computing efficiently the lattice width in any dimension we provide an algorithm for the exact computation of the lattice width of a set of points k in z 2 in linear time with respect to the size of k this method consists in computing a particular surrounding polygon from this polygon we deduce a set of candidate vectors allowing the computation of the lattice width moreover we describe how this new algorithm can be extended to an arbitrary dimension thanks to a greedy and practical approach to compute a surrounding polytope indeed this last computation is very efficient in practice as it processes only a few linear time iterations whatever the size of the set of points hence it avoids complex geometric processings

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

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

a probabilistic comparison of split and type 1 triangle cuts for two row mixed integer programs we provide a probabilistic comparison of split and type 1 triangle cuts for mixed integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff under a specific probabilistic model of the problem parameters we show that for the above measure the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut

discontinuous galerkin isogeometric analysis of elliptic diffusion problems on segmentations with gaps we propose a new discontinuous galerkin dg isogeometric analysis iga technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with nonmatching interfaces due to an incorrect segmentation procedure it may happen that the interfaces of adjacent subdomains do not coincide in this way gap regions which are not present in the original physical domain are created in this paper the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains we apply a multipatch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem the last formulation is further modified by deriving interface conditions without the presence of the solution in the gap finally the solution of this modified problem is approximated b

a polynomial algorithm for multiprocessor scheduling with two job lengths the following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft each maintenance period is a job and the maintenance facilities are machines in this context there are very few different types of maintenances performed so it is natural to consider the problem with only a small fixed number c of different types of jobs each job type has a processing time and each machine is available for the same length of time a machine can handle at most one job at a time all jobs are released at time zero there are no due dates or precedence constraints and preemption is not allowed the question is whether it is possible to finish all jobs we call this problem the multiprocessor scheduling problem with c job lengths mspc scheduling problems such as mspc where we can partition the jobs into relatively few types such that all jobs of a certain type are identical are often called high multiplicity problems high multiplicity problems are interesting because their input is very compact the input to mspc consists of only 2c 2 numbers for the case c 2 we present a polynomial time algorithm we show that this algorithm guarantees a schedule that uses the minimum possible number of different one machine schedules namely three further we extend this algorithm to the case of machine dependent deadlines uniform parallel machines to a multi parametric case that contains the case of unrelated parallel machines and to some related covering problems finally we give some counterexamples showing why our results do not extend to the case c 2

discontinuous galerkin isogeometric analysis of elliptic problems on segmentations with non matching interfaces abstract in this paper we develop a discontinuous galerkin isogeometric analysis method for solving elliptic problems on decompositions of the computational domain into volumetric patches with non matching parametrized interfaces we specially focus on high order numerical solutions for complex gap regions and extend ideas from our previous work on simple gap regions for the communication of the numerical solution between the subdomains which are separated by the gap region discontinuous galerkin numerical fluxes are constructed taking into account the diametrically opposite points on the boundary of the gap due to lack of information on the behavior of the solution in the gap region the fluxes coming from the interior of the gap are approximated by taylor expansions with respect to the adjacent subdomain solutions we follow the same ideas of our previous work and show a priori error estimates in the dg norm with respect to the mesh size and the gap distance numerical examples performed for two three and even four dimensional computational domains demonstrate the robustness of the proposed numerical method and validate the estimates predicted by the theory

cut generation through binarization for a mixed integer linear program where all integer variables are bounded we study a reformulation introduced by roy that maps general integer variables to a collection of binary variables we study theoretical properties and empirical strength of rank 2 simple split cuts of the reformulation we show that for a pure integer problem with two integer variables these cuts are sufficient to obtain the integer hull of the problem but that this does not generalize to problems in higher dimensions we also give an algorithm to compute an approximation of the rank 2 simple split cut closure we report empirical results on 22 benchmark instances showing that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice free cut closure it gives a better bound than the split closure on 6 instances while it is weaker on 9 instances for an average gap closed 3 8 smaller than the one for the split closure

a geometric perspective on lifting recently it has been shown that minimal inequalities for a continuous relaxation of mixed integer linear programs are associated with maximal lattice free convex sets in this paper we show how to lift these inequalities for integral nonbasic variables by considering maximal lattice free convex sets in a higher dimensional space we apply this approach to several examples in particular we identify cases in which the lifting is unique

inexact feti dp methods inexact feti dp domain decomposition methods are considered preconditioners based on formulations of feti dp as a saddle point problem are used which allow for an inexact solution of the coarse problem a positive definite reformulation of the preconditioned saddle point problem which also allows for approximate solvers is considered as well in the formulation that iterates on the original feti dp saddle point system it is also possible to solve the local neumann subdomain problems inexactly given good approximate solvers for the local and coarse problems convergence bounds of the same quality as for the standard feti dp methods are obtained numerical experiments which compare the convergence of the inexact methods with that of standard feti dp are shown for 2d and 3d elasticity using gmres and cg as krylov space methods based on parallel computations a comparison of one variant of the inexact feti dp algorithms and the standard feti dp method is carried out and similar parallel performance is achieved parallel scalability of the inexact variant is also demonstrated it is shown that for a very large number of subdomains and a very large coarse problem the inexact method can be superior copyright 2006 john wiley sons ltd

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

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

a polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed we prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d dimensional euclidean space previously such algorithms were known for dimensions d 1 2 3 and 4 only

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

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

constrained infinite group relaxations of mips recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized in this paper we consider a stronger relaxation of general mixed integer sets by allowing constraints such as bounds on the free integer variables in the continuous group relaxation we generalize a number of results for the continuous infinite group relaxation to this stronger relaxation and characterize the extreme inequalities when there are two integer variables

cutting planes for optimization of convex functions over nonconvex sets we derive linear inequality characterizations for sets of the form mathop rm conv x q in mathbb r d times mathbb r q ge q x x in mathbb r d mathop rm int p where q is convex and differentiable and p subset mathbb r d we show that in several cases our characterization leads to polynomial time separation algorithms that operate in the original space of variables in particular when q is a positive definite quadratic and p is a polyhedron or an ellipsoid

on the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs we compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints two unrestricted integer variables and any number of nonnegative continuous variables in particular we prove that the closure of type 2 triangle resp type 3 triangle quadrilateral inequalities are all within a factor of 1 5 1 5 of the integer hull and provide examples showing that the approximation factor is not less than 1 125 1 125 there is no fixed approximation ratio for split or type 1 triangle inequalities however

parallelization of continuous and discontinuous galerkin dual primal isogeometric tearing and interconnecting methods abstract in this paper we investigate the parallelization of dual primal isogeometric tearing and interconnecting ieti dp type methods for solving large scale continuous and discontinuous galerkin systems of equations arising from isogeometric analysis of elliptic boundary value problems these methods are extensions of the finite element tearing and interconnecting methods to isogeometric analysis the algorithms are implemented by means of energy minimizing primal subspaces we discuss how these methods can efficiently be parallelized in a distributed memory setting weak and strong scaling studies presented for two and three dimensional problems show an excellent parallel efficiency

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

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

two dimensional lattice free cuts and asymmetric disjunctions for mixed integer polyhedra in this paper we study the relationship between 2d lattice free cuts the family of cuts obtained by taking two row relaxations of a mixed integer program mip and applying intersection cuts based on maximal lattice free sets in r 2 and various types of disjunctions recently li and richard 2007 studied disjunctive cuts obtained from t branch split disjunctions of mixed integer sets these cu ts generalize split cuts balas 2009 initiated the study of cuts for the two row continuous group relaxation obtained from 2 branch split disjunctions we study these cuts and call them cross cuts for the two row continuous group relaxation and for general mips we also consider cuts obtained from asymmetric 2 branch disjunctions which we call crooked cross cuts for the two row continuous group relaxation we show that unimodular cross cuts the coefficients of the two split inequalities fo rm a unimodular matrix are equivalent to the cuts obtained from maximal lattice free sets other than type 3 triangles we also prove that all 2d lattice free cuts and their s free extensions ar e crooked cross cuts for general mixed integer sets we show that crooked cross cuts can be generated from a structured three row relaxation finally we show that for the corner relaxation of an mip every crooked cross cut is a 2d lattice free cut

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

cut generating functions for integer variables for an integer linear program gomory s corner relaxation is obtained by ignoring the nonnegativity of the basic variables in a tableau formulation in this paper we do not relax these nonnegativity constraints we generalize a classical result of gomory and johnson characterizing minimal cut generating functions in terms of subadditivity symmetry and periodicity our result is based on the notion of generalized symmetry condition we also prove a 2 slope theorem for extreme cut generating functions in our setting in the spirit of the 2 slope theorem of gomory and johnson

on minimal valid inequalities for mixed integer conic programs we study disjunctive conic sets involving a general regular closed convex full dimensional and pointed cone x1d4a6 such as the nonnegative orthant the lorentz cone or the positive semidefinite cone in a unified framework we introduce x1d4a6 minimal inequalities and show that under mild assumptions these inequalities together with the trivial cone implied inequalities are sufficient to describe the convex hull we focus on the properties of x1d4a6 minimal inequalities by establishing algebraic necessary conditions for an inequality to be x1d4a6 minimal this characterization leads to a broader algebraically defined class of x1d4a6 sublinear inequalities we demonstrate a close connection between x1d4a6 sublinear inequalities and the support functions of convex sets with a particular structure this connection results in practical ways of verifying x1d4a6 sublinearity and or x1d4a6 minimality of inequalities our study generalizes some of the results from the mixed integer linear case it is well known that the minimal inequalities for

toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations the solution of nonlinear problems e g in material science requires fast and highly scalable parallel solvers finite element tearing and interconnecting dual primal feti dp domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements recently nonlinear versions of the well known feti dp methods for linear problems have been introduced in these methods the nonlinear problem is decomposed before linearization this approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of feti dp methods toward extreme scale supercomputers here a recent nonlinear feti dp method is combined with an approach that allows an inexact solution of the feti dp coarse problem we combine the nonlinear feti dp domain decomposition method with an algebraic multigrid amg method and thus obtain a hybrid nonlinear domain decomposition multigrid method we consider scalar nonlinear problems as well as nonlinear hyp

maximal lattice free convex sets in linear subspaces we consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice free convex sets in an affine subspace we also show that these sets are polyhedra the latter result extends a theorem of lovasz characterizing maximal lattice free convex sets in rn

an algorithm for the separation of two row cuts we consider the question of finding deep cuts from a model with two rows of the type p i x s in mathbb z 2 times mathbb r n x f rs to do that we show how to reduce the complexity of setting up the polar of mathop mathrm conv p i from a quadratic number of integer hull computations to a linear number of integer hull computations furthermore we present an algorithm that avoids computing all integer hulls a polynomial running time is not guaranteed but computational results show that the algorithm runs quickly in practice

on the rank of mixed 0 1 polyhedra for a polytope in the 0 1 n cube eisenbrand and schulz showed recently that the maximum chvatal rank is bounded above by o n 2logn and bounded below by 1 e n for some e 0 chvatal cuts are equivalent to gomory fractional cuts which are themselves dominated by gomory mixed integer cuts what do these upper and lower bounds become when the rank is defined relative to gomory mixed integer cuts an upper bound of n follows from existing results in the literature in this note we show that the lower bound is also equal to n this result still holds for mixed 0 1 polyhedra with n binary variables

an asymptotically exact algorithm for the high multiplicity bin packing problem the bin packing problem consists of finding the minimum number of bins of given capacity d required to pack a set of objects each having a certain weight we consider the high multiplicity version of the problem in which there are only c different weight values we show that when c 2 the problem can be solved in time o log d for the general case we give an algorithm which provides a solution requiring at most c 2 bins more than the optimal solution i e an algorithm that is asymptotically exact for fixed c the complexity of the algorithm is o poly log d where poly is a polynomial function not depending on c

master corner polyhedron vertices abstract we focus on the vertices of the master corner polyhedron mcp a fundamental object in the theory of integer linear programming we introduce two combinatorial operations that transform vertices to their neighbors this implies that each mcp can be defined by the initial vertices regarding these operations we call them support vertices we prove that the class of support vertices of all mcps over a group is invariant under automorphisms of this group and describe mcp vertex bases among other results we characterize its irreducible points establish relations between a vertex and the nontrivial facets that pass through it and prove that this polyhedron is of diameter 2

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

computing two dimensional integer hulls an optimal algorithm is presented for computing the smallest set of linear inequalities that define the integer hull of a possibly unbounded two dimensional convex polygon r input to the algorithm is a set of linear inequalities defining r and the integer hull computed is the convex hull of the integer points of r it is proven that the integer hull has at most o n log amax inequalities where n is the number of input inequalities and amax is the magnitude of the largest input coefficient it is shown that the algorithm presented has complexity o n log amax and that this is optimal by proving that the integer hull may have omega n log a max inequalities in the worst case

a feti dp preconditioner for a composite finite element and discontinuous galerkin method in this paper a nitsche type discretization based on a discontinuous galerkin dg method for an elliptic two dimensional problem with discontinuous coefficients is considered the problem is posed on a polygonal region omega which is a union of n disjoint polygonal subdomains omega i of diameter o h i the discontinuities of the coefficients possibly very large are assumed to occur only across the subdomain interfaces partial omega i inside each subdomain a conforming finite element space on a quasi uniform triangulation with mesh size o h i is introduced to handle the nonmatching meshes across the subdomain interfaces a dg discretization is applied only on the interfaces for solving the resulting discrete system a feti dp type method is proposed and analyzed it is established that the condition number of the preconditioned linear system is estimated by c 1 max i log h i h i 2 with a constant c independent of h i h i h j h i and the jumps of coefficients the

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

dual primal isogeometric tearing and interconnecting solvers for multipatch dg iga equations abstract in this paper we consider a new version of the dual primal isogeometric tearing and interconnecting ieti dp method for solving large scale linear systems of algebraic equations arising from discontinuous galerkin dg isogeometric analysis of diffusion problems on multipatch domains with non matching meshes the dg formulation is used to couple the local problems across patch interfaces the purpose of this paper is to present this new method and provide numerical examples indicating a polylogarithmic condition number bound for the preconditioned system and showing an incredible robustness with respect to large jumps in the diffusion coefficient across the interfaces

unique lifting of integer variables in minimal inequalities this paper contributes to the theory of cutting planes for mixed integer linear programs milps minimal valid inequalities are well understood for a relaxation of an milp in tableau form where all the nonbasic variables are continuous they are derived using the gauge function of maximal lattice free convex sets in this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities we characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting this result first appeared in the proceedings of ipco 2010 the answer is a nonconvex region that can be obtained as a finite union of convex polyhedra we then establish a necessary and sufficient condition for the uniqueness of the lifting function

on the relationship between standard intersection cuts lift and project cuts and generalized intersection cuts we examine the connections between the classes of cuts in the title we show that lift and project l p cuts from a given disjunction are equivalent to generalized intersection cuts from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction while l p cuts from split disjunctions are known to be equivalent to standard intersection cuts sics from the strip obtained by complementing the terms of the split we show that l p cuts from more general disjunctions may not be equivalent to any sic in particular we give easily verifiable necessary and sufficient conditions for a l p cut from a given disjunction d to be equivalent to a sic from the polyhedral counterpart of d irregular l p cuts i e those that violate these conditions have interesting properties for instance unlike the regular ones they may cut off part of the corner polyhedron associated with the lp solution from which they are derived furthermore they are not exceptional their frequency exceeds that of regular cuts a numerical example illustrates some of the above properties

adaptive selection of primal constraints for isogeometric bddc deluxe preconditioners isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer aided design cad software and the discretization of variational problems of continuum mechanics in contrast with the finite element case the basis functions of isogeometric analysis are often not nodal as a consequence there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains building on earlier work on the deluxe version of the bddc balancing domain decomposition by constraints family of domain decomposition algorithms several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global coarse component of these preconditioners numerical experiments provide evidence that this work can be successful yielding scalable and quasi optimal adaptive bddc algorithms for isogeometric discretizations

on the facets of mixed integer programs with two integer variables and two constraints in this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables nonnegative continuous variables and two equality constraints and we give a complete characterization of its facets we also derive an analogous characterization of the facets of the underlying finite integer program

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

minimal cut generating functions are nearly extreme we study continuous strongly minimal cut generating functions for the model where all variables are integer we consider both the original gomory johnson setting as well as a recent extension by yildiz and cornuejols math oper res 41 1381 1403 2016 we show that for any continuous minimal or strongly minimal cut generating function there exists an extreme cut generating function that approximates the strongly minimal function as closely as desired in other words the extreme functions are dense in the set of continuous strongly minimal functions

cut generating functions and s free sets we consider the separation problem for sets x that are pre images of a given set s by a linear mapping classical examples occur in integer programming as well as in other optimization problems such as complementarity one would like to generate valid inequalities that cut off some point not lying in x without reference to the linear mapping to this aim we introduce a concept cut generating functions cgf and we develop a formal theory for them largely based on convex analysis they are intimately related to s free sets and we study this relation disclosing several definitions for minimal cgf s and maximal s free sets our work unifies and puts into perspective a number of existing works on s free sets in particular we show how cgf s recover the celebrated gomory cuts

split rank of triangle and quadrilateral inequalities a simple relaxation consisting of two rows of a simplex tableau is a mixed integer set with two equations two free integer variables and nonnegative continuous variables recently andersen et al and cornuejols and margot showed that the facet defining inequalities of this set are either split cuts or intersection cuts obtained from lattice free triangles and quadrilaterals from an example given by cook et al it is known that one particular class of facet defining triangle inequality does not have finite split rank in this paper we show that all other facet defining triangle and quadrilateral inequalities have finite split rank

equivariant perturbation in gomory and johnson s infinite group problem i the one dimensional case we give an algorithm for testing the extremality of minimal valid functions for gomory and johnson s infinite group problem that are piecewise linear possibly discontinuous with rational breakpoints this is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions we also present an extreme function that is a piecewise linear function with some irrational breakpoints whose extremality follows from a new principle

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

the value of continuity refined isogeometric analysis and fast direct solvers abstract we propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers starting from a highly continuous isogeometric analysis iga discretization we introduce c 0 separators to reduce the interconnection between degrees of freedom in the mesh by doing so both the solution time and best approximation errors are simultaneously improved we call the resulting method refined isogeometric analysis riga to illustrate the impact of the continuity reduction we analyze the number of floating point operations flops computational times and memory required to solve the linear system obtained by discretizing the laplace problem with structured meshes and uniform polynomial orders theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 with p being the polynomial order of the discretization numerical results indicate that our proposed refined isogeometric analysis delivers a speed up factor proportional to p 2 in a 2 d mesh with four million elements and p 5 the linear system resulting from riga is solved 22 times faster than the one from highly continuous iga in a 3 d mesh with one million elements and p 3 the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis

origin and early evolution of corner polyhedra abstract corner polyhedra are a natural intermediate step between linear programming and integer programming this paper first describes how the concept of corner polyhedra arose unexpectedly from a practical operations research problem and then describes how it evolved to shed light on fundamental aspects of integer programming and to provide a great variety of cutting planes for integer programming

a warped product splitting theorem the celebrated cheeger gromoll splitting theorem see chgr and also eshe wu implies that a connected complete riemannian manifold having nonnegative ricci curvature and two ends is isometric to a riemannian product where one factor is a line there is a version of this theorem for compact connected riemannian manifolds m with two smooth boundary components it states that if m has nonnegative ricci curvature and two boundary components both of which have nonnegative mean curvature with respect to their inward normals then m is isometric to the riemannian product of n1 with a closed interval in particular n1 is isometric to n2 in this paper we will prove an analog of this result for manifolds with ricci curvature n 1 since our argument parallels the proof of the splitting theorem for manifolds with nonnegative ricci curvature we present the proofs simultaneously by letting be either zero or one in what follows

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

cutting planes from extended lp formulations given a mixed integer set defined by linear inequalities and integrality requirements on some of the variables we consider extended formulations of its continuous lp relaxation and study the effect of adding cutting planes in the extended space in terms of optimization extended lp formulations do not lead to better bounds as their projection onto the original space is precisely the original lp relaxation however adding cutting planes in the extended space can lead to stronger bounds in this paper we show that for every 0 1 mixed integer set with n integer and k continuous variables there is an extended lp formulation with 2n k 1 2n k 1 variables whose elementary split closure is integral the proof is constructive but it requires an inner description of the lp relaxation we then extend this idea to general mixed integer sets and construct the best extended lp formulation for such sets with respect to lattice free cuts we also present computational results on the two row continuous group relaxation showing the strength of cutting planes derived from extended lp formulations

index parity of closed geodesics and rigidity of hopf fibrations the diameter rigidity theorem of gromoll and grove 1987 states that a riemannian manifold with sectional curvature 1 and diameter 2 is either homeomorphic to a sphere locally isometric to a rank one symmetric space or it has the cohomology ring of the cayley plane ca the reason that they were only able to recognize the cohomology ring of ca is due to an exceptional case in another theorem gromoll and grove 1988 a riemannian submersion m b b with connected fibers that is defined on the euclidean sphere m is metrically congruent to a hopf fibration unless possibly m b 15 8 we will rule out the exceptional cases in both theorems our argument relies on a rather unusal application of morse theory for that purpose we give a general criterion which allows to decide whether the morse index of a closed geodesic is even or odd

focal radius rigidity and lower curvature bounds this is the accepted version of the following article luis guijarro and frederick wilhelm focal radius rigidity and lower curvature bounds which has been published in final form at https doi org 10 1112 plms 12113

two row mixed integer cuts via lifting recently andersen et al 1 borozan and cornuejols 6 and cornuejols and margot 9 have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non negative continuous variables these inequalities are either split cuts or intersection cuts derived using maximal lattice free convex sets in order to use these inequalities to obtain cuts from two rows of a general simplex tableau one approach is to extend the system to include all possible non negative integer variables giving the two row 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 study the characteristics of these lifting functions we show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice free triangle with multiple integer points in the relative interior of one of its sides or a maximal lattice free triangle with integral vertices and one integer point in the relative interior of each side in the other cases maximal lattice free triangles with one integer point in the relative interior of each side and non integral vertices and maximal lattice free quadrilaterals non unique lifting functions may yield distinct extreme inequalities for the latter family of triangles we present sufficient conditions to yield an extreme inequality for the two row mixed integer infinite group problem

ranking functions for linear constraint loops in this article we study the complexity of the problems given a loop described by linear constraints over a finite set of variables is there a linear or lexicographical linear ranking function for this loop while existence of such functions implies termination these problems are not equivalent to termination when the variables range over the rationals or reals it is known that both problems are ptime decidable however when they range over the integers whether for single path or multipath loops the complexity has not yet been determined we show that both problems are conp complete however we point out some special cases of importance of ptime complexity we also present complete algorithms for synthesizing linear and lexicographical linear ranking functions both for the general case and the special ptime cases moreover in the rational setting our algorithm for synthesizing lexicographical linear ranking functions extends existing ones because our definition for such functions is more general yet it has ptime complexity

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

a note on the split rank of intersection cuts in this note we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts as a first step of this argument a polyhedral subset of the lattice free convex set that is used to generate the intersection cut is constructed we call this subset the restricted lattice free set it is then shown that lceil log 2 l rceil is a lower bound on the split rank of the intersection cut where l is the number of integer points lying on the boundary of the restricted lattice free set satisfying the condition that no two points lie on the same facet of the restricted lattice free set the use of this result is illustrated by obtaining a lower bound of lceil log 2 n 1 rceil on the split rank of n row mixing inequalities

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

the master equality polyhedron with multiple rows the master equality polyhedron mep is a canonical set that generalizes the master cyclic group polyhedron mcgp of gomory we recently characterized a nontrivial polar for the mep i e a polyhedron t such that an inequality defines a nontrivial facet of the mep if and only if its coefficient vector forms a vertex of t in this paper we study the mep when it is defined by m 1 rows we define the notion of a polaroid a set containing all nontrivial facet defining inequalities we show how to use linear programming lp to efficiently solve the separation problem for the mep when the polaroid has a compact polyhedral description we obtain such descriptions via subadditivity conditions when m 2 or m 3 and using lp duality show how to efficiently optimize over the mep these results yield a pseudo polynomial time lp based algorithm to solve the problem min cx ax b x geq bf 0 x in mathbb z n when a has at most three constraints for the mcgp and the mep defined by a single constraint the notions of two term subadditivity and valid inequalities for mep are essentially equivalent we show this is not true in the case of the mep when m 3 in fact we prove that subadditivity conditions with a sub exponential number of terms do not imply validity in particular when m 3 we show that four term subadditivity conditions are necessary and sufficient for validity

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

sufficiency of cut generating functions the concept of cut generating function has its origin in the work of gomory and johnson from the 1970s it has received renewed attention in the past few years recently conforti cornuejols daniilidis lemarechal and malick proposed a general framework for studying cut generating functions however they gave an example showing that not all cuts can be produced by cut generating functions in this framework they conjectured a natural condition under which cut generating functions might be sufficient this note settles this open problem

mixed integer sets from two rows of two adjacent simplex bases in andersen et al lecture notes in computer science vol 4513 springer berlin pp 1 15 2007 we studied a mixed integer set arising from two rows of a simplex tableau we showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables in this paper we generalize our findings and show that when upper bounds on the non basic variables are also considered further classes of facets arise that cannot be obtained from triangles and quadrilaterals specifically when exactly one upper bound on a non basic variable is introduced stronger inequalities that can be derived from pentagons involving up to six variables also appear