Rational Polyhedron Research Articles

Minimizing Convex Functions with Rational Minimizers

Given a separation oracle SO for a convex function f defined on ℝ n that has an integral minimizer inside a box with radius R , we show how to find an exact minimizer of f using at most O(n (n log log (n)/ log (n) + log ( R ))) calls to SO and poly ( n , log ( R )) arithmetic operations, or O(n log (nR) calls to SO and exp ( O(n) ) ⋅ poly (log (R) ) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n 2 ( n + log ( R ))) for polynomial time algorithms and O(n 2 log ( nR ) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n 3 log log ( n )/log ( n )) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n 2 log ( n )) calls to an evaluation oracle. These improve upon the previously best O(n 3 log 2 ( n )) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n 3 log ( n )) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.

Polyhedra, lattice structures, and extensions of semigroups

For an arbitrary rational polyhedron we consider its decompositions into Minkowski summands and, dual to this, the free extensions of the associated pair of semigroups. Being free for the pair of semigroups is equivalent to flatness for the corresponding algebras. Our main result is phrased in this dual setup: the category of free extensions always contains an initial object, which we describe explicitly. These objects seem to be related to unique liftings in log geometry. Further motivation comes from the deformation theory of the associated toric singularity.

On the complexity of recognizing integrality and total dual integrality of the {0,1/2}-closure

The {0,12}-closure of a rational polyhedron {x:Ax≤b} is obtained by adding all Gomory-Chvátal cuts that can be derived from the linear system Ax≤b using multipliers in {0,12}. We show that deciding whether the {0,12}-closure coincides with the integer hull is strongly NP-hard. A direct consequence of our proof is that, testing whether the linear description of the {0,12}-closure derived from Ax≤b is totally dual integral, is strongly NP-hard.

On a generalization of the Chvátal–Gomory closure

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvátal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf, 2012) considered a strengthened version of Chvátal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints.

Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups

Abstract Several recent papers have examined a rational polyhedron Pm whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m. A combinatorial description of the faces of Pm was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of Pm containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of Pm . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of Pm .

Generalized Chvátal-Gomory closures for integer programs with bounds on variables

Integer programming problems that arise in practice often involve decision variables with one or two sided bounds. In this paper, we consider a generalization of Chvatal-Gomory inequalities obtained by strengthening Chvatal-Gomory inequalities using the bounds on the variables. We prove that the closure of a rational polyhedron obtained after applying the generalized Chvatal-Gomory inequalities is also a rational polyhedron. This generalizes a result of Dunkel and Schulz on 0–1 problems to the case when some of the variables have upper or lower bounds or both while the rest of them are unbounded.

Exact Augmented Lagrangian Duality for Mixed Integer Quadratic Programming

Mixed integer quadratic programming (MIQP) is the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. This paper focuses on the augmented Lagrangian dual (ALD) for MIQP. ALD augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomially bound on the weight on the penalty term to obtain a zero gap.

Lattice closures of polyhedra

Given \(P\subset {\mathbb {R}}^n\), a mixed-integer set \(P^I=P\cap ({\mathbb {Z}}^{t}\times {\mathbb {R}}^{n-t}\)), and a k-tuple of n-dimensional integral vectors \((\pi _1, \ldots , \pi _k)\) where the last \(n-t\) 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 et al. (Math Program 47:155–174, 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 \(k\ge 2\). We also construct a polyhedral mixed-integer 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, full-dimensional, convex lattice-free sets.

On the NP-hardness of deciding emptiness of the split closure of a rational polytope in the 0,1 hypercube

Split cuts are prominent general-purpose cutting planes in integer programming. The split closure of a rational polyhedron is what is obtained after intersecting the half-spaces defined by all the split cuts for the polyhedron. In this paper, we prove that deciding whether the split closure of a rational polytope is empty is NP-hard, even when the polytope is contained in the unit hypercube. As a direct corollary, we prove that optimization and separation over the split closure of a rational polytope in the unit hypercube are NP-hard, extending an earlier result of Caprara and Letchford.

When the Gomory–Chvátal closure coincides with the integer hull

Gomory–Chvátal cuts are prominent in integer programming. The Gomory–Chvátal closure of a polyhedron is the intersection of the half spaces defined by all its Gomory–Chvátal cuts. We prove that it is NP-hard to decide whether the Gomory–Chvátal closure of a rational polyhedron P is identical to the integer hull of P. An earlier version of this paper appeared in the proceedings of IPCO 2016.

Fans, decision problems and generators of free abelian ℓ\ell-groups

Abstract Let t 1 , … , t n {t_{1},\ldots,t_{n}} be ℓ {\ell} -group terms in the variables X 1 , … , X m {X_{1},\ldots,X_{m}} . Let t ^ 1 , … , t ^ n {\hat{t}_{1},\ldots,\hat{t}_{n}} be their associated piecewise homogeneous linear functions. Let G be the ℓ {\ell} -group generated by t ^ 1 , … , t ^ n {\hat{t}_{1},\ldots,\hat{t}_{n}} in the free m-generator ℓ {\ell} -group 𝒜 m {\mathcal{A}_{m}} . We prove: (i) the problem whether G is ℓ {\ell} -isomorphic to 𝒜 n {\mathcal{A}_{n}} is decidable; (ii) the problem whether G is ℓ {\ell} -isomorphic to 𝒜 l {\mathcal{A}_{l}} (l arbitrary) is undecidable; (iii) for m = n {m=n} , the problem whether { t ^ 1 , … , t ^ n } {\{\hat{t}_{1},\ldots,\hat{t}_{n}\}} is a free generating set is decidable. In view of the Baker–Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the ℓ {\ell} -group G. We make pervasive use of fans and their stellar subdivisions.

On the Polyhedrality of Closures of Multibranch Split Sets and Other Polyhedra with Bounded Max-Facet-Width

For a fixed integer $t > 0$, we say that a $t$-branch split set (the union of $t$ split sets) is dominated by another one on a polyhedron $P$ if all cuts for $P$ obtained from the first $t$-branch split set are implied by cuts obtained from the second one. We prove that given a rational polyhedron $P$, any arbitrary family of $t$-branch split sets has a finite subfamily such that each element of the family is dominated on $P$ by an element from the subfamily. The result for $t=1$ (i.e., for split sets) was proved by Averkov [Discrete Optim., 9 (2012), pp. 209--215] extending results in Andersen, Cornuejols, and Li [Math. Program, 102 (2005), pp. 457--493]. Our result implies that the closure of $P$ with respect to any family of $t$-branch split sets is a polyhedron. We extend this result by replacing split sets with bounded max-facet-width polyhedra as building blocks, and show that any family of $t$-branch sets where each set is the union of $t$ polyhedral sets that have bounded max-facet-width has a fin...

On the polyhedrality of cross and quadrilateral closures

Split cuts form a well-known class of valid inequalities for mixed-integer programming problems. Cook et al. (Math Program 47:155---174, 1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We then use this result to prove that cross cuts yield closures that are rational polyhedra. Cross cuts are a generalization of split cuts introduced by Dash et al. (Math Program 135:221---254, 2012). Finally, we show that the quadrilateral closure of the two-row continuous group relaxation is a polyhedron, answering an open question in Basu et al. (Math Program 126:281---314, 2011).

Surface subgroups from linear programming

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive b2 obtained by doubling free groups along collections of subgroups and groups obtained by “random” ascending HNN (Higman–Neumann–Neumann) extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank 2 free group sending a to ab and b to ba; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension 2) of a double of a free group along a collection of subgroups is a finite-sided rational polyhedron and that every rational class is virtually represented by an extremal surface subgroup. These results are obtained by a mixture of combinatorial, geometric, and linear programming techniques.

Invariant Measure Under the Affine Group Over

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d &lt; n then λd(P)&gt;0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

Rademacher–Carlitz polynomials

We introduce and study the \emph{Rademacher-Carlitz polynomial} \[ \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1} u^{\fl{\frac{ka + t}{b}}} v^k \] where $a, b \in \Z_{>0}$, $s, t \in \R$, and $u$ and $v$ are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view $\RC(u, v, s, t, a, b)$ as a polynomial analogue (in the sense of Carlitz) of the \emph{Dedekind-Rademacher sum} \[ \r_t(a,b) := \sum_{k=0}^{b-1}\left(\left(\frac{ka+t}{b} \right)\right) \left(\left(\frac{k}{b} \right)\right), \] which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms \[ \sigma(x,y):=\sum_{(j,k) \in \mathcal{P}\cap \Z^2} x^j y^k \] of any rational polyhedron $\mathcal{P}$, and we derive a novel reciprocity theorem for Dedekind-Rademacher sums, which follows naturally from our setup.

On the Convergence of the Affine Hull of the Chvátal--Gomory Closures

Given an integral polyhedron $P\subseteq\mathbb{R}^n$ and a rational polyhedron $Q\subseteq\mathbb{R}^n$ containing the same integer points as $P$, we investigate how many iterations of the Chvátal--Gomory closure operator have to be performed on $Q$ to obtain a polyhedron contained in the affine hull of $P$. We show that if $P$ contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on $n$. On the other hand, we prove that if $P$ is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.

On finitely generated closures in the theory of cutting planes

Let P be a rational polyhedron in Rd and let L be a class of d-dimensional maximal lattice-free rational polyhedra in Rd. For L∈L by RL(P) we denote the convex hull of points belonging to P but not to the interior of L. Andersen, Louveaux and Weismantel showed that if the so-called max-facet-width of all L∈L is bounded from above by a constant independent of L, then ⋂L∈LRL(P) is a rational polyhedron. We give a short proof of a generalization of this result. We also give a characterization for the boundedness of the max-facet-width on L. The presented results are motivated by applications in cutting-plane theory from mixed-integer optimization.

Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three

A convex set with nonempty interior is maximal lattice-free if it is inclusion maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron P in ℝd is the smallest natural number s such that sP is an integral polyhedron. In this paper we show that, up to affine mappings preserving ℤd, the number of maximal lattice-free rational polyhedra of a given precision s is finite. Furthermore, we present the complete list of all maximal lattice-free integral polyhedra in dimension three. Our results are motivated by recent research on cutting plane theory in mixed-integer linear optimization.

On convergence in mixed integer programming

Let $${P \subseteq {\mathbb R}^{m+n}}$$ be a rational polyhedron, and let P I be the convex hull of $${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$$ . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in $${{\mathbb R}^m}$$ . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvatal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P I .