Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

Abstract

In this paper, we study the relationship between 2D lattice-free cuts, the family of cuts obtained by taking two-row relaxations of a mixed-integer program (MIP) and applying intersection cuts based on maximal lattice-free sets in R 2 , and various types of disjunctions. Recently, Li and Richard (2007) studied disjunctive cuts obtained from t-branch split disjunctions of mixed-integer sets (these cuts generalize split cuts). Balas (2009) initiated the study of cuts for the two-row continuous group relaxation obtained from 2-branch split disjunctions. We study these cuts (and call them cross cuts) for the two-row continuous group relaxation, and for general MIPs. We also consider cuts obtained from asymmetric 2-branch disjunctions which we call crooked cross cuts. For the two-row continuous group relaxation, we show that unimodular cross cuts (the coefficients of the two split inequalities form a unimodular matrix) are equivalent to the cuts obtained from maximal lattice-free sets other than type 3 triangles. We also prove that all 2D lattice-free cuts and their S-free extensions are crooked cross cuts. For general mixed integer sets, we show that crooked cross cuts can be generated from a structured three-row relaxation. Finally, we show that for the corner relaxation of an MIP, every crooked cross cut is a 2D lattice-free cut.

Computational Experiments with Cross and Crooked Cross Cuts

“…The cross disjunction (see Dash et al (2012a)) associated to π 1 , π 2 ∈ Z n 1 \ {0}, and γ 1 , γ 2 ∈ Z, is given by…”

Corner polyhedron and intersection cuts

“…Dash, Dey and Günlük, [38] proved an intriguing result for model (3) when |B| = p = 2. They showed that corner(B) is defined entirely by disjunctive cuts from crooked cross disjunctions of the form {x ∈ R 2 : π 1 x ≤ π 1 0 , (π 2 − π 1 )x ≤ π 2 0 − π 1 0 } ∨ {x ∈ R 2 : π 1 x ≤ π 1 0 , (π 2 − π 1 )x ≥ π 2 0 − π 1 0 + 1} ∨ {x ∈ R 2 : π 1 x ≥ π 1 0 + 1, π 2 x ≤ π 2 0 } ∨ {x ∈ R 2 : π 1 x ≥ π 1 0 + 1, π 2 x ≥ π 2 0 + 1} where π 1 , π 2 ∈ Z 2 and π 1 0 , π 2 0 ∈ Z.…”

On mixed-integer sets with two integer variables

“…. , 4, then it is called a CC cut for P obtained from the disjunctionD(π 1 , π 2 , γ 1 , γ 2 ), see [3]. In other words a linear inequality is called a CC cut if it is valid for P LP \ int(C).…”

“…The extension of D(π 1 , π 2 , γ 1 , γ 2 ) is defined to beD(π 1 , π 2 , γ 1 , γ 2 ) = {(x, y) ∈ R n 1 +n 2 : x ∈ D(π 1 , π 2 , γ 1 , γ 2 )} and it is called a crooked cross (CC) disjunction for Z n 1 × R n 2 , see [3]. The extensions of the sets in (1)- (4), are defined similarly and each such extension is called an atom of the disjunctionD(π 1 , π 2 , γ 1 , γ 2 ).…”

“…

AbstractWe show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined recently in [3]). We then extend this observation to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables provided that the coefficients of the integer variables form a matrix of rank 2.

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