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Sanjeeb Dash (sanjeebdus.ibm.com) Abstract: Given $P\subset\R^n$, a mixed integer set $P^I=P\cap (\Z^{t}\times\R^{nt}$), and a $k$tuple of $n$dimensional integral vectors $(\pi_1, \ldots, \pi_k)$ where the last $nt$ entries of each vector is zero, we consider the relaxation of $P^I$ obtained by taking the convex hull of points $x$ in $P$ for which $ \pi_1^Tx,\ldots,\pi^T_kx$ are integral. We then define the $k$dimensional lattice closure of $P^I$ to be the intersection of all such relaxations obtained from $k$tuples of $n$dimensional vectors. When $P$ is a rational polyhedron, we show that given any collection of such $k$tuples, there is a finite subcollection that gives the same closure; more generally, we show that any $k$tuple is dominated by another $k$tuple coming from the finite subcollection. The $k$dimensional lattice closure contains the convex hull of $P^I$ and is equal to the split closure when $k=1$. Therefore, a result of Cook, Kannan, and Schrijver (1990) implies that when $P$ is a rational polyhedron, the $k$dimensional lattice closure is a polyhedron for $k=1$ and our finiteness result extends this to all $k\ge2$. We also construct a polyhedral mixedinteger set with $n$ integer variables and one continuous variable such that for any $k < n$, finitely many iterations of the $k$dimensional lattice closure do not give the convex hull of the set. Our result implies that $t$branch split cuts cannot give the convex hull of the set, nor can valid inequalities from unbounded, fulldimensional, convex latticefree sets. Keywords: integer programming, cutting planes, split cuts, split closure, lattice closure Category 1: Integer Programming ((Mixed) Integer Linear Programming ) Category 2: Integer Programming (Cutting Plane Approaches ) Citation: Download: [PDF] Entry Submitted: 10/30/2016 Modify/Update this entry  
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