matthias

doi:10.1007/s10107-016-1064-9

Cited by 10 publications

(15 citation statements)

References 24 publications

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“…This was later generalized to an arbitrary number of equations [96,47]. For the case of 2 equations, Basu et al [44,50] provide algorithms for testing whether a minimal valid function is extreme.…”

Section: Theoretical Analysis Of the Strength Of Cutting-planesmentioning

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

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While many classes of cutting-planes are at the disposal of integer programming solvers, our scientific understanding is far from complete with regards to cutting-plane selection, i.e., the task of selecting a portfolio of cutting-planes to be added to the LP relaxation at a given node of the branch-and-bound tree. In this paper we review the different classes of cutting-planes available, known theoretical results about their relative strength, important issues pertaining to cut selection, and discuss some possible new directions to be pursued in order to accomplish cutting-plane selection in a more principled manner. Finally, we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection. * santanu.dey@isye.gatech.edu † molinaro@inf.puc-rio.brIn particular, general-purpose cutting-planes were seen as mostly of theoretical interest. However, in the mid 90s a breakthrough came with a series of papers by Balas, Ceria, Cornuéjols, and Natraj, where they obtained striking computational results using Gomory Mixed Integer cuts and the then newly discovered Lift-and-Project cuts, within the branch-and-bound framework [29,30]. These papers show the efficacy of general-purpose cutting-planes, when properly selected and employed.Today, even more families of cutting-planes are known. While we do not attempt to survey the vast literature on cutting-planes, we discuss the broad perspectives from which they are derived, with some representative examples.and therefore valid inequalities for P \ (int(Q) × R q ) are valid for P ∩ (Z n × R q ). We call such cuts geometric cuts. A set Q is a maximal lattice-free convex set if it is lattice-free, convex, and no other lattice-free convex set properly contains it. A beautiful structural result is that all maximal lattice-free convex sets are polyhedral [169,38]. Note that when Q is a polyhedral} one can generate a geometric cut by ensuring its validity for all the disjunctions identified with the facets of Q, i.e.,Famous geometric cuts include the Chvátal-Gomory (CG) cuts [131,65,164,120], implied bounds [145], and more generally split cuts [27,85,34]. For all these cuts, the underlying lattice-free set is the so-called split set: {x ∈ R n | π 0 ≤ π ⊤ x ≤ π 0 + 1}, where π 0 ∈ Z and π ∈ Z n . More general geometric cuts include cuts from maximal lattice-free convex sets [12,62] (see the excellent survey [39]) and the closely related intersection cuts developed by Balas in 1970s [25]. Geometric cuts called multi-branch split cuts [165,99,104] are based on non-convex lattice-free sets obtained as the union of split sets.Structured relaxations. Given a mixed-integer set P ∩ (Z n × R q ), the basic idea is to construct a relaxation of P , say R ⊇ P , and analyze the mixed-integer set R ∩ (Z n × R q ). Valid inequalities for the relaxation then serve as cutting-planes for the original set, and the additional structure of R eases the search for facets or strong inequalities. Two excellent review articles discu...

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Section: Theoretical Analysis Of the Strength Of Cutting-planesmentioning

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

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“…Remark 5.3. As shown in [3] (for a stronger statement, see [5, Theorem 8.6]), the family F 1 of continuous piecewise linear functions with rational breakpoints is such a subfamily where existence of an effective perturbation implies existence of a piecewise linear effective perturbation.…”

Section: Weak Facetsmentioning

confidence: 99%

“…The extremality proof appears as [18, Example 7.2]; it can also be verified using the software [20]. 3 The function ψ is piecewise linear on a complex P, which is illustrated in Figure 3 (left). Consider the minimal valid function ψ = discontinuous facets paper example psi prime() defined by…”

Section: Weak Facetsmentioning

confidence: 99%

On the Notions of Facets, Weak Facets, and Extreme Functions of the Gomory–Johnson Infinite Group Problem

Köppe

Zhou

2017

Integer Programming and Combinatorial Optimization

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Abstract. We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.

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“…Our proof of Proposition 4.1 is based on the Facet Theorem, which gives a sufficient condition for a function to be a facet [6,11], and the Interval Lemma, which first appeared in [13], and was subsequently elaborated upon in [10,9,7,3]; see also the survey [4,5].…”

Section: A Construction Of K-slope Functions π Kmentioning

confidence: 99%

Extreme Functions with an Arbitrary Number of Slopes

Basu

Conforti

Summa

et al. 2016

Integer Programming and Combinatorial Optimization

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For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number k ≥ 2, there is a function in the sequence with k slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson stating that all extreme functions are piecewise linear. These constructions are extended to obtain functions for the higher dimensional group problems via the sequential-merge operation of Dey and Richard.

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Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$k=2$$ k = 2

Cited by 10 publications

References 24 publications

Theoretical challenges towards cutting-plane selection

Theoretical challenges towards cutting-plane selection

On the Notions of Facets, Weak Facets, and Extreme Functions of the Gomory–Johnson Infinite Group Problem

Extreme Functions with an Arbitrary Number of Slopes

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