Integer and combinatorial optimization
Integer and combinatorial optimization
Chva´tal closures for mixed integer programming problems
Mathematical Programming: Series A and B
On the relative strength of split, triangle and quadrilateral cuts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Minimal Valid Inequalities for Integer Constraints
Mathematics of Operations Research
A Geometric Perspective on Lifting
Operations Research
Lifted Tableaux Inequalities for 0--1 Mixed-Integer Programs: A Computational Study
INFORMS Journal on Computing
Split Rank of Triangle and Quadrilateral Inequalities
Mathematics of Operations Research
Experiments with Two-Row Cuts from Degenerate Tableaux
INFORMS Journal on Computing
Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three
Mathematics of Operations Research
Constrained Infinite Group Relaxations of MIPs
SIAM Journal on Optimization
Intersection Cuts with Infinite Split Rank
Mathematics of Operations Research
On lifting integer variables in minimal inequalities
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Experiments with two row tableau cuts
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Unique Minimal Liftings for Simplicial Polytopes
Mathematics of Operations Research
On degenerate multi-row Gomory cuts
Operations Research Letters
Computing with multi-row Gomory cuts
Operations Research Letters
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Recently, Andersen et al. [1] and Borozan and Cornuéjols [3] characterized the minimal inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these minimal inequalities to obtain cuts from two rows of a general simplex tableau, it is necessary to extend the system to include integer variables (giving the two-dimensional mixed integer infinite group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we analyze the lifting of minimal inequalities derived from lattice-free triangles. Maximal lattice-free triangles in R2 can be classified into three categories: those with multiple integral points in the relative interior of one of its sides, those with integral vertices and one integral point in the relative interior of each side, and those with non integral vertices and one integral point in the relative interior of each side. We prove that the lifting functions are unique for each of the first two categories such that the resultant inequality is minimal for the mixed integer infinite group problem, and characterize them. We show that the lifting function is not necessarily unique in the third category. For this category we show that a fill-in inequality (Johnson [11]) yields minimal inequalities for mixed integer infinite group problem under certain sufficiency conditions. Finally, we present conditions for the fill-in inequality to be extreme.