non connected toric hilbert schemes we construct small 50 and 26 points respectively point sets in dimension 5 whose graphs of triangulations are not connected these examples improve our construction in j amer math soc 13 3 2000 611 637 not only in size but also in that the associated toric hilbert schemes are not connected either a question left open in that article additionally the point sets can easily be put into convex position providing examples of 5 dimensional polytopes with non connected graph of triangulations

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

subset algebra lift operators for 0 1 integer programming we extend the sherali adams lovasz schrijver balas ceria cornuejols and lasserre lift and project methods for 0 1 optimization by considering liftings to subset algebras our methods yield polynomial time algorithms for solving a relaxation of a set covering problem at least as strong as that given by the set of all valid inequalities with small coefficients and more generally all valid inequalities where the right hand side is not very large relative to the positive coefficients in the left hand side applied to generalizations of vertex packing problems our methods yield in polynomial time relaxations that have unbounded rank using for example the n operator

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

cellular resolutions of cointerval ideals minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed this class of d uniform hypergraphs coincides with the complements of interval graphs for the case d 2 and strictly contains the class of strongly stable hypergraphs corresponding to pure shifted simplicial complexes the polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms and are realized as subcomplexes of mixed subdivisions of the minkowski sums of simplices resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs

decomposing the efficient frontier of the dea production possibility set into a smallest number of convex polyhedrons by mixed integer programming abstract this paper extends the works by olesen and petersen 2003 russell and schworm 2006 and cooper et al 2007 about describing the efficient frontier of a production possibility set by the intersection of a finite number of closed halfspaces in several ways first we decompose the efficient frontier into a smallest number of convex polyhedrons or equivalently into a new class of efficient faces called maximal efficient faces mefs second we show how to identify all mefs even if full dimensional efficient faces do not exist third by applying the mef decomposition to various real world data sets we demonstrate the validity of the mef decomposition and how it can contribute to the dea literature finally we illustrate how to use the identified mefs in practice

matching fields and lattice points of simplices abstract we show that the chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points and we demonstrate how one can recover the linkage matching field from this bijection this resolves two open questions from sturmfels and zelevinsky 1993 26 on linkage matching fields for this we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation as additional results we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex finally we relate our findings to transversal matroids through the tropical stiefel map

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

fuzzy data envelopment analysis and its application to location problems in this paper fuzzy dea data envelopment analysis models are proposed for evaluating the efficiencies of objects with fuzzy input and output data the obtained efficiencies are also fuzzy numbers that reflect the inherent ambiguity in evaluation problems under uncertainty an aggregation model for integrating fuzzy attribute values is provided in order to rank objects objectively using the proposed method a case study involving a restaurant location problem is analyzed in detail rent of establishment traffic amount level of security consumer consumption level and competition level are identified as the primary factors in determining an ideal location for a japanese style rotisserie restaurant based on field investigation the uncertain information on primary factors is represented by fuzzy numbers using the fuzzy aggregation model the best location of restaurant is determined the case study shows that fuzzy dea models can be quite useful for solving business problems under uncertainty

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

testing ignorable missingness in estimating equation approaches for longitudinal data summary we address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches this involves testing for whether or not the zero mean property of estimating equations holds true chen little 1999 proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data whereas we propose a more unified generalised score type test this avoids exhaustive estimation of parameters for each missing data pattern testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern specific estimating equations are zero comparisons are made for the two approaches with both simulations and real data examples

input output substitutability and strongly monotonic p norm least distance dea measures abstract in dea there are two frameworks for efficiency assessment and targeting the greatest and the least distance framework the greatest distance framework provides us with the efficient targets that are determined by the farthest projections to the assessed decision making unit via maximization of the p norm relative to either the strongly efficient frontier or the weakly efficient frontier non radial measures belonging to the class of greatest distance measures are the slacks based measure sbm and the range adjusted measure ram whereas these greatest distance measures have traditionally been utilized because of their computational ease least distance projections are quite often more appropriate than greatest distance projections from the perspective of managers of decision making units because closer efficient targets may be attained with less effort in spite of this desirable feature of the least distance framework the least distance in efficiency versions of the additive measure sbm and ram do not even satisfy weak monotonicity in this study therefore we introduce and investigate least distance p norm inefficiency measures that satisfy strong monotonicity over the strongly efficient frontier in order to develop these measures we extend a free disposable set and introduce a tradeoff set that implements input output substitutability

flag arrangements and triangulations of products of simplices abstract we investigate the line arrangement that results from intersecting d complete flags in c n we give a combinatorial description of the matroid t n d that keeps track of the linear dependence relations among these lines we prove that the bases of the matroid t n 3 characterize the triangles with holes which can be tiled with unit rhombi more generally we provide evidence for a conjectural connection between the matroid t n d the triangulations of the product of simplices n 1 d 1 and the arrangements of d tropical hyperplanes in tropical n 1 space our work provides a simple and effective criterion to ensure the vanishing of many schubert structure constants in the flag manifold and a new perspective on billey and vakil s method for computing the non vanishing ones

chance constrained efficiency evaluation a model for efficiency evaluation based upon the theory of chance constrained programming is developed the model uses a piecewise linear envelopment of confidence regions for observed stochastic multiple input multiple output combinations in the data envelopment analysis dea tradition the model allows for an exogenous decomposition of the total variation in data for each decision making unit dmu by varying certain probability levels the model can provide estimates of the sensitivity of efficiency scores regarding an unknown amount of noise in date an application of the method in an evaluation of the research activities in economic departments at danish universities is presented

highly efficient aggregate unbiased estimating functions approach for correlated data with missing at random we develop a consistent and highly efficient marginal model for missing at random data using an estimating function approach our approach differs from inverse weighted estimating equations robins rotnitzky and zhao 1995 and the imputation method paik 1997 in that our approach does not require estimating the probability of missing or imputing the missing response based on assumed models the proposed method is based on an aggregate unbiased estimating function approach which does not require the likelihood function however it is equivalent to the score equation if the likelihood is known the aggregate unbiased approach is based on the best linear approximation of efficient scores from the full dataset we provide comparisons of the three approaches using simulated data and also a human immunodeficiency virus hiv data example

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

sensitivity analysis of the additive model in data envelopment analysis abstract sensitivity analysis of the additive model in data envelopment analysis is studied in this paper sufficient conditions for simultaneous change of all outputs and all inputs of an efficient decision making unit which preserves efficiency are established a numerical illustration is provided

monotonicity of minimum distance inefficiency measures for data envelopment analysis abstract this research explores the minimum distance inefficiency measure for the data envelopment analysis dea model a critical issue is that this measure does not satisfy monotonicity i e the measure may provide a better evaluation score to an inferior decision making unit dmu than to a superior one to overcome this a variant called the extended facet approach has been introduced this approach however requires a certain regularity condition to be met we discuss several special classes of the dea model and show that for these models the minimum distance inefficiency measure satisfies the monotonicity property without the regularity condition moreover we conducted computational experiments using real world data sets from these special classes and demonstrated that the extended facet approach may overestimate the performance of a dmu

on measuring the inefficiency with the inner product norm in data envelopment analysis abstract a technique for assessing the sensitivity of efficiency classifications in data envelopment analysis dea is presented it extends the technique proposed by charnes et al a charnes j j rousseau j h semple journal of productivity analysis 7 1996 5 18 an organization s input output vector serves as the center for a cell within which the organization s classification remains unchanged under perturbations of the data the maximal radius among such cells can be interpreted as a stability measure of the classification our approach adopts the inner product norm for the radius while the previous work does the polyhedral norms for an efficient organization the maximal radius problem is a convex program on the other hand for an inefficient organization it is reduced to a nonconvex program whose feasible region is the complement of a convex polyhedral set we show that the latter nonconvex problem can be transformed into a linear reverse convex program our formulations and algorithms are valid not only in the ccr model but in its variants

tropical hyperplane arrangements and oriented matroids we study the combinatorial properties of a tropical hyperplane arrangement we define tropical oriented matroids and prove that they share many of the properties of ordinary oriented matroids we show that a tropical oriented matroid determines a subdivision of a product of two simplices and conjecture that this correspondence is a bijection

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

composite indicators as generalized benefit of the doubt weighted averages abstract composite indicators cis are usually computed as arithmetic weighted averages of often normalized sub indicators several studies criticized this procedure for implying requirements and properties that are often hard to maintain in practical applications recent studies explored the multiplicative aggregation and more specifically the geometric weighted average as aggregator in ci building this paper takes this exploration one step further by considering other members of the family of generalized weighted averages as aggregator in the construction of cis it is argued that the choice for a particular version of the generalized weighted average enables to reflect decision makers attitudes in the evaluation for the human development index results show that the choice for a specific version of the generalized weighted average as aggregator in ci construction impacts country rankings

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

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

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

a test of missing completely at random for generalised estimating equations with missing data we consider inference from generalised estimating equations when data are incomplete a test for missing completely at random is proposed to help decide whether or not we should adjust estimating equations to correct the possible bias introduced by a missing data mechanism that is not missing completely at random likelihood ratio tests have been introduced to test the missing completely at random hypothesis fuchs 1982 little 1988 for the estimating equation setting following the basic idea of little 1988 we propose a wald type test based on an information decomposition and recombination procedure which also provides an alternative method for estimating parameters one application of the test is to assess the adequacy of the marginal generalised estimating equation for longitudinal data with missing values simulations are done to evaluate its performance

stability regions for maintaining efficiency in data envelopment analysis abstract this paper develops a procedure for performing a sensitivity analysis of the efficient decision making units dmus within the charnes et al ccr european journal of operational research 2 1978 429 444 model of data envelopment analysis dea the procedure yields an exact input stability region and output stability region within which the efficiency of a specific efficient dmu remains unchanged such stability regions are simply characterized by optimal solutions of modified ccr models which are easily computed in contrast to existing sufficient conditions for the preservation of efficiency under changes in inputs or outputs the paper provides both necessary and sufficient conditions for an efficient dmu to remain efficient the procedure is illustrated by numerical examples somewhat surprisingly for real world data sets for most of the efficient dmus the amount of some individual input can be infinitely increased when keeping other inputs and all outputs constant this indicates that these efficient dmus are located at extreme positions

choosing weights from alternative optimal solutions of dual multiplier models in dea in this paper we propose a two step procedure to be used for the selection of the weights that we obtain from the multiplier model in a dea efficiency analysis it is well known that optimal solutions of the envelopment formulation for extreme efficient units are often highly degenerate and consequently have alternate optima for the weights different optimal weights may then be obtained depending for instance on the software used the idea behind the procedure we present is to explore the set of alternate optima in order to help make a choice of optimal weights the selection of weights for a given extreme efficient point is connected with the dimension of the efficient facets of the frontier our approach makes it possible to select the weights associated with the facets of higher dimension that this unit generates and in particular it selects those weights associated with a full dimensional efficient facet fdef if any in this sense the weights provided by our procedure will have the maximum support from the production possibility set we also look for weights that maximize the relative value of the inputs and outputs included in the efficiency analysis in a sense to be described in this article

a connection between cutting plane theory and the geometry of numbers in this paper we relate several questions about cutting planes to a fundamental problem in the geometry of numbers namely the closest vector problem using this connection we show that the dominance membership and validity problems are np complete for chvatal and split cuts

on the separation of split cuts and related inequalities the split cuts of cook kannan and schrijver are general purpose valid inequalities for integer programming which include a variety of other well known cuts as special cases to detect violated split cuts one has to solve the associated separation problem the complexity of split cut separation was recently cited as an open problem by cornuejols li cl01 in this paper we settle this question by proving strong completeness of separation for split cuts as a by product we also show completeness of separation for several other classes of inequalities including the mir inequalities of nemhauser and wolsey and some new inequalities which we call balanced split cuts and binary split cuts we also strengthen completeness results of caprara fischetti cf96 for cuts and eisenbrand e99 for chvatal gomory cuts to compensate for this bleak picture we also give a positive result for the symmetric travelling salesman problem we show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle

assessing the validity of weighted generalized estimating equations the inverse probability weighted generalized estimating equations approach robins et al 1994 robins et al 1995 effectively removes bias and provides valid statistical inference for regression parameter estimation in marginal models when longitudinal data contain missing values the validity of the weighted generalized estimating equations regarding consistent estimation depends on whether the underlying missing data process is properly modelled however there is little work available to examine whether or not this condition holds in this paper we propose a test constructed from two sets of estimating equations one set is known to be unbiased but the other set is not known we utilize the quadratic inference function qu et al 2000 method to assess their compatibility which is equivalent to testing for the validity of the weighted generalized estimating equations approach we conduct simulation studies to assess the performance of the proposed method the test procedure is illustrated through a real data example

a point set whose space of triangulations is disconnected in this paper we explicitly construct a triangulation of a 6 dimensional point configuration of 324 points which admits no geometric bistellar operations or flips for short this triangulation is an isolated element in the graph of triangulations of the point configuration it has been a central open question in polytope combinatorics in the last decade whether point configurations exist for which this graph is not connected see e g 37 question 1 2 and 48 challenge 3 we also construct a 234 dimensional polytope with 552 vertices whose graph of triangulations has an isolated element our construction is likely to have an impact in algebraic geometry too via the connections between lattice polytopes and toric varieties 21 23 31 43 for example in 2 section 2 and 24 section 4 the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration the connectivity of these schemes and of the graph of triangulations are equivalent see section 4 3 in particular corollary 4 9 the graph of triangulations is also related to the baues poset which appears in oriented matroid theory zonotopal tilings and hyperplane arrangements so our result has implications in these areas let a be a finite point set in the real affine space r a polyhedral subdivision of a is a geometric polyhedral complex with vertices in a which covers the convex hull of a if all the cells are simplices then it is a triangulation more combinatorial definitions are convenient if a is not in convex position i e if some element of a is not a vertex of the convex hull see definitions 4 1 and 1 1 for details and also 6 21 chapter 7 36 47 chapter 9 or the monograph in preparation 14 there are at least the following three ways to give a structure to the collection of all triangulations of a point configuration a a flips geometric bistellar operations or flips are the minimal changes which can be made in a triangulation of a to produce a new one see definition 1 3 a particular case is the familiar diagonal edge flip in two dimensional triangulations of frequent use in computational geometry and geometric combinatorics

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

linear production functions and the data envelopment analysis abstract this paper shows that the data envelopment analysis or farrell method is not nonparametric in the sense that the method implicitly employs linear production functions frontiers to measure efficiency deriving positive multiplier linear production frontiers and grouping the dmus by their corresponding frontiers can simplify the estimation procedures when there are new dmus included and can easily determine how many inputs should be reduced or and how many outputs should be increased to make an inefficient dmu efficiently this procedure can also circumvent the problems of the non archimedean infinitesmal in the ccr literature it is also shown that because of the linear production functions both banker s and the banker charnes cooper methods fail in testing for returns to scale

indicators of ill conditioned data sets and model misspecification in data envelopment analysis an extended facet approach date envelopment analysis dea employs mathematical programming to measure the relative efficiency of decision making units dmus this paper is concerned with development of indicators to determine whether or not the specification of the input and output space is supported by data in the sense that the variation in data is sufficient for estimation of a frontier of the same dimension as the input output space insufficient variation in data implies that some inputs outputs can be substituted along the efficient frontier but only in fixed proportions data thus locally supports variation in a subspace of a lower dimension rather than in the input output space of full dimension each segment of the efficient frontier is in this sense subject to local collinearity insufficient variation in data provides a bound on admissible disaggregations in cases where substitution in fixed proportions is incompatible with a priori information concerning the production process a data set incapable of estimating a fro

projected chvatal gomory cuts for mixed integer linear programs recent experiments by fischetti and lodi show that the first chvatal closure of a pure integer linear program ilp often gives a surprisingly tight approximation of the integer hull they optimize over the first chvatal closure by modeling the chvatal gomory cg separation problem as a mixed integer linear program milp which is then solved by a general purpose milp solver unfortunately this approach does not extend immediately to the gomory mixed integer gmi closure of an milp since the gmi separation problem involves the solution of a nonlinear mixed integer program or a parametric milp in this paper we introduce a projected version of the cg cuts and study their practical effectiveness for milp problems the idea is to project first the linear programming relaxation of the milp at hand onto the space of the integer variables and then to derive chvatal gomory cuts for the projected polyhedron though theoretically dominated by gmi cuts projected cg cuts have the advantage of producing a separation model very similar to the one introduced by fischetti and lodi which can typically be solved in a reasonable amount of computing time

a note on the bias of estimators with missing data it is well known that many standard analyses including maximum likelihood estimation and the generalized estimating equation approach liang and zeger 1986 biometrika 73 13 22 can result in biased estimation when there are missing observations in such cases it is of interest to calculate the magnitude of the bias incurred under specific assumptions about the process generating the full data and the nonresponse mechanism in this paper we give a condition that identifies the limit in probability of estimators that are solutions of estimating equations computed from the incomplete data with discrete data this condition suggests a simple algorithm to compute the asymptotic bias of these estimators that can be easily implemented with existing statistical software we illustrate our approach with asthma prevalence data in children

an autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes we are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient and many patients drop out with a reason related to the response in this paper we focus on a model that assumes a dropout process is missing at random mar in this dropout process we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified however parsimonious covariance structures for the profiles approaching asymptotes are unclear an autoregressive linear mixed effects model can express the profile with random individual asymptotes we show that this model provides a new parsimonious covariance structure the covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points in simulation studies the estimate of the asymptote is unbiased in mar dropouts but biased in non ignorable dropouts we also applied this model to actual schizophrenia trial data

strong formulations of robust mixed 0 1 programming we introduce strong formulations for robust mixed 0 1 programming with uncertain objective coefficients we focus on a polytopic uncertainty set described by a budget constraint for allowed uncertainty in the objective coefficients we show that for a robust 0 1 problem there is an tight linear programming formulation with size polynomial in the size of an tight linear programming formulation for the nominal 0 1 problem we give extensions to robust mixed 0 1 programming and present computational experiments with the proposed formulations

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

a geometric littlewood richardson rule we describe a geometric littlewood richardson rule interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties there are no restrictions on the base eld and all multiplicities arising are 1 this is important for applications this rule should be seen as a generalization of pieri s rule to arbitrary schubert classes by way of explicit homotopies it has straightforward bijections to other littlewood richardson rules such as tableaux and knutson and tao s puzzles this gives the rst geometric proof and interpretation of the littlewood richardson rule geometric consequences are described here and in v2 kv1 kv2 v3 for example the rule also has an interpretation in k theory suggested by buch which gives an extension of puzzles to k theory

a nonparametric test of missing completely at random for incomplete multivariate data missing data occur in many real world studies knowing the type of missing mechanisms is important for adopting appropriate statistical analysis procedure many statistical methods assume missing completely at random mcar due to its simplicity therefore it is necessary to test whether this assumption is satisfied before applying those procedures in the literature most of the procedures for testing mcar were developed under normality assumption which is sometimes difficult to justify in practice in this paper we propose a nonparametric test of mcar for incomplete multivariate data which does not require distributional assumptions the proposed test is carried out by comparing the distributions of the observed data across different missing pattern groups we prove that the proposed test is consistent against any distributional differences in the observed data simulation shows that the proposed procedure has the type i error well controlled at the nominal level for testing mcar and also has good power against a variety of non mcar alternatives

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

geometric mean quantity index numbers with benefit of the doubt weights abstract geometric mean index numbers are a multiplicative aggregation of price or quantity ratios with their importance exponents weights derived from one or more observed budget shares in the specific context of composite indicator construction we propose to use the budget shares as naturally generated by the linear benefit of the doubt model this approach is directly inspired by the literature on index number theory our basic model is easily extended to provide transitive composite indicator orderings in a multilateral setting also a multi factor decomposition is proposed to explain the intertemporal evolution of a single entity we illustrate our results with social inclusion data for the eu countries

an occurrence of multiple projections in dea based measurement of technical efficiency theoretical comparison among dea models from desirable properties this study discusses nine desirable properties that a measure of technical efficiency te needs to satisfy from the perspective of production economics and optimization seven data envelopment analysis dea models are theoretically compared from a viewpoint of nine te criteria all the seven dea models suffer from a problem of multiple projections even though a unique projection for efficiency comparison is one of the nine desirable properties furthermore all the dea models violate the property on aggregation of inputs and outputs thus the seven dea models do not satisfy all desirable te properties in addition the comparison provides us with the following guidelines a the additive model violates all desirable te properties b russell measure and sbm ergm perform as well as ram as a non radial measure if we are interested in strict monotonicity the two models outperform the other dea models including ram in contrast if we are interested in translation invariance ram is better than russell measure and sbm ergm c the radial measures ccr and bcc have the property of linear homogeneity d the ccr model is useful for measuring a frontier shift among different periods e if a data set contains a negative value ram becomes a dea model to handle the negative value because it has the property of translation invariance after examining the desirable te properties this study proposes a new approach to deal with an occurrence of multiple projections the proposed approach includes a test to examine an occurrence of multiple projections a mathematical expression of a projection set and a selection process of a unique reference set as the largest one covering all the possible reference sets

missing data mechanisms and homogeneity of means and variances covariances unless data are missing completely at random mcar proper methodology is crucial for the analysis of incomplete data consequently methods for effectively testing the mcar mechanism become important and procedures were developed via testing the homogeneity of means and variances covariances across the observed patterns e g kim bentler in psychometrika 67 609 624 2002 little in j am stat assoc 83 1198 1202 1988 the current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data and the normal distribution based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random although the values may not equal those of the underlying population distribution the results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the mcar mechanism even when the population distribution is multivariate normal

on the dea total weight flexibility and the aggregation in cross efficiency evaluations abstract this paper discusses the dea total weight flexibility in the context of the cross efficiency evaluation the dmus in dea are often assessed with unrealistic weighting schemes in their attempt to achieve the best ratings in their self evaluation we claim here that in a peer appraisal like the cross efficiency evaluation the cross efficiencies provided by such weights cannot play the same role as those obtained with more reasonable weights to address this issue we propose to calculate the cross efficiency scores by means of a weighted average of cross efficiencies instead of with the usual arithmetic mean so the aggregation weights reflect the disequilibrium in the profiles of dea weights that are used thus the cross efficiencies provided by profiles with large differences in their weights especially those obtained with zero weights would be attached lower aggregation weights less importance than those provided by more balanced profiles of weights

testing for order restricted hypotheses in longitudinal data an inherent ordering we propose a quadratic score test qst based on a quadratic inference function for detecting an order in treatment effects for correlated data the quadratic inference function is similar to the negative of a log likelihood and it provides test statistics in the spirit of a x2 test for testing nested hypotheses as well as for assessing the goodness of fit of model assumptions under the null hypothesis of no order restriction it is shown that the qst statistic has a wald type asymptotic representation and that the asymptotic distribution of the qst statistic is a weighted x2 distribution furthermore an asymptotic distribution of the qst statistic under an arbitrary convex cone alternative is provided the performance of the qst is investigated through monte carlo simulation experiments analysis of the polyposis data demonstrates that the qst outperforms the wald test when data are highly correlated with a small sample size and there is a significant amount of missing data with a small number of clusters the proposed test statistic accommodates both time dependent and time independent covariates in a model

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

a multiplier bound approach to assess relative efficiency in dea without slacks abstract in this paper we propose a new approach to deal with the non zero slacks in data envelopment analysis dea assessments that is based on restricting the multipliers in the dual multiplier formulation of the used dea model it guarantees strictly positive weights which ensures reference points on the pareto efficient frontier and consequently zero slacks we follow a two step procedure which after specifying some weight bounds results in an assurance region type model that will be used in the assessment of the efficiency the specification of these bounds is based on a selection criterion among the optimal solutions for the multipliers of the unbounded dea models that tries to avoid the extreme dissimilarity between the weights that is often found in dea applications the models developed do not have infeasibility problems and we do not have problems with the alternate optima in the choice of weights that is made to use our multiplier bound approach we do not need a priori information about substitutions between inputs and outputs and it is not required the existence of full dimensional efficient facets on the frontier either as is the case of other existing approaches that address this problem

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