Corner polyhedron and intersection cuts

Conforti, Michele; Cornuéjols, Gérard; Zambelli, Giacomo

doi:10.1016/j.sorms.2011.03.001

Cited by 42 publications

(68 citation statements)

References 85 publications

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“…As pointed out in [8], the nonnegativity assumption in the definition of a valid function might seem artificial at first. Although there exist valid inequalities r∈R π(r)s(r) ≥ α for R f (R, Z) such that π(r) < 0 for some r ∈ R, it can be shown that π must be nonnegative over all rational r ∈ Q.…”

Section: Valid Inequalities and Valid Functionsmentioning

confidence: 99%

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

Basu¹,

Hildebrand²,

Köppe³

2016

Math. Program.

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Abstract. We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, we present the general regular solution to Cauchy's additive functional equation on restricted lower-dimensional convex domains. This provides a k-dimensional generalization of the so-called Interval Lemma, allowing us to deduce affine properties of the function from certain additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions.We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.

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Section: Valid Inequalities and Valid Functionsmentioning

confidence: 99%

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

Basu¹,

Hildebrand²,

Köppe³

2016

Math. Program.

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“…Now, the set of variables G does not contain any of the u t j with j ∈ J \ M t , t ∈ T . Consequently, (14) can be rewritten as…”

Section: Sic and Landp Cutsmentioning

confidence: 99%

“…Restricted intersection cuts have recently been shown [13,14] to dominate all valid inequalities for corner polyhedra. More specifically, every nontrivial minimal valid inequality for a nonempty corner polyhedron is an intersection cut from some lattice-free set (Theorem 1 of [13]).…”

Section: Landp Cuts Disjunctive Hulls and Corner Polyhedramentioning

confidence: 99%

“…as cuts obtained by intersecting the extreme rays of the cone C(J) with the boundary of some P I -free set S, were introduced in the early 1970's [3,4]. More recently, intersection cuts became the focus of renewed interest in the context of cut generation from multiple rows of the simplex tableau (see [1], [16], [14]). However, this more recent literature uses a narrower definition of intersection cuts, namely as cuts obtained by intersecting the extreme rays of C(J) with the boundary of some convex set S ′ whose interior contains no integer point (feasible or not).…”

Section: Introductionmentioning

confidence: 99%

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On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts

Balas

Kis

2016

Math. Program.

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We examine the connections between the classes of cuts in the title. We show that lift-and-project (L&P) cuts from a given disjunction are equivalent to generalized intersection cuts (GICs) from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction. While L&P cuts from split disjunctions are known to be equivalent to standard intersection cuts (SICs) from the strip obtained by complementing the terms of the split, we show that L&P cuts from more general disjunctions may not be equivalent to any SIC. In particular, we give easily verifiable necessary and sufficient conditions for a L&P cut from a given disjunction D to be equivalent to a SIC from the polyhedral counterpart of D. Irregular L&P cuts, i.e. those that violate these conditions, have interesting properties. For instance, unlike the regular ones, they may cut off part of the corner polyhedron associated with the LP solution from which they are derived. Furthermore, they are not exceptional: their frequency exceeds that of regular cuts. A numerical example illustrates some of the above properties.

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“…With a view toward applications in the cutting-plane theory from integer and mixed-integer optimization it is desirable to have upper bounds on f ((Z d × R n ) ∩ C), where d ∈ N, n ∈ N 0 and C ⊆ R d+n is convex. See also [1,4,6,12,13,17,31] for information on application of S-free sets for generation of cutting planes. The main topic of this manuscript is the study of the relationship between f (S) and the Helly number h(S) for the family of of S-convex sets.…”

Section: Introductionmentioning

confidence: 99%

On Maximal $S$-Free Sets and the Helly Number for the Family of $S$-Convex Sets

Averkov¹

2013

SIAM J. Discrete Math.

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We study two combinatorial parameters, which we denote by f (S) and h(S), associated to an arbitrary set S ⊆ R d , where d ∈ N. In the nondegenerate situation, f (S) is the largest possible number of facets of a d-dimensional polyhedron L such that the interior of L is disjoint with S and L is inclusion-maximal with respect to this property. The parameter h(S) is the Helly number of the family of all sets that can be given as the intersection of S with a convex subset of R d . We obtain the inequality f (S) ≤ h(S) for an arbitrary S and the equality f (S) = h(S) for every discrete S. Furthermore, motivated by research in integer and mixed-integer optimization, we show that 2 d is the sharp upper bound on

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Corner polyhedron and intersection cuts

Cited by 42 publications

References 85 publications

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

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

On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts

On Maximal $S$-Free Sets and the Helly Number for the Family of $S$-Convex Sets

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