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

• Published in: Mathematical programming Vol. 135; no. 1-2; pp. 221 - 254
• Main Authors: Dash, Sanjeeb, Dey, Santanu S., Günlük, Oktay
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2012; Springer; Springer Nature B.V
• Subjects: Applied sciences; Asymmetry; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Corners; Equivalence; Euclidean space; Exact sciences and technology; Full Length Paper; Linear programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Mixed integer; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Polyhedrons; Springs; Theoretical; Two dimensional; Variables; 90C57; 90C11; Branching; Unimodular matrix; Mixed integer programming; Cut generation; Lattice; Polyhedron; Integer programming; Disjunction; Asymmetry; Disjunctive form; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0455-1

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 , and various types of disjunctions. Recently Li and Richard ( 2008 ), studied disjunctive cuts obtained from t -branch split disjunctions of mixed-integer sets (these cuts generalize split cuts). Balas (Presentation at the Spring Meeting of the American Mathematical Society (Western Section), San Francisco, 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.