non linear frequency response characteristics of an impact pair abstract the method of harmonic balance is used to develop approximate analytical solutions to the equations of motion of a single impact pair including a general discussion of the existence and stability of these solutions non linear frequency response characteristics are obtained for a single frequency excitation and primary resonance no subharmonic or superharmonic response and the results are validated using an analog computer simulation it has been found that the use of digital simulation numerical integration may lead to inaccurate results over certain frequency ranges it is also shown that although the shapes of the force displacement relationship for clearance non linearity and the classic cubic non linearity are similar the resulting non linear frequency response curves can be quite different various issues related to the clearance non linearity problem are discussed

effect of nonlinear impact damping on the frequency response of a torsional system with clearance in this article we examine a particular nonlinear impact damping model that has been well accepted by the scientific community first we consider the damping element with a linear spring and examine the nonlinear frequency response characteristics of a torsional system second the impact damping is examined along with the clearance nonlinearity including the backlash problem three semi analytical methods namely the describing function method multi term harmonic balance method and stochastic linearization method are utilized to investigate the effect of impact damping feasibility of such methods is confirmed by comparing predictions with results yielded by a numerical method the differences between impact damping and linear viscous damping mechanism are studied in terms of frequency responses time histories and the minimum number of harmonics required to simulate the behavior given sinusoidal excitations furthermore the effect of impact damping on sub and super harmonic responses and quasi periodic regimes are examined

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

multivariate logistic models summary the multivariate logistic transform is a reparameterisation of cell probabilities in terms of marginal logistic contrasts it is known that an arbitrary set of logistic contrasts may not correspond to a valid joint distribution in this paper we present an efficient algorithm for detecting whether or not the inverse transform exists and for computing it if it does

effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor the response of a balanced horizontal rigid motor rotor supported by a deep groove ball bearing is theoretically simulated the effect of radial internal clearance of the ball bearing on the dynamic response of the rotor is studied the system equations have been numerically integrated the results of which have been validated with harmonic balance alternating frequency time domain method variation of radial internal clearance shifts the peak response as the speed is changed over a range the results of a parametric study done by taking radial internal clearance have resulted in the observation of a third region of instability which has not been reported in literature the appearance of regions of periodic subharmonic and chaotic behavior is seen to be strongly dependent on the radial internal clearance the system response is analyzed for stability and nature with the help of floquets method for stability analysis and generation of higher order poincare maps the bearing stiffness is estimated experimentally and the effect of variation in radial internal clearance on the bearing stiffness is studied

cyclic group and knapsack facets any integer program may be relaxed to a group problem we define the master cyclic group problem and several master knapsack problems show the relationship between the problems and give several classes of facet defining inequalities for each problem as well as a set of mappings that take facets from one type of master polyhedra to another

clustering of multivariate binary data with dimension reduction via l1 regularized likelihood maximization clustering methods with dimension reduction have been receiving considerable wide interest in statistics lately and a lot of methods to simultaneously perform clustering and dimension reduction have been proposed this work presents a novel procedure for simultaneously determining the optimal cluster structure for multivariate binary data and the subspace to represent that cluster structure the method is based on a finite mixture model of multivariate bernoulli distributions and each component is assumed to have a low dimensional representation of the cluster structure this method can be considered as an extension of the traditional latent class analysis sparsity is introduced to the loading values which produces the low dimensional subspace for enhanced interpretability and more stable extraction of the subspace an em based algorithm is developed to efficiently solve the proposed optimization problem we demonstrate the effectiveness of the proposed method by applying it to a simulation study and real datasets highlightswe propose a novel clustering method of multivariate binary data the proposed method provides a low dimensional representation of clusters the proposed method overcomes the conventional tandem analysis the proposed method conducts feature selection for clustering

binary models for marginal independence summary log linear models are a classical tool for the analysis of contingency tables in particular the subclass of graphical log linear models provides a general framework for modelling conditional independences however with the exception of special structures marginal independence hypotheses cannot be accommodated by these traditional models focusing on binary variables we present a model class that provides a framework for modelling marginal independences in contingency tables the approach that is taken is graphical and draws on analogies with multivariate gaussian models for marginal independence for the graphical model representation we use bidirected graphs which are in the tradition of path diagrams we show how the models can be parameterized in a simple fashion and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm finally we consider combining these models with symmetry restrictions

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

non linear dynamics of gear pair systems with periodic stiffness and backlash the present work investigates dynamics of a gear pair system involving backlash and time dependent mesh stiffness in addition the system is under the action of external excitation caused by torsional moments and gear geometry errors first the equation of motion is established in a strongly non linear form then the emphasis is laid on a specific forcing frequency range corresponding to conditions of simultaneous fundamental parametric resonance and principal external resonance for these conditions several types of periodic steady state response are identified and determined by employing suitable methodologies including techniques applicable to piecewise linear systems and to oscillators with time periodic coefficients moreover these methodologies are complemented by appropriate procedures revealing the stability properties of the located periodic solutions in the second part of the work numerical results are presented these results verify the validity and effectiveness of the new analytical methodology and provide information on the gear pair dynamics first series of typical response diagrams are obtained illustrating the effect of the mesh stiffness variation the damping and the forcing parameters on the gear pair periodic response these response diagrams are accompanied by results obtained with direct integration of the equation of motion in this way it is demonstrated that for some parameter combinations the dynamical system examined can exhibit more complicated and irregular response including crises and intermittent chaos

cyclic group blocking polyhedra gomory linear algebra appl 2 451 558 1969 gave a subadditive characterization of the facets of the group polyhedron although there are exponentially many facets see gomory and johnson in math program 3 359 389 1972 example 4 6 and exponentially many vertices for the group polyhedron for the master cyclic group problem gomory s characterization of the non trivial facets has polynomially many subadditive inequalities in fact of order g 2 for a finite abelian group g we reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non trivial facets the system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed our minimal representation leads to a characterization of the vertices of length three gomory et al math program 96 321 339 2003 introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets we develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples hunsaker 2003 gave a knapsack shooting experiment which we use to find a simple pattern for the most hit knapsack facets

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

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

on the extreme inequalities of infinite group problems infinite group relaxations of integer programs ip were introduced by gomory and johnson math program 3 23 85 1972 to generate cutting planes for general ips these valid inequalities correspond to real valued functions defined over an appropriate infinite group among all the valid inequalities of the infinite group relaxation extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group theoretic framework however very few properties of extreme inequalities of infinite group relaxations are known in particular it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems in this paper we describe new properties of extreme functions of infinite group problems in particular we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems using these results we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems this class of extreme functions is the generalization of the functions given by letchford and lodi oper res lett 30 2 74 82 2002 dash and gunluk proceedings 10th conference on integer programming and combinatorial optimization springer heidelberg pp 33 45 2004 math program 106 29 53 2006 and richard et al math program 2008 to appear we also present several other new classes of discontinuous extreme functions surprisingly we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

the structure of the infinite models in integer programming the infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions in this paper we study the relationships between these two descriptions our results have implications for finite dimensional corner polyhedra one consequence is that nonnegative continuous functions suffice to describe finite dimensional corner polyhedra with rational data we also discover new facts about corner polyhedra with non rational data

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

mixed integer cuts from cyclic groups we analyze a separation procedure for mixed integer programs related to the work of gomory and johnson on interpolated subadditive functions this approach has its roots in the gomory johnson characterization on the master cyclic group polyhedron to our knowledge the practical benefit that can be obtained by embedding interpolated subadditive cuts in a cutting plane algorithm was not investigated computationally by previous authors in this paper we compute for the first time the lower bound value obtained when adding implicitly all the interpolated subadditive cuts that can be derived from the individual rows of an optimal lp tableau thus approximating the optimization over the intersection of the gomory corner polyhedron with the lp relaxation of the original problem formulation the computed bound is compared with that obtained when only gomory mixed integer cuts are used on a very large test bed of mip instances

valid inequalities based on simple mixed integer sets in this paper we use facets of simple mixed integer sets with three variables to derive a parametric family of valid inequalities for general mixed integer sets we call these inequalities two step mir inequalities as they can be derived by applying the simple mixed integer rounding mir principle of wolsey 1998 twice the two step mir inequalities define facets of the master cyclic group polyhedron of gomory 1969 in addition they dominate the strong fractional cuts of letchford and lodi 2002

subadditive approaches in integer programming linear programming duality is well understood and the reduced cost of a column is frequently used in various algorithms on the other hand for integer programs it is not clear how to define a dual function even though the subadditive dual theory has been developed a long time ago in this work we propose a family of computationally tractable subadditive dual functions for integer programs we develop a solution methodology that computes an optimal primal solution and an optimal subadditive dual function we present computational experiments which show that the new algorithm is tractable

valid inequalities based on the interpolation procedure we study the interpolation procedure of gomory and johnson 1972 which generates cutting planes for general integer programs from facets of cyclic group polyhedra this idea has recently been re considered by evans 2002 and gomory johnson and evans 2003 we compare inequalities generated by this procedure with mixed integer rounding mir based inequalities discussed in dash and gunluk 2003 we first analyze and extend the shooting experiment described in gomory johnson and evans we show that mir based inequalities dominate inequalities generated by the interpolation procedure in some important cases we also show that the gomory mixed integer cut is likely to dominate any inequality generated by the interpolation procedure in a certain probabilistic sense we also generalize a result of cornuejols li and vandenbussche 2003 on comparing the strength of the gomory mixed integer cut with related inequalities

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

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

a counterexample to a conjecture of gomory and johnson in mathematical programming 2003 gomory and johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions in this paper we give an example showing that the gomory johnson conjecture is false

association models for a multivariate binary response models for a multivariate binary response are parameterized by univariate marginal probabilities and dependence ratios of all orders the w order dependence ratio is the joint success probability of w binary responses divided by the joint success probability assuming independence this parameterization supports likelihood based inference for both regression parameters relating marginal probabilities to explanatory variables and association model parameters relating dependence ratios to simple and meaningful mechanisms five types of association models are proposed where responses are 1 independent given a necessary factor for the possibility of a success 2 independent given a latent binary factor 3 independent given a latent beta distributed variable 4 follow a markov chain and 5 follow one of two first order markov chains depending on the realization of a binary latent factor these models are illustrated by reanalyzing three data sets foremost a set of binary time series on auranofin therapy against arthritis likelihood based approaches are contrasted with approaches based on generalized estimating equations association models specified by dependence ratios are contrasted with other models for a multivariate binary response that are specified by odds ratios or correlation coefficients

period one motions of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness the dynamic behavior of a piecewise nonlinear mechanical oscillator with parametric and external excitations is investigated the viscously damped single degree of freedom oscillator is subjected to a periodically time varying piecewise nonlinear restoring function defined by a clearance surrounded by continuously nonlinear quadratic and cubic regions typical applications represented by this oscillator are highlighted a multi term harmonic balance formulation is used in conjunction with discrete fourier transforms and a parametric continuation scheme to determine steady state period 1 motions of the system due to both parametric and external excitations the accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions floquet theory is applied to determine the stability of the steady state harmonic balance solutions at the end detailed parametric studies are presented to quantify the combined influence of clearance quadratic and cubic nonlinearities within reasonable ranges of all other system parameters a comparison between time varying and time invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise nonlinear system

valid inequalities for mips and group polyhedra from approximate liftings in this paper we present an approximate lifting scheme to derive valid inequalities for general mixed integer programs and for the group problem this scheme uses superadditive functions as the building block of integer and continuous lifting procedures it yields a simple derivation of new and known families of cuts that correspond to extreme inequalities for group problems this new approximate lifting approach is constructive and potentially efficient in computation

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

on the augmented lagrangian dual for integer programming we consider the augmented lagrangian dual for integer programming and provide a primal characterization of the resulting bound as a corollary we obtain proof that the augmented lagrangian is a strong dual for integer programming we are able to show that the penalty parameter applied to the augmented lagrangian term may be placed at a fixed large value and still obtain strong duality for pure integer programs

a k 1 slope theorem for the k dimensional infinite group relaxation we prove that any minimal valid function for the k dimensional innite group relaxation that is piecewise linear with at most k 1 slopes and does not factor through a linear map with non trivial kernel is extreme this generalizes a theorem of gomory and johnson for k 1 and cornu ejols and molinaro for k 2

multivariate logistic models summary when data composed of several categorical responses together with categorical or continuous predictors are observed the multivariate logistic transform introduced by mccullagh and nelder can be used to define a class of regression models that is in many applications particularly suitable for relating the joint distribution of the responses to predictors in this paper we give a general definition of this class of models and study their properties a computational scheme for performing maximum likelihood estimation for data sets of moderate size is described and a system of model formulae that succinctly define particular models is introduced applications of these models to longitudinal problems are illustrated by numerical examples

large amplitude non linear normal modes of piecewise linear systems a numerical method for constructing non linear normal modes nnms for piecewise linear autonomous systems is presented these nnms are based on the concept of invariant manifolds and are obtained using a galerkin based solution of the invariant manifold s non linear partial differential equations the accuracy of the constructed non linear modes is checked by the comparison of the motion on the invariant manifold to the exact solution in both time and frequency domains it is found that this construction approach can accurately capture the nnms over a wide range of amplitudes including those with strong non linear effects several interesting dynamic characteristics of the non linear modal motion are found and compared to those of linear modes a two degree of freedom example is used to illustrate the technique the existence stability and bifurcations of the nnms for this example are investigated

interpretation of regressive logistic regression coefficients in analyses of familial data regressive logistic regression was introduced by bonney in 1986 and due to its easy implementation using standard logistic regression computer packages has been widely used in the analysis of binary familial disease traits associations between binary outcomes for different family members are measured by regression coefficients included in the corresponding logit models the interpretation of these coefficients is examined in detail revealing a number of features of the method that are worthy of comment

extreme functions with an arbitrary number of slopes for the one dimensional infinite group relaxation we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number k ge 2 there is a function in the sequence with k slopes this settles an open question in this area regarding a universal bound on the number of slopes for extreme functions the function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes this provides a new and more refined counterexample to an old conjecture of gomory and johnson stating that all extreme functions are piecewise linear these constructions are extended to obtain functions for the higher dimensional group problems via the sequential merge operation of dey and richard

a class of regression models for multivariate categorical responses summary when data composed of several categorical responses together with categorical or continuous predictors are observed it is often useful to relate the response probabilities to the predictors via a generalised linear model with a composite link function this paper discusses a class of link functions that lie between the two extremes of the multivariate logistic transform of mccullagh nelder 1989 and the log linear decomposition of contingency table analysis the models derived from these link functions are shown to inherit various desirable properties of both the multivariate logistic regression models and the log linear regression models a computational scheme for implementing these models is derived and they are demonstrated to be computationally more tractable than the multivariate logistic regression models their application is illustrated in a numerical example

on the strength of gomory mixed integer cuts as group cuts gomory mixed integer gmi cuts generated from optimal simplex tableaus are known to be useful in solving mixed integer programs further it is well known that gmi cuts can be derived from facets of gomory s master cyclic group polyhedron and its mixed integer extension studied by gomory and johnson in this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra group cuts do not seem to be as useful when used in conjunction with gmi cuts for many practical problem instances we numerically show that once gmi cuts from different rows of the optimal simplex tableau are added to the formulation all other group cuts from the same tableau rows are satisfied

log mean linear models for binary data this paper is devoted to the theory and application of a novel class of models for binary data which we call log mean linear lml models the characterizing feature of these models is that they are specied by linear constraints on the

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

marginal regression analysis of a multivariate binary response we propose the use of the mean parameter for regression analysis of a multivariate binary response we model the association using dependence ratios defined in terms of the mean parameter the components of which are the joint success probabilities of all orders this permits flexible modelling of higher order associations using maximum likelihood estimation we reanalyse two data sets one with variable cluster size and the other a longitudinal data set with constant cluster size

n step mingling inequalities new facets for the mixed integer knapsack set the n step mixed integer rounding mir inequalities of kianfar and fathi math program 120 2 313 346 2009 are valid inequalities for the mixed integer knapsack set that are derived by using periodic n step mir functions and define facets for group problems the mingling and 2 step mingling inequalities of atamturk and gunluk math program 123 2 315 338 2010 are also derived based on mir and they incorporate upper bounds on the integer variables it has been shown that these inequalities are facet defining for the mixed integer knapsack set under certain conditions and generalize several well known valid inequalities in this paper we introduce new classes of valid inequalities for the mixed integer knapsack set with bounded integer variables which we call n step mingling inequalities for positive integer n these inequalities incorporate upper bounds on integer variables into n step mir and therefore unify the concepts of n step mir and mingling in a single class of inequalities furthermore we show that for each n the n step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions for n 2 we extend the results of atamturk and gunluk on facet defining properties of 2 step mingling inequalities for n 3 we present new facets for the mixed integer knapsack set as a special case we also derive conditions under which the n step mir inequalities define facets for the mixed integer knapsack set in addition we show that n step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets

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

frequency response characteristics of a multi degree of freedom system with clearances abstract in an earlier study the frequency response characteristics of a single degree of freedom system with a clearance non linearity were studied the current study an extension of the earlier work is concerned with the frequency response characteristics of a multi degree of freedom system with clearances the method of harmonic balance is used to develop approximate analytical solutions of the undamped equations of motion of a multi degree of freedom system composed of three coupled non linear oscillators for primary resonances and a harmonic excitation general formulations are presented which can be used to study both the existence and the stability of the solutions these general formulations are used to discuss a number of modal spacing and modal coupling issues an analysis methodology for multi degree of freedom systems is introduced and illustrated by two special but practical cases a strongly non linear system and a weakly non linear system the results of the analysis of the special cases are validated by using analog simulation it is shown that an analysis of the special cases of the general multi degree of freedom non linear formulation can provide not only an improved understanding of the dynamic behavior of non linear systems but also can be used as the basis for the development of simplified approximate solutions the limitations of the current study and areas for further research are discussed

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

generating facets for finite master cyclic group polyhedra using n step mixed integer rounding functions the n step mixed integer rounding mir functions generate n step mir inequalities for mip problems and are facets for the infinite group problems we show that the n step mir functions also directly generate facets for the finite master cyclic group polyhedra especially in many cases where the breakpoints of the n step mir function are not necessarily at the elements of the group hence the linear interpolation of the facet coefficients obtained has more than two slopes

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

joint regression and association modeling of longitudinal ordinal data we propose models for longitudinal or otherwise clustered ordinal data the association between subunit responses is characterized by dependence ratios ekholm smith and mcdonald 1995 biometrika 82 847 854 which are extended from the binary to the multicategory case the joint probabilities of the subunit responses are expressed as explicit functions of the marginal means and the dependence ratios of all orders obtaining a computational advantage for likelihood based inference equal emphasis is put on finding regression models for the univariate cumulative probabilities and on deriving the dependence ratios from meaningful association generating mechanisms a data set on the effects of treatment with fluvoxamine which has been analyzed in parts before molenberghs kenward and lesaffre 1997 biometrika 84 33 44 is analyzed in its entirety selection models are used for studying the sensitivity of the results to drop out

super and sub harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method abstract this article presents a new multi term harmonic balance method hbm for nonlinear frequency response calculations of a torsional sub system containing a clearance type nonlinearity the ability of the simplified subsystem to capture the salient behavior of the larger system is verified by the comparison of results to experimental data unlike previous analytical and numerical methods the proposed hbm includes adaptive arc length continuation and stability calculation capabilities to find periodic solutions in multi valued nonlinear frequency response regimes as well as to improve convergence essential steps of the proposed hbm calculations are introduced and it is validated by comparing time and frequency domain predictions with those yielded by numerical solutions experimental studies or analog simulations for several examples then nonlinear frequency response characteristics of an oscillator with clearance nonlinearity are examined with focus on super and sub harmonics we also explore some issues that are not fully resolved in the literature for instance the effect of mean operating point is examined for 0 and 0 18 0 25 cases where is the stiffness ratio of the piecewise linear elastic function in addition the number of harmonic terms that must be included in the hbm response calculations given sinusoidal excitation has been investigated finally some simple analytical predictions for super and sub harmonic resonances are presented

generalized mixed integer rounding inequalities facets for infinite group polyhedra we present a generalization of the mixed integer rounding mir approach for generating valid inequalities for mixed integer programming mip problems for any positive integer n we develop n facets for a certain n 1 dimensional single constraint polyhedron in a sequential manner we then show that for any n the last of these facets which we call the n step mir facet can be used to generate a family of valid inequalities for the feasible set of a general mixed ip constraint which we refer to as the n step mir inequalities the gomory mixed integer cut and the 2 step mir inequality of dash and gunluk math program 105 1 29 53 2006 are the first two families corresponding to n 1 2 respectively the n step mir inequalities are easily produced using periodic functions which we refer to as the n step mir functions none of these functions dominates the other on its whole period finally we prove that the n step mir inequalities generate two slope facets for the infinite group polyhedra and hence are potentially strong

t space and cutting planes in this paper we show how knowledge about t space translates directly into cutting planes for general integer programming problems after providing background on corner polyhedra and on t space this paper examines t space in some detail it gives a variety of constructions for t space facets all of which translate into cutting planes and introduces continuous families of facets in view of the great variety of possible facets no one of which can be dominated either by any other or by any combination of the others a measure of merit is introduced to provide guidance on their usefulness t spaces based on higher dimensional groups are discussed briefly as is the idea of going beyond cutting planes to iterated approximations of corner polyhedra

relations between facets of low and high dimensional group problems one dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher dimensional groups there are no known facets for infinite group problems whose dimension is larger than two in this paper we introduce an operation that we call sequential merge we prove that the sequential merge operator creates a very large family of facet defining inequalities for high dimensional infinite group problems using facet defining inequalities of lower dimensional group problems further they exhibit two properties that reflect the benefits of using facets of high dimensional group problems they have continuous variables coefficients that are not dominated by those of the constituent low dimensional cuts and they can produce cutting planes that do not belong to the first split closure of mips further we introduce a general scheme for generating valid inequalities for lower dimensional group problems using valid inequalities of higher dimensional group problems we present conditions under which this construction generates facet defining inequalities when applied to sequential merge inequalities we show that this procedure yields some two step mir inequalities of dash and gunluk

on a generalization of the master cyclic group polyhedron we study the master equality polyhedron mep which generalizes the master cyclic group polyhedron mcgp and the master knapsack polyhedron mkp we present an explicit characterization of the polar of the nontrivial facet defining inequalities for mep this result generalizes similar results for the mcgp by gomory 1969 and for the mkp by araoz 1974 furthermore this characterization gives a polynomial time algorithm for separating an arbitrary point from mep we describe how facet defining inequalities for the mcgp can be lifted to obtain facet defining inequalities for mep and also present facet defining inequalities for mep that cannot be obtained in such a way finally we study the mixed integer extension of mep and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of mep

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

using cuts for mixed integer knapsack sets to generate cuts for mixed integer polyhedral conic sets based on a bijective mapping between two mixed integer sets we introduce a new perspective on developing cuts for the mixed integer polyhedral conic mipc set by establishing a one to one correspondence between the cuts for this set and those for a related mixed integer knapsack mik set the face facet defining properties of the corresponding cuts are identical for their respective sets we further show that the cut generation approach for the mipc set resulting from this new perspective always produces cuts that dominate those generated based on any of the two individual mik constraints corresponding to the mipc constraint our computational results show this dominance can be quite significant as a special case of this new perspective the conic mir inequality of atamturk and narayanan for the mipc set and its properties can be directly derived from the mir inequality for the mik set and its properties we also generalize these cuts to the n step conic mir inequalities which are directly derived form the n step mir inequalities for the mik set

the worst case analysis of strong knapsack facets in this paper we identify strong facet defining inequalities for the master knapsack polytope our computational experiments for small master knapsack problems show that 1 k facets for small values of k k le 4 k 4 are strong facets for the knapsack polytope we show that this finding is robust by proving that the removal of these facets from the master knapsack polytope significantly weakens the resulting relaxation in the worst case we show that the 1 k facets for k 1 k 1 are the strongest in that their removal from the master knapsack polytope weakens the relaxation by a factor of 3 2 in the worst case we then show that the 1 k facets with k 3 k 3 or 4 are the next strongest we also show that the strength of the 1 k facets weakens as k grows and that the 1 k facets with k even are stronger than the 1 k facets with k odd

a dependence measure for multivariate and spatial extreme values properties and inference we present properties of a dependence measure that arises in the study of extreme values in multivariate and spatial problems for multivariate problems the dependence measure characterises dependence at the bivariate level for all pairs and all higher orders up to and including the dimension of the variable necessary and sufficient conditions are given for subsets of dependence measures to be self consistent that is to guarantee the existence of a distribution with such a subset of values for the dependence measure for pairwise dependence these conditions are given in terms of positive semidefinite matrices and non differentiable positive definite functions we construct new nonparametric estimators for the dependence measure which unlike all naive nonparametric estimators impose these self consistency properties as the new estimators provide an improvement on the naive methods both in terms of the inferential and interpretability properties their use in exploratory extreme value analyses should aid the identification of appropriate dependence models the methods are illustrated through an analysis of simulated multivariate data which shows that a lack of self consistency is frequently a problem with the existing estimators and by a spatial analysis of daily rainfall extremes in south west england which finds a smooth decay in extremal dependence with distance

n n step cycle inequalities facets for continuous multi mixing set and strong cuts for multi module capacitated lot sizing problem in this paper we introduce a generalization of the continuous mixing set which we refer to as the continuous multi mixing set this set is closely related to the feasible set of the multi module capacitated lot sizing mml problem with out backlogging we present a family of valid inequalities for the continuous multi mixing set and identify conditions under which they are facet defining the cycle inequalities n n step mir inequalities and mixed n n step mir inequalities are special cases of these inequalities we also present an exact separation algorithm for our inequalities we then use these inequalities to generate valid inequalities for mml with out backlogging our computational results show that our cuts are very effective in solving the mml instances resulting in substantial reduction in the integrality gap number of nodes and total solution time

corner polyhedra and their connection with cutting planes we review the necessary background on corner polyhedra and use this to show how knowledge about corner polyhedra and subadditive functions translates into a great variety of cutting planes for general integer programming problems experiments are described that indicate the dominance of a relatively small number of the facets of corner polyhedra this has implications for their value as cutting planes

on a level set characterization of the value function of an integer program and its application to stochastic programming we propose a level set approach to characterize the value function of a pure linear integer program with inequality constraints we study theoretical properties of our characterization and show how they can be exploited to optimize a class of stochastic integer programs through a value function reformulation specifically we develop algorithmic approaches that solve two stage multidimensional knapsack problems with random budgets yielding encouraging computational results

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

mingling mixed integer rounding with bounds mixed integer rounding mir is a simple yet powerful procedure for generating valid inequalities for mixed integer programs when used as cutting planes mir inequalities are very effective for mixed integer programming problems with unbounded integer variables for problems with bounded integer variables however cutting planes based on lifting techniques appear to be more effective this is not surprising as lifting techniques make explicit use of the bounds on variables whereas the mir procedure does not in this paper we describe a simple procedure which we call mingling for incorporating variable bound information into mir by explicitly using the variable bounds the mingling procedure leads to strong inequalities for mixed integer sets with bounded variables we show that facets of mixed integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure in particular the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of marchand and wolsey math program 85 15 33 1999 as well as the continuous integer knapsack cover and pack inequalities of atamturk math program 98 145 175 2003 ann oper res 139 21 38 2005 in addition mingling inequalities give a generalization of the two step mir inequalities of dash and gunluk math program 105 29 53 2006 under some conditions

on a class of subadditive duals for the uncapacitated facility location problem a family of subadditive functions which contains the optimal solution of subadditive duals for integer programs has been presented by klabjan 2007 in this paper we present a subadditive dual ascent procedure to find an optimal subadditive dual function based on klabjan s generator subadditive function then we use the proposed method to solve the so called uncapacitated facility location problem uflp establishing an explicit formulation for generator subadditive functions in the columns of the coefficient matrix of the uflp we derive some variable fixing rules for the problem some computational results are also presented

logistic regression for autocorrelated data with application to repeated measures summary a stochastic model is proposed for the study of the influence of time dependent covariates on the marginal distribution of the binary response in serially correlated binary data markov chains are expressed in terms of transitional rather than marginal probabilities we show how to construct the model so that the covariates relate only to the mean value of the process independently of the association parameter after formulating the stochastic model for a simple sequence of data with possibly missing data the same approach is applied to a repeated measures setting and illustrated with a real data example

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

on a direct method of analyzing impacting mechanical systems the paper presents an analytic method of investigating impacting mechanical systems the method is based on presentation of impact forces as sequences of dirac delta functions periodic vibrations are described by particular solutions of the equations of motion resulting from trigonometric expansions of sequences of dirac distributions the fourier form of the solution is avoided by establishing the initial state vector which ensures the occurrence of periodic motion and allows one to determine the solution in a compact form the infinite series describing elements of the initial state vector are convergent and their sums have been determined forced vibrations of the impacting systems are considered for dissipative and conservative systems the method is illustrated by an example of an impacting system with three degrees of freedom

integer programming duality price functions and sensitivity analysis recently a duality theory for integer programming has been developed here we examine some of the economic implications of this theory in particular the necessity of using price functions in place of prices and the possibility of carrying out sensitivity analysis of optimal solutions in addition we consider the form of price functions that are generated by known algorithms for integer programming

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

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