On the relative strength of different generalizations of split cuts

Dash, Sanjeeb; Günlük, Oktay; Molinaro, Marco

doi:10.1016/j.disopt.2014.12.003

Cited by 9 publications

(5 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Cornuéjols and Li [90] provide comparisons between several basic closures, including the split, L&P, and RLT closures. The papers [165,104] show that the t-branch closure can be strictly stronger than infinitely many iterations of the k-branch closure whenever t > k. See also [99,108,105]. Laurent [162] compares RLT, N + , and the Lasserre closures for 0/1 polytopes and shows that the Lasserre construction provides the tightest relaxations.…”

Section: Theoretical Analysis Of the Strength Of Cutting-planesmentioning

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Theoretical Analysis Of the Strength Of Cutting-planesmentioning

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

Self Cite

View full text Add to dashboard Cite

show abstract

“…e fulldimensional case = ℝ is Lemma 3.1 of [14], and the general case follows directly by applying it to the affine subspace .…”

Section: Abstract Branch-and-bound Trees and Notions Of Hardnessmentioning

confidence: 99%

Lower Bounds on the Size of General Branch-and-Bound Trees

Dey¹,

Dubey²,

Molinaro³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form ⊤ ≤ 0 ∨ ⊤ ≥ 0 + 1, where is an integer vector and 0 is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to the coefficients of the cross polytope, thus showing that polynomial-size "smoothed analysis" upper bound is not possible. e results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. [7], who proved lower bounds for the Chvátal-Gomory rank.

show abstract

“…Further, it is shown in Dash et al (2011) that P CC ⊆ P C , but the containment is not strict (Dash et al 2012b). …”

Section: Crooked Cross Disjunctions and Crooked Cross Cutsmentioning

confidence: 99%

“…To achieve this we use the following proposition, which shows that certain rank-2 split cuts are also cross cuts. We note that not all rank-2 split cuts are cross cuts and these two cut families do not dominate each other (Dash et al 2012b). …”

Section: Separating Cross Cuts Using Rank-2 Split Cutsmentioning

confidence: 99%

“…Recently, Balas and Saxena (2008) and approximately optimized over the split closure of practical MIP instances and obtained very strong bounds on the optimal values of such instances. There 3 not true (Dash et al 2012b). Their computational experiments are performed on randomly generated problems, where they get a nontrivial improvement over one round of GMI cuts, but their procedure has limited success on practical MIPs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computational Experiments with Cross and Crooked Cross Cuts

Dash

Günlük

Vielma

2014

INFORMS Journal on Computing

Self Cite

View full text Add to dashboard Cite

In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by GMI cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical MIPs and can yield a non-trivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the t-branch split cuts studied by Li and Richard (2008). Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Dey and Günlük (2010) and were shown to dominate cross cuts in Dash, Günlük and Molinaro (2012).

show abstract

On the relative strength of different generalizations of split cuts

Cited by 9 publications

References 28 publications

Theoretical challenges towards cutting-plane selection

Theoretical challenges towards cutting-plane selection

Lower Bounds on the Size of General Branch-and-Bound Trees

Computational Experiments with Cross and Crooked Cross Cuts

Contact Info

Product

Resources

About