We 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. We also present an alternative characterization of the crooked cross cut closure of mixed-integer sets similar to the one about the equivalence of different definitions of split cuts presented in Cook, Kannan, and Schrijver [4]. This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard [6]. Finally, we extend our results to mixed-integer sets that are defined as the set of points (with some components being integral) inside a general convex set.
By: Sanjeeb Dash; Santanu S. Dey; Oktay Günlük
Published in: RC25013 in 2010
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