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 polynomial interpolation at the points of a geometric progression this note pursues two aims the first is historical and the second is factual 1 we present james stirling s discovery 1730 that newton s general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression for its most important case of extrapolation at the origin karl schellbach 1864 develops his algorithm of q differences that also leads naturally to theta functions carl runge 1891 solves the same extrapolation at the origin without referring to the stirling schellbach algorithm instead runge uses richardson s deferred approach to the limit 20 years before richardson 2 recently the author found a close connection to romberg s quadrature formula in terms of binary trapezoidal sums it is shown that the problems of stirling schellbach and runge are elegantly solved by romberg s algorithm numerical examples are given briefly fuller numerical details can be found in the author s mrc t s report 2173 december 1980 madison wisconsin thanks are due to the referee for suggesting the present stream lined version

the triangle closure is a polyhedron recently cutting planes derived from maximal lattice free convex sets have been studied intensively by the integer programming community an important question in this research area has been to decide whether the closures associated with certain families of lattice free sets are polyhedra for a long time the only result known was the celebrated theorem of cook kannan and schrijver who showed that the split closure is a polyhedron although some fairly general results were obtained by andersen et al math oper res 35 1 233 256 2010 and averkov discret optimiz 9 4 209 215 2012 some basic questions have remained unresolved for example maximal lattice free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron in this paper we show that when the number of integer variables m 2 the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data the techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet defining due to cornuejols and margot math program 120 429 456 2009 and obtain polynomial complexity results about the mixed integer hull

intersection cuts with infinite split rank we consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables when this model only has two integer variables dey and louveaux characterized the intersection cuts that have infinite split rank we show that for any number of integer variables the split rank of an intersection cut generated from a rational lattice free polytope l is finite if and only if the integer points on the boundary of l satisfy a certain 2 hyperplane property the dey louveaux characterization is a consequence of this more general result

polynomial interpolation at points of a geometric mesh on a triangle in an earlier paper 8 i j schoenberg discussed polynomial interpolation in one dimension at the points of a geometric progression which was originally proposed by james stirling in the present paper these ideas are generalised to two dimensional polynomial interpolation at the points of a geometric mesh on a triangle a lagrange form is obtained for this interpolating polynomial and an algorithm is derived for evaluating it efficiently

facets of two dimensional infinite group problems in this paper we lay the foundation for the study of the two dimensional mixed integer infinite group problem 2dmiigp we introduce tools to determine if a given continuous and piecewise linear function over the two dimensional infinite group is subadditive and to determine whether it defines a facet of 2dmiigp we then present two different constructions that yield the first known families of facet defining inequalities for 2dmiigp the first construction uses valid inequalities of the one dimensional integer infinite group problem 1diigp as building blocks for creating inequalities for the two dimensional integer infinite group problem 2diigp we prove that this construction yields all continuous piecewise linear facets of the two dimensional group problem that have exactly two gradients the second construction we present has three gradients and yields facet defining inequalities of 2dmiigp whose continuous coefficients are not dominated by those of facets of the one dimensional mixed integer infinite group problem 1dmiigp

notions of maximality for integral lattice free polyhedra the case of dimension three lattice free sets and their applications for cutting plane methods in mixed integer optimization have been studied in recent literature the family of all integral lattice free polyhedra that are not properly contained in another integral lattice free polyhedron has been of particular interest we call these polyhedra d maximal for fixed d the family of d maximal integral lattice free polyhedra is finite up to unimodular equivalence in view of possible applications in cutting plane theory one would like to have a classification of this family this is a challenging task already for small dimensions in contrast the subfamily of all integral lattice free polyhedra that are not properly contained in any other lattice free set which we call d maximal lattice free polyhedra allow a rather simple geometric characterization hence the question was raised for which dimensions the notions of d maximality and d maximality are equivalent this was known to be the case for dimensions one and two on the

lattice free sets multi branch split disjunctions and mixed integer programming in this paper we study the relationship between valid inequalities for mixed integer sets lattice free sets associated with these inequalities and the multi branch split cuts introduced by li and richard discret optim 5 724 734 2008 by analyzing n dimensional lattice free sets we prove that for every integer n there exists a positive integer t such that every facet defining inequality of the convex hull of a mixed integer polyhedral set with n integer variables is a t branch split cut we use this result to give a finite cutting plane algorithm to solve mixed integer programs we also show that the minimum value t for which all facets of polyhedral mixed integer sets with n integer variables can be generated as t branch split cuts grows exponentially with n in particular when n 3 we observe that not all facet defining inequalities are 6 branch split cuts

on an analysis of the strength of mixed integer cutting planes from multiple simplex tableau rows in this paper we derive valid inequalities for general mixed integer linear programs milps by considering several simplex tableau rows simultaneously we explore links between these cutting planes and lattice free convex sets which have a representation as the sum of a polytope and a linear space the codimension of the linear space of such a set is called split dimension we show that in terms of a strength measure introduced by goemans math programming 69 1995 pp 335 349 only lattice free convex sets with full split dimension give rise to a good approximation of the mixed integer hull whereas lattice free convex sets with low split dimension might approximate the mixed integer hull arbitrarily badly in general however we also show that ordinary split cuts approximate the mixed integer hull to within a constant factor when the size of the input data is given

largest integral simplices with one interior integral point solution of hensley s conjecture and related results for each dimension d d dimensional integral simplices with exactly one interior integral point have bounded volume this was first shown by hensley explicit volume bounds were determined by hensley lagarias and ziegler pikhurko and averkov in this paper we determine the exact upper volume bound for such simplices and characterize the volume maximizing simplices we also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point this result confirms a conjecture of hensley from 1983 moreover for an integral simplex with precisely one interior integral point we give bounds on the volumes of its faces the barycentric coordinates of the interior integral point and its number of integral points furthermore we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points the presented results have applications in toric geometry and in integer optimization

an analysis of mixed integer linear sets based on lattice point free convex sets a maximal lattice free polyhedron l has max facet width equal to w if documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document max x in l pi t x min x in l pi t x leq w end document for all facets documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document pi t x leq pi 0 end document of l and documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usep

lifting properties of maximal lattice free polyhedra we study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice free polyhedra we prove a basic invariance property of unique minimal liftings for general maximal lattice free polyhedra this generalizes a previous result by basu et al math oper res 37 2 346 355 2012 for simplicial maximal lattice free polytopes thus completely settling this fundamental question about lifting for maximal lattice free polyhedra we further give a very general iterative construction to get maximal lattice free polyhedra with the unique lifting property in arbitrary dimensions this single construction not only obtains all previously known polyhedra with the unique lifting property but goes further and vastly expands the known list of such polyhedra finally we extend characterizations from basu et al 2012 about lifting with respect to maximal lattice free simplices to more general polytopes these nontrivial generalizations rely on a number of results from discrete geometry including the venkov alexandrov mcmullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces

integral polyhedra in three space in a series of recent papers i have introduced a particular class of convex polyhedra in rn arising in the study of integer programming problems with n variables in the present paper a detailed analysis of these polyhedra will be given for the case in which n 3 the analysis is based on an unpublished theorem demonstrated several years ago by roger howe which seems to have no immediate generalization to higher values of n

minimal valid inequalities for integer constraints in this paper we consider a semi infinite relaxation of mixed integer linear programs we show that minimal valid inequalities for this relaxation correspond to maximal lattice free convex sets and that they arise from nonnegative piecewise linear positively homogeneous convex functions

projecting lattice polytopes without interior lattice points we show that up to unimodular equivalence in each dimension there are only finitely many lattice polytopes without interior lattice points that do not admit a lattice projection onto a lower dimensional lattice polytope without interior lattice points this was conjectured by treutlein treutlein j 2008 3 dimensional lattice polytopes without interior lattice points september 10 http arxiv org abs 0809 1787 as an immediate corollary we get a short proof of a recent result of averkov wagner and weismantel averkov g c wagner r weismantel 2010 maximal lattice free polyhedra finiteness and an explicit description in dimension three math oper res forthcoming namely the finiteness of the number of maximal lattice polytopes without interior lattice points moreover we show that in dimension four and higher some of these finitely many polytopes are not maximal as convex bodies without interior lattice points

homogeneous polynomial splines we construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions these give a rich and elegant theory which combines properties of polynomial box splines see 6 and the references therein with the explicit representation of simple exponential box splines 11 while enjoying complete symmetry in the three variables by a linear transformation followed by a projection on suitable planes one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines the vertices of these pencils may be finite or one or two may be infinite i e the corresponding pencils may comprise parallel lines as a limiting case all three vertices become infinite and one recovers polynomial box splines on a three direction mesh

efficient estimation of semiparametric multivariate copula models we propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions a joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions this class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable and its circumvention of the curse of dimensionality associated with purely nonparametric multivariate distributions we show that the plug in sieve maximum likelihood estimators mles of all smooth functionals including the finite dimensional copula parameters and the unknown marginal distributions are semiparametrically efficient and that their asymptotic variances can be estimated consistently moreover prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve fu

blowing up convex sets in the plane abstract let k be a convex set in r 2 such that every line in r 2 meets k z 2 we prove that k z 2 r 2 for 1 2 3 3 and that this bound is best possible thus solving a problem of kannan and lovasz

on convergence in mixed integer programming let p subseteq mathbb r m n be a rational polyhedron and let pi 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 pi 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 pi

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

a geometric approach to cut generating functions the cutting plane approach to integer programming was initiated more than 40 years ago gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and balas introduced intersection cuts for the corner polyhedron this line of research was left dormant for several decades until relatively recently when a paper of andersen louveaux weismantel and wolsey generated renewed interest in the corner polyhedron and intersection cuts recent developments rely on tools drawn from convex analysis geometry and number theory and constitute an elegant bridge between these areas and integer programming we survey these results and highlight recent breakthroughs in this area