A computational study of the cutting plane tree algorithm for general mixed-integer linear programs

Chen, Binyuan; Küçükyavuz, Simge; Sen, Suvrajeet

doi:10.1016/j.orl.2011.10.009

Cited by 8 publications

(4 citation statements)

References 30 publications

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“…For some families of cutting-planes, such as L&P, it is possible to write the separation problem as an LP. An important design choice is the selection of normalization used in the resulting LP; see for example [61,28,70]. In a recent paper, Wolsey and Conforti [80] address the problem of efficiently finding an inequality that is violated and either defines an improper face or a facet by the proper use of normalization.…”

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

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

confidence: 99%

Theoretical challenges towards cutting-plane selection

Dey

Molinaro

2018

Math. Program.

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“…This finding happens to be critical for a general SMIP algorithm because one can now support the claim that the value function of a mixed-integer programming problem can be represented by generating a sequence of cutting planes, some of which are intended to refine an approximation of the two-stage MIP polyhedron, whereas other cutting planes are used to refine the value function approximation of each MIP in the scenario set. The representability result appears in [16] and its computability appears in [17]. Incidentally, these cuts are based on multi-term disjunctions which we refer to as Cutting Plane Tree (CPT) cuts.…”

Section: Bendersmentioning

confidence: 99%

“…Based on the development mentioned above, Qi and Sen [44] use CPT cuts to develop two versions of the ABC algorithm: one based purely on a polyhedral approximation of the feasible set for each scenario, and another based on a disjunctive representation of each scenario. The first option (creating a polyhedral approximation), referred to as ABC (CPT-D), uses the algorithm in [17] to create cuts in the (x, y) space. These iterations are carried out in two phases: for a given x k , we perform a sequence of iterations indexed by d which create improved approximations by deleting points x k , {y d }.…”

Section: Bendersmentioning

confidence: 99%

An Introduction to Two-Stage Stochastic Mixed-Integer Programming

Küçükyavuz

Sen

2017

The Operations Research Revolution

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This paper provides an introduction to algorithms for two-stage stochastic mixed-integer programs. Our focus is on methods which decompose the problem by scenarios representing randomness in the problem data. The design of these algorithms depend on where the uncertainty appears (right-hand-side, recourse matrix and/or technology matrix) and where the continuous and discrete decision variables are (first-stage and/or secondstage). In addition we provide computational evidence that, similar to other classes of stochastic programming problems, decomposition methods can provide desirable theoretical properties (such as finite convergence) as well as enhanced computational performance when compared to solving a deterministic equivalent formulation using an advanced commercial MIP solver.

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“…Note that while the convex hull was generated at termination in this example, this generally need not be the case for the algorithm to terminate. Computational results with a branch-andcut algorithm based on cuts from the CPT are presented in [33]. Jörg [34] proposes another finitely convergent pure cutting-plane algorithm for general MIP using multiterm disjunctions in conjunction with Gomory cuts.…”

Section: Cutting-plane-tree Algorithmmentioning

confidence: 99%