-

 

 

 




Optimization Online





 

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

Sanjeeb Dash (sanjeebd***at***us.ibm.com)
Santanu S. Dey (sdey30***at***isye.gatech.edu)
Oktay Gunluk (gunluk***at***us.ibm.com)

Abstract: In this paper, we study the relationship between {\em 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 {\em 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 {\em crooked cross} cuts. For the two-row continuous group relaxation, we show that {\em 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 triangles with one integer point in the relative interior of each side. We also show that all 2D lattice-free cuts are crooked cross cuts, and that some cutting planes in the literature for variants of the two-row continuous group relaxation are crooked cross cuts. For general mixed integer sets, we extend results in Nemhauser and Wolsey~\cite{NW90} on the equivalence of split cuts and mixed-integer rounding cuts. Finally, we show that for the corner relaxation of an MIP, every crooked cross cut is a 2D lattice-free cut.

Keywords: disjunctive cuts, lattice-free cuts, continous group relaxations, cross cuts

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 2: Integer Programming (Cutting Plane Approaches )

Citation:

Download: [PDF]

Entry Submitted: 03/28/2010
Entry Accepted: 03/29/2010
Entry Last Modified: 01/03/2011

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Programming Society