bayesian analysis of dichotomous quantal response models abstract qualitative response models qrm s are analyzed from the bayesian point of view using diffuse and informative prior distributions exact finite sample bayesian and large sample bayesian and non bayesian estimation results are compared in addition the paper provides 1 plots and discussion of the properties of likelihood functions for qrm s 2 posterior distributions for logit models derivatives and elasticities 3 bayesian prediction procedures for qrm s 4 new estimates for the median and other fractiles of the logistic distribution 5 posterior odds ratios for model selection problems and 6 comparisons of two alternative monte carlo numerical integration procedures it is concluded that asymptotic approximations are not accurate for small to moderate sized samples even when only a single input variable is used and that operational bayesian methods are available for providing both exact small sample and large sample approximate inferences for drm s

tight formulations for some simple mixed integer programs and convex objective integer programs we study the polyhedral structure of simple mixed integer sets that generalize the two variable set s z r 1 z 1 s z b these sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs for four such sets we give a complete description by valid inequalities and or an integral extended formulation and we also indicate what constraints can be added without destroying integrality we apply these results to provide tight formulations for certain piecewise linear convex objective integer programs and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot sizing problems

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 the exact separation of mixed integer knapsack cuts during the last decades much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets which we call knapsack cuts bixby et al the sharpest cut the impact of manfred padberg and his work mps siam series on optimization pp 309 326 2004 empirically observe that within the context of branch and cut algorithms to solve mixed integer programming problems the most important inequalities are knapsack cuts derived by the mixed integer rounding mir procedure in this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set the main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships the exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them using these benchmarks on miplib 3 0 and miplib 2003 instances we analyze the performance of mir inequalities our computations which are performed in exact arithmetic are surprising in the vast majority of the instances in which knapsack cuts yield bound improvements mir cuts alone achieve over 87 of the observed gain

a new general purpose algorithm for mixed integer bilevel linear programs bilevel optimization problems are very challenging optimization models arising in many important practical contexts including pricing mechanisms in the energy sector airline and telecommunication industry transportation networks critical infrastructure defense and machine learning in this paper we consider bilevel programs with continuous and discrete variables at both levels with linear objectives and constraints continuous upper level variables if any must not appear in the lower level problem we propose a general purpose branch and cut exact solution method based on several new classes of valid inequalities which also exploits a very effective bilevel specific preprocessing procedure an extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances our new algorithm consistently outperforms often by a large margin alternative state of the art metho

on the separation of disjunctive cuts disjunctive cuts for mixed integer linear programs mips were introduced by egon balas in the late 1970s and have been successfully exploited in practice since the late 1990s in this paper we investigate the main ingredients of a disjunctive cut separation procedure and analyze their impact on the quality of the root node bound for a set of instances taken from miplib library we compare alternative normalization conditions and try to better understand their role in particular we point out that constraints that become redundant because of the disjunction used can produce over weak cuts and analyze this property with respect to the normalization used finally we introduce a new normalization condition and analyze its theoretical properties and computational behavior along the way we make use of a number of small numerical examples to illustrate some basic and often misinterpreted disjunctive programming features

pivot and reduce cuts an approach for improving gomory mixed integer cuts abstract gomory mixed integer cuts are of crucial importance in solving large scale mixed integer linear programs recently there has been considerable research particularly on the strengthening of these cuts we present a new heuristic algorithm which aims at improving gomory mixed integer cuts our approach is related to the reduce and split cuts these cuts are based on an algorithm which tries to reduce the coefficients of the continuous variables by forming integer linear combinations of simplex tableau rows our algorithm is designed to achieve the same result by performing pivots on the simplex tableau we give a detailed description of the algorithm and its implementation finally we report on computational results with our approach and analyze its performance the results indicate that our algorithm can enhance the performance of the gomory mixed integer cuts

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

computing bayes factors using a generalization of the savage dickey density ratio abstract we present a simple method for computing bayes factors the method derives from observing that in general a bayes factor can be written as the product of a quantity called the savage dickey density ratio and a correction factor both terms are easily estimated from posterior simulation in some cases it is possible to do these computations without ever evaluating the likelihood

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

binary extended formulations of polyhedral mixed integer sets we analyze different ways of constructing binary extended formulations of polyhedral mixed integer sets with bounded integer variables and compare their relative strength with respect to split cuts we show that among all binary extended formulations where each bounded integer variable is represented by a distinct collection of binary variables what we call unimodular extended formulations are the strongest we also compare the strength of some binary extended formulations from the literature finally we study the behavior of branch and bound on such extended formulations and show that branching on the new binary variables leads to significantly smaller enumeration trees in some cases

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

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

split closure and intersection cuts in the seventies balas introduced intersection cuts for a mixed integer linear program milp and showed that these cuts can be obtained by a closed form formula from a basis of the standard linear programming relaxation in the early nineties cook kannan and schrijver introduced the split closure of a milp and showed that the split closure is a polyhedron in this paper we show that the split closure can be obtained using only intersection cuts we give two different proofs of this result one geometric and one algebraic the result is then used to provide a new proof of the fact that the split closure of a milp is a polyhedron finally we extend the result to more general disjunctions

on cutting planes for cardinality constrained linear programs we derive cutting planes for cardinality constrained linear programs these inequalities can be used to separate any basic feasible solution of an lp relaxation of the problem assuming that this solution violates the cardinality requirement to derive them we first relax the given simplex tableau into a disjunctive set expressed in the space of nonbasic variables we establish that coefficients of valid inequalities for the closed convex hull of this set obey ratios that can be computed directly from the simplex tableau we show that a transportation problem can be used to separate these inequalities we then give a constructive procedure to generate violated facet defining inequalities for the closed convex hull of the disjunctive set using a variant of prim s algorithm

on mixing inequalities rank closure and cutting plane proofs we study the mixing inequalities that were introduced by gunluk and pochet math program 90 2001 pp 429 457 we show that a mixing inequality which mixes n mir inequalities has mir rank at most n if it is a type i mixing inequality and at most n 1 if it is a type ii mixing inequality we also show that these bounds are tight for n 2 given a mixed integer set p i p cap z i where p is a polyhedron and z i x in mathbb r n x i in mathbb z forall i in i we define mixing inequalities for p i we show that the elementary mixing closure of p with respect to i can be described using a bounded number of mixing inequalities each of which has a bounded number of terms this implies that the elementary mixing closure of p is a polyhedron finally we show that any mixing inequality can be derived via a polynomial length mir cutting plane proof combined with results of dash on the complexity of cutting plane proofs using split cuts ibm research report rc 24082 oct 2006 and pudlak j symbolic logic 62 1997 pp 981 998 this implies that there are valid inequalities for a certain mixed integer set that cannot be obtained via a polynomial size mixing cutting plane proof

unique minimal liftings for simplicial polytopes for a minimal inequality derived from a maximal lattice free simplicial polytope in rn we investigate the region where minimal liftings are uniquely defined and we characterize when this region covers rn we then use this characterization to show that a minimal inequality derived from a maximal lattice free simplex in rn with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points

computing deep facet defining disjunctive cuts for mixed integer programming the problem of separation is to find an affine hyperplane or cut that lies between the origin o and a given closed convex set q in a euclidean space we study cuts which are deep in a well defined geometrical sense and facet defining the cases when the deepest cut is decomposable as a combination of facet defining cuts are characterized using the reverse polar set when q is a disjunctive polyhedron a description of the reverse polar linked to the so called cut generating linear program of lift and project techniques is given a successive projections algorithm onto the reverse polar is proposed that computes the decomposition of the deepest cut into facet defining cuts illustrative numerical experiments show how these cuts compare with the deepest cut and with the most violated cut

a recursive procedure to generate all cuts for 0 1 mixed integer programs we study several ways of obtaining valid inequalities for mixed integer programs we show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to gomory s mixed integer cuts we also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities

on the spectrum of six vertex corner transfer matrices we consider the eigenstates of the six vertex corner transfer matrix using its generator which is an operator of heisenberg xxz type in the ordered regime we construct a complete set of fundamental eigenstates and some bethe ansatz eigenstates as well in the disordered critical regime the spectrum of the finite size system collapses in the thermodynamic limit and we show how this depends on in m the size of the system and

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

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

disjunctive programming and relaxations of polyhedra given a polyhedron l with h facets whose interior contains no integral points and a polyhedron p recent work in integer programming has focused on characterizing the convex hull of p minus the interior of l we show that to obtain such a characterization it suffices to consider all relaxations of p defined by at most n h 1 among the inequalities defining p this extends a result by andersen cornuejols and li

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

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

finite disjunctive programming characterizations for general mixed integer linear programs in this paper we give a finite disjunctive programming procedure to obtain the convex hull of general mixed integer linear programs milp with bounded integer variables we propose a finitely convergent convex hull tree algorithm that constructs a linear program that has the same optimal solution as the associated milp in addition we combine the standard notion of sequential cutting planes with ideas underlying the convex hull tree algorithm to help guide the choice of disjunctions to use within a cutting plane method this algorithm which we refer to as the cutting plane tree algorithm is shown to converge to an integral optimal solution in finitely many iterations finally we illustrate the proposed algorithm on three well known examples in the literature that require an infinite number of elementary or split disjunctions in a rudimentary cutting plane algorithm

reverse split rank the reverse split rank of an integral polyhedron p p is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is p p already in mathbb r 3 r3 there exist polyhedra with infinite reverse split rank we give a geometric characterization of the integral polyhedra in mathbb r n rn with infinite reverse split rank

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

a disjunctive cutting plane procedure for general mixed integer linear programs in this paper we develop a cutting plane algorithm for solving mixed integer linear programs with general integer variables a novel feature of the algorithm is that it generates inequalities at all optimal vertices of the lp relaxation at each iteration the cutting planes generated in the procedure are found by considering a natural generalization of the 0 1 disjunction used by balas ceria and cornuejols in the context of solving binary mixed integer linear programs 3 4

reduce and split cuts improving the performance of mixed integer gomory cuts mixed integer gomory cuts have become an integral part of state of the art software for solving mixed integer linear programming problems therefore improvements in the performance of these cutting planes can be of great practical value in this paper we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed integer gomory cuts this is motivated by the fact that in a mixed integer gomory cut the coefficient of an integer variable lies between 0 and 1 whereas for a continuous variable there is no upper bound the heuristic tries to reduce the coefficients of the continuous variables we call the resulting cuts reduce and split cuts we found that on several test problems reduce and split cuts can substantially enhance the performance of a branch and bound algorithm

corner transfer matrix of generalized free fermion vertex systems the hamiltonian limit of the corner transfer matrix ctm of a generalized free fermion vertex system of finite size leads to a quantum spin hamiltonian of the particular form hn sigma n 1n 1 n sigma nx sigma n 1x lambda sigma ny sigma n 1y h sigma nz sigma sub n 1z diagonalization may be achieved for all pairs of parameters lambda h with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of ctm

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

the ancestral benders cutting plane algorithm with multi term disjunctions for mixed integer recourse decisions in stochastic programming this paper focuses on solving two stage stochastic mixed integer programs smips with general mixed integer decision variables in both stages we develop a decomposition algorithm in which the first stage approximation is solved by a branch and bound algorithm with its nodes inheriting benders cuts that are valid for their ancestor nodes in addition we develop two closely related convexification schemes which use multi term disjunctive cuts to obtain approximations of the second stage mixed integer programs we prove that the proposed methods are finitely convergent one of the main advantages of our decomposition scheme is that we use a benders based branch and cut approach in which linear programming approximations are strengthened sequentially moreover as in many decomposition schemes these subproblems can be solved in parallel we also illustrate these algorithms using several variants of an smip example from the literature as well as a new set of test problems which we refer to as stochastic server location and sizing finally we present our computational experience with previously known examples as well as the new collection of smip instances our experiments reveal that our algorithm is able to produce provably optimal solutions within an hour of cpu time even in instances for which a highly reliable commercial mip solver is unable to provide an optimal solution within an hour of cpu time

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 j 1 n and b 1 b n the set s generalizes the mixed integer rounding mir set of nemhauser and wolsey and the mixing mir set of gunluk and pochet in addition it arises as a substructure in general mixed integer programming mip such as in lot sizing despite its importance a number of basic questions about s remain unanswered including the tractability of optimization over s and how to efficiently find a most violated cutting plane valid for p conv s we address these questions by analyzing the extreme points and extreme rays of p we give all extreme points and extreme rays of p in the worst case the number of extreme points grows exponentially with n however we show that in some interesting cases it is bounded by a polynomial of n in such cases it is possible to derive strong cutting planes for p efficiently finally we use our results on the extreme points of p to give a polynomial time algorithm for solving optimization over s

diagonalisation of corner transfer matrix by orthogonal polynomials the authors discuss the generator of baxter s corner transfer matrix 1982 for a critical ising model of finite size they perform its diagonalisation in terms of special orthogonal polynomials and give an asymptotic expression of the eigenvalues for large size which agrees with the conformal predictions

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

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

scalable bayesian model averaging through local information propagation this article shows that a probabilistic version of the classical forward stepwise variable inclusion procedure can serve as a general data augmentation scheme for model space distributions in generalized linear models this latent variable representation takes the form of a markov process thereby allowing information propagation algorithms to be applied for sampling from model space posteriors in particular we propose a sequential monte carlo method for achieving effective unbiased bayesian model averaging in high dimensional problems using proposal distributions constructed using local information propagation the method called lips for local information propagation based sampling is illustrated using real and simulated examples with dimensionality ranging from 15 to 1000 and its performance in estimating posterior inclusion probabilities and in out of sample prediction is compared to those of several other methods namely mcmc bas ibma and lasso in addition it is shown that the latent variab

intersection cuts for nonlinear integer programming convexification techniques for structured sets we study the generalization of split k branch split and intersection cuts from mixed integer linear programming to the realm of mixed integer nonlinear programming constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure we introduce two techniques to give precise characterizations of such convex hulls and use them to construct split k branch split and intersection cuts for several classes of non polyhedral sets in particular we give simple formulas for split cuts for essentially all convex sets described by a single conic quadratic inequality we also give simple formulas for k branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets

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

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

posterior odds ratios for regression hypotheses general considerations and some specific results abstract much work is econometrics and statistics has been concerned with comparing bayesian and non bayesian estimation results while much less has involved comparisons of bayesian and non bayesian analyses of hypotheses some issues arising in this latter area that are mentioned and discussed in the paper are 1 is it meaningful to associate probabilities with hypotheses 2 what concept of probability is to be employed in analyzing hypotheses 3 is a separate theory of testing needed 4 must a theory of testing be capable of treating both sharp and non sharp hypotheses 5 how is prior information incorporated in testing 6 does the use of power functions in practice necessitate the use of prior information 7 how are significance levels determined when sample sizes are large and what are the interpretations of p values and tail areas 8 how are conflicting results provided by asymptotically equivalent testing procedures to be reconciled 9 what is the rationale for the 5 accept reject syndrome that afflicts econometrics and applied statistics 10 does it make sense to test a null hypothesis with no alternative hypothesis present and 11 how are the results of analyses of hypotheses to be combined with estimation and prediction procedures brief discussions of these issues with references to the literature are provided since there is much controversy concerning how hypotheses are actually analyzed in applied work the results of a small survey relating to 22 articles employing empirical data published in leading economic and econometric journals in 1978 are presented the major results of this survey indicate that there is wide spread use of the 1 and 5 levels of significance in non bayesian testing with no systematic relation between choice of significance level and sample size also power considerations are not generally discussed in empirical studies in fact there was a discussion of power in only one of the articles surveyed further there was very little formal or informal use of prior information employed in testing hypotheses and practically no attention was given to the effects of tests or pre tests on the properties of subsequent tests or estimation results these results indicate that there is much room for improvement in applied analyses of hypotheses given the findings of the survey of applied studies it is suggested that bayesian procedures for analyzing hypotheses may be helpful in improving applied analyses in this connection the paper presents a review of some bayesian procedures and results for analyzing sharp and non sharp hypotheses with explicit use of prior information in general bayesian procedures have good sampling properties and enable investigators to compute posterior probabilities and posterior odds ratios associated with alternative hypotheses quite readily the relationships of several posterior odds ratios to usual non bayesian testing procedures is clearly demonstrated also a relation between the p value or tail area and a posterior odds ratio is described in detail in the important case of hypotheses about a mean of a normal distribution other examples covered in the paper include posterior odds ratios for the hypotheses that 1 i and i where i is a regression coefficient 2 data are drawn from either of two alternative distributions 3 0 and where is the mean of a normal distribution 4 0 and 0 where is a vector of regression coefficients 5 2 0 vs 2 0 where 1 2 is a vector regression coefficients and 1 s value is unrestricted in several cases is a vector of regression coefficients and 1 s value is unrestricted in several cases tabulations of odds ratios are provided bayesian versions of the chow test for equality of regression coefficients and of the goldfeld quandt test for equality of disturbance variances are given also an application of bayesian posterior odds ratios to a regression model selection problem utilizing the hald data is reported in summary the results reported in the paper indicate that operational bayesian procedures for analyzing many hypotheses encountered in model selection problems are available these procedures yield posterior odds ratios and posterior probabilities for competing hypotheses these posterior odds ratios represent the weight of the evidence supporting one model or hypothesis relative to another given a loss structure as is well known one can choose among hypotheses so as to minimize expected loss also with posterior probabilities available and an estimation or prediction loss function it is possible to choose a point estimate or prediction that minimizes expected loss by averaging over alternative hypotheses or models thus it is seen that the bayesian approach for analyzing competing models or hypotheses provides a unified framework that is extremely useful in solving a number of model selection problems

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

the continuous mixing polyhedron we analyze the polyhedral structure of the setspcmix s r z r xr nxz n s r j z j f j j 1 n andp cmix pcmixn s 0 the setp cmix is a natural generalization of the mixing set studied by pochet and wolsey 15 16 and ga nla k and pochet 8 and recently has been introduced by miller and wolsey 12 we introduce a new class of valid inequalities that has proven to be sufficient for describing conv pcmix we give an extended formulation of sizeo n xo n2 variables and constraints and indicate how to separate over conv pcmix ino n3 time finally we show how the mixed integer rounding mir inequalities of nemhauser and wolsey 14 and the mixing inequalities of ga nla k and pochet 8 constitute special cases of the cycle inequalities

a precise correspondence between lift and project cuts simple disjunctive cuts and mixed integer gomory cuts for 0 1 programming we establish a precise correspondence between lift and project cuts for mixed 0 1 programs simple disjunctive cuts intersection cuts and mixed integer gomory cuts the correspondence maps members of one family onto members of the others it also maps bases of the higher dimensional cut generating linear program cglp into bases of the linear programming relaxation it provides new bounds on the number of facets of the elementary closure and on the rank of the standard linear programming relaxation of the mixed 0 1 polyhedron with respect to the above families of cutting planes

polyhedra for lot sizing with wagner whitin costs we examine the single item lot sizing problem with wagner whitin costs over ann period horizon i e pt ht pt 1 fort 1 n 1 wherept ht are the unit production and storage costs in periodt respectively so it always pays to produce as late as possible

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

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

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

alternative tests of rational expectations models the case of the term structure a linearized version of the rational expectations models of the term structure is put forth in terms of a complete vector of equally spaced observations along the yield curve a data series on intermediate maturity yields which meets the specifications of the model is presented the model is tested against a specific and easily interpreted alternative earlier studies of rational expectations models which used volatility tests or likelihood ratio tests are discussed this abstract was borrowed from another version of this item

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

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

finite size scaling of corner transfer matrices for the two dimensional ising model the eigenvalue spectra of generalised corner transfer matrices are calculated for an ising model on a square lattice the predictions of conformal invariance at the critical point are thereby verified

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

model choice in generalised linear models a bayesian approach via kullback leibler projections summary we propose a general bayesian method of comparing models the approach is based on the kullback leibler distance between two families of models one nested within the other for each parameter value of a full model we compute the projection of the model to the restricted parameter space and the corresponding minimum distance from the posterior distribution of the minimum distance we can judge whether or not a more parsimonious model is appropriate we show how the projection method can be implemented for generalised linear model selection and we propose some markov chain monte carlo algorithms for its practical implementation in less tractable cases we illustrate the method with examples

on the rank of disjunctive cuts let l be a family of lattice free polyhedra in rm containing the splits given a polyhedron p in rm n we characterize when a valid inequality for p zm rn can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l we also characterize the lattice free polyhedra m such that all the disjunctive cuts corresponding to m can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l for every polyhedron p our results imply interesting consequences related to split rank and to integral lattice free polyhedra that extend recent research findings

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

mir closures of polyhedral sets we study the mixed integer rounding mir closures of polyhedral sets the mir closure of a polyhedral set is equal to its split closure and the associated separation problem is np hard we describe a mixed integer programming mip model with linear constraints and a non linear objective for separating an arbitrary point from the mir closure of a given mixed integer set we linearize the objective using additional variables to produce a linear mip model that solves the separation problem exactly using a subset of these additional variables yields an mip model which solves the separation problem approximately with an accuracy that depends on the number of additional variables used our analysis yields an alternative proof of the result of cook et al 1990 that the split closure of a polyhedral set is again a polyhedron we also discuss a heuristic to obtain mir cuts based on our approximate separation model and present some computational results

generalized intersection cuts and a new cut generating paradigm intersection cuts are generated from a polyhedral cone and a convex set s whose interior contains no feasible integer point we generalize these cuts by replacing the cone with a more general polyhedron c the resulting generalized intersection cuts dominate the original ones this leads to a new cutting plane paradigm under which one generates and stores the intersection points of the extreme rays of c with the boundary of s rather than the cuts themselves these intersection points can then be used to generate in a non recursive fashion cuts that would require several recursive applications of some standard cut generating routine a procedure is also given for strengthening the coefficients of the integer constrained variables of a generalized intersection cut the new cutting plane paradigm yields a new characterization of the closure of intersection cuts and their strengthened variants this characterization is minimal in the sense that every one of the inequalities it uses defines a facet of the closure

invariant bayesian inference in regression models that is robust against the jeffreys lindley s paradox we obtain the prior and posterior probability of a nested regression model as the hausdorff integral of the prior and posterior on the parameters of an encompassing linear regression model over a lower dimensional set that represents the nested model the hausdorff integral is invariant and therefore avoids the borel kolmogorov paradox basing priors and prior probabilities of nested regression models on the prior on the parameters of an encompassing linear regression model reduces the discrepancies between classical and bayesian inference like the jeffreys lindley s paradox we illustrate the analysis with examples of linear restrictions i e a linear regression model and non linear restrictions i e a cointegration and an autoregressive moving average model on the parameters of an encompassing linear regression model

conformal invariance and critical spectrum of corner transfer matrices the finite size scaling spectra of ising model corner transfer matrices and their generators are studied at criticality the generators are diagonalised using fermion algebra the low lying eigenvalues given by the zeros of meixner polynomials are equally spaced and collapse like 1 log n for large n as predicted by conformal invariance the asymptotics are obtained using a generalised euler maclaurin summation formula the shift in the largest eigenvalue is given analytically as pi c 6 log n with central charge c 1 2 the spectrum generating functions for both fixed and free boundary conditions are expressed simply in terms of the c 1 2 virasoro characters chi delta q with modular parameter q exp pi log n and conformal dimensions delta 0 1 2 1 16

outer product free sets for polynomial optimization and oracle based cuts this paper introduces cutting planes that involve minimal structural assumptions enabling the generation of strong polyhedral relaxations for a broad class of problems we consider valid inequalities for the set s cap p where s is a closed set and p is a polyhedron given an oracle that provides the distance from a point to s we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron these cuts are generated from convex forbidden zones or s free sets derived from the oracle we also consider the special case of polynomial optimization accordingly we develop a theory of outer product free sets where s is the set of real symmetric matrices of the form xx t all maximal outer product free sets of full dimension are shown to be convex cones and we identify several families of such sets these families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently computational experiments demonstrate the promise of our approach

bayes factors for nonlinear hypotheses and likelihood distributions summary new methods are proposed which allow bayes factors to be computed for hypotheses which impose nonlinear restrictions on the parameters projection methods are used to induce the prior distribution over the restricted parameter space which is required for computation of the bayes factor various distance metrics are introduced to define the projection including a utility based metric which gives kullback leibler divergence as a special case draws from the restricted and unrestricted prior distributions are used to construct marginal distributions of the likelihood which is shown to have additional diagnostic value over and above the bayes factor these methods are applied to hypotheses statisticians are frequently asked to entertain hypotheses which are nonlinear in the model parameters nonlinear hypotheses arise both from subject matter theory as well as from the process of model selection to address nonlinear hypotheses researchers use frequentist methods which are usually implemented using asymptotic approximations berger 1987 zellner 1971 1978 1984 and others have discussed the weaknesses of the classical significance testing approach and suggest bayesian alternatives such as odds ratios the odds ratio is the ratio of posterior probabilities of two competing hypotheses and is the product of the prior odds ratio and the bayes factor the odds ratio has the conceptual advantages of providing a natural probability metric for assessing the strength of data and prior beliefs because of the difficulties of assessing priors under the nonlinear restrictions and the computation of the required integrals few bayes factors have been computed for nonlinear hypotheses see lindley 1988 for an exception in the present paper we outline a general method for computing bayes factors which is flexible conceptually appealing and computationally tractable the literature on bayes factors starts with the work of jeffreys 1961 other references include berger delampady 1987 dickey 1971 good 1985 kass vaidyanathan 1992 smith spiegelhalter 1980 and zellner 1971 1978 1984 these papers deal with the testing of linear restrictions on the parameter space often in the context of linear models even the widely used schwarz 1978 model selection criterion considers only linear restrictions in models of the exponential family form the key difficulty which has kept researchers from developing bayes factors for nonlinear hypotheses is the

corner transfer matrices for the ising model we consider corner transfer matrices for the regular ising model and show that finding a diagonal representation is a soluble problem in interacting fermions we show how the requirement of having an invertible diagonalising transformation determines which representation must be used and explain why baxter s original solution of the problem did not produce such an invertible transformation

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

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

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

facet separation with one linear program given polyhedron p and and a point x the separation problem for polyhedra asks to certify that x in p and if not to determine an inequality that is satisfied by p and violated by x this problem is repeatedly solved in cutting plane methods for integer programming and the quality of the violated inequality is an essential feature in the performance of such methods in this paper we address the problem of finding efficiently an inequality that is violated by x and either defines an improper face or a facet of p we show that by solving a single linear program one almost surely obtains such an improper face or facet

characterization of the split closure via geometric lifting abstract we analyze split cuts from the perspective of cut generating functions via geometric lifting we show that cuts a natural higher dimensional generalization of the k cuts of cornuejols et al give all the split cuts for the mixed integer corner relaxation as an immediate consequence we obtain that the k cuts are equivalent to split cuts for the 1 row mixed integer relaxation further we show that split cuts for finite dimensional corner relaxations are restrictions of split cuts for the infinite dimensional relaxation in a final application of this equivalence we exhibit a family of pure integer programs whose split closure has arbitrarily bad integrality gap this complements the mixed integer example provided by basu et al 2011

corner transfer matrix of a critical free fermion system in order to diagonalize the generator of the corner transfer matrix of a critical free fermion 8 vertex system with magnetic field the authors derive an integral representation for the eigenvector components of the generator at large n they numerically verify the predictions of conformal invariance for critical systems

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

optimizing over the split closure the polyhedron defined by all the split cuts obtainable directly i e without iterated cut generation from the lp relaxation p of a mixed integer program mip is termed the elementary or rank 1 split closure of p this paper deals with the problem of optimizing over the elementary split closure this is accomplished by repeatedly solving the following separation problem given a fractional point say x find a rank 1 split cut violated by x or show that none exists following caprara and letchford 17 we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program pmilp with a single parameter in the objective function and the right hand side we develop an algorithmic framework to deal with the resulting pmilp by creating and maintaining a dynamically updated grid of parameter values and use the corresponding mixed integer programs to generate rank 1 split cuts our approach was implemented in the coin or framework using cplex 9 0 as a general purpose mip solver we report our computational results on well known benchmark instances from miplib 3 0 and several classes of structured integer and mixed integer problems our computational results show that rank 1 split cuts close more than 98 of the duality gap on 15 out of 41 mixed integer instances from miplib 3 0 more than 75 of the duality gap can be closed on an additional 10 instances the average gap closed over all 41 instances is 72 78 in the pure integer case rank 1 split cuts close more than 75 of the duality gap on 13 out of 24 instances from miplib 3 0 on average rank 1 split cuts close about 72 of the duality gap on these 24 instances we also report results on several classes of structured problems capacitated versions of warehouse location single source facility location p median fixed charge network flow multi commodity network design with splittable and unsplittable flows and lot sizing the fraction of the integrality gap closed varies for these problem classes between 100 and 67 we also gathered statistics on the average coefficient size absolute value of the disjunctions generated they turn out to be surprisingly small