Minimal Cut-Generating Functions are Nearly Extreme

Basu, Amitabh; Hildebrand, Robert; Molinaro, Marco

doi:10.1007/978-3-319-33461-5_17

Cited by 7 publications

(20 citation statements)

References 17 publications

(37 reference statements)

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“…Note that a polyhedron P has a finite number of facet-defining inequalities and, excluding degenerate cases, infinitely many minimal inequalities; thus, restricting to facet-defining inequalities really narrows down the inequalities under consideration. However, interestingly, for the infinite group relaxation (which is infinite dimensional), it was recently proved that the facet-defining inequalities form a dense set within the set of minimal inequalities [48,163]. This casts doubts on whether being a facet-defining inequality is really useful in this context.…”

Section: Theoretical Analysis Of the Strength Of Cutting-planesmentioning

confidence: 99%

“…As mentioned above, other recent papers [48,163] considered the question of separating faces versus facet-defining inequalities in the context of the (infinite) group relaxation. Surprisingly, it was shown that for any minimal inequality, there exists an extreme inequality that approximates the minimal inequality as closely as desired.…”

Section: Cutting-plane Selection: the Need To Ask Different Questionsmentioning

confidence: 99%

“…The study of the group relaxation has been very popular in the last 10-15 years [136,137,14,102,175,118,32,123,76,41,22,49,119,22,156,157]. Many beautiful and non-trivial structural results have been proved regarding valid inequalities for these relaxations [47,44,48,122,209]. See [190,39,45,46] for reviews on the group relaxation and its variants.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Cutting-plane Selection: the Need To Ask Different Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

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“…In terms of Gomory-Johnson cut-generating functions, the 2-slope theorem is a famous result of Gomory and Johnson's masterpiece [10,11]. Basu et al [8] proved that the 2-slope extreme Gomory-Johnson cut-generating functions are dense in the set of continuous minimal functions. We show that any 2-slope maximal general DFF with one slope value 0 is extreme.…”

Section: Introductionmentioning

confidence: 99%

“…This result is a key step in our approximation theorem, which indicates that almost all continuous maximal general DFFs can be approximated by extreme (2-slope) general DFFs as close as we desire. Unlike the 2-slope fill-in procedure Basu et al [8] used, we always use 0 as one slope value in our fill-in procedure, which is necessary since the 2-slope theorem of general DFFs requires 0 to be one slope value. This paper is structured as follows.…”

Section: Introductionmentioning

confidence: 99%

Characterization and Approximation of Strong General Dual Feasible Functions

Köppe

Wang

2018

Lecture Notes in Computer Science

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Dual feasible functions (DFFs) have been used to provide bounds for standard packing problems and valid inequalities for integer optimization problems. In this paper, the connection between general DFFs and a particular family of cut-generating functions is explored. We find the characterization of (restricted/strongly) maximal general DFFs and prove a 2-slope theorem for extreme general DFFs. We show that any restricted maximal general DFF can be well approximated by an extreme general DFF.

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Piecewise smooth extreme functions are piecewise linear

Summa

2018

Math. Program.

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Minimal Cut-Generating Functions are Nearly Extreme

Cited by 7 publications

References 17 publications

Theoretical challenges towards cutting-plane selection

Theoretical challenges towards cutting-plane selection

Characterization and Approximation of Strong General Dual Feasible Functions

Piecewise smooth extreme functions are piecewise linear

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