dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

finite sample adjustments in estimating equations and covariance estimators for intracluster correlations bias corrected covariance estimators are introduced in the context of an estimating equations approach for intracluster correlations among binary outcomes simulation study results show that the bias corrected covariance estimators perform better than uncorrected sandwich estimators in terms of bias and coverage probabilities additionally introduction of a matrix based bias correction into the estimating equations considerably improves point and interval estimation for the intracluster correlations the methods are illustrated using data from a nested cross sectional cluster trial on reducing underage drinking

model comparison of generalized linear mixed models generalized linear mixed models glmms have been widely appreciated in biological and medical research maximum likelihood estimation has received a great deal of attention comparatively not much has been done on model comparison or hypotheses testing in this article we propose a path sampling procedure to compute the observed data log likelihood function so that the bayesian information criterion bic can be applied to model comparison or hypothesis testing advantages of the proposed path sampling procedure are discussed two medical data sets are analysed for providing illustrative examples of the proposed methodology

dispersive blowup of solutions of generalized korteweg de vries equations abstract the strong effect of dispersion on short wavelength disturbances featured by the korteweg de vries equation and some of its generalizations is exploited to provide solutions of these equations that correspond to infinitely smooth initial data which exhibit a specified loss of spatial smoothness at particular times the points in space time at which smoothness is lost may even comprise an arbitrary discrete subset of the upper half plane x t x r t 0 our results are related to recent work on smoothing of solutions of such equations some of which are sharpened here and they show that in certain aspects these earlier results are not far from being optimal the theory makes use of new results concerning well posedness of such equations in weighted sobolov spaces and some detailed analysis of the linear korteweg de vries equation

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

a non homogeneous boundary value problem for the korteweg de vries equation posed on a finite domain ii abstract studied here is an initial and boundary value problem for the korteweg de vries equation posed on a bounded interval with nonhomogeneous boundary conditions this particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed it is shown here that this initial boundary value problem is globally well posed in the l 2 based sobolev space h s 0 1 for any s 0 in addition the mapping that associates to appropriate initial and boundary data the corresponding solution is shown to be analytic as a function between appropriate banach spaces

on the quadratic eigenvalue complementarity problem over a general convex cone the solution of the conic quadratic eigenvalue complementarity problem cqeicp is first investigated without assuming symmetry on the matrices defining the problem a new sufficient condition for existence of solutions of cqeicp is presented extending to arbitrary pointed closed and convex cones a condition known to hold when the cone is the nonnegative orthant we also address the symmetric cqeicp where all its defining matrices are symmetric we show that assuming that two of its defining matrices are positive definite this symmetric cqeicp reduces to the computation of a stationary point of an appropriate merit function on a convex set furthermore we discuss the use of the so called spectral projected gradient spg algorithm for solving cqeicp when the cone of interest is the second order cone socqeicp a new algorithm is designed for the computation of the projections required by the spg method to deal with socqeicp numerical results are included to illustrate the efficiency of the spg method and the new projection technique in practice

a goodness of fit approach to inference procedures for the kappa statistic confidence interval construction significance testing and sample size estimation we propose a new procedure for constructing a confidence interval about the kappa statistic in the case of two raters and a dichotomous outcome the procedure is based on a chi square goodness of fit test as applied to a model frequently used for clustered binary data the procedure provides coverage levels that are accurate in samples of smaller size than those required for other procedures the procedure also has use for significance testing and the planning of corresponding sample size requirements

complementary eigenvalues of graphs abstract in this paper we study the eigenvalue complementarity problem eicp when its matrix a belongs to the class s g a a i j a i j a j i 0 iff i j e where g v e is a connected graph it is shown that if all nondiagonal elements of a s g are nonpositive then a has a unique complementary eigenvalue which is the smallest eigenvalue of a in particular zero is the unique complementary eigenvalue of the laplacian and the normalized laplacian matrices of a connected graph the number c g of complementary eigenvalues of the adjacency matrix of a connected graph g is shown to be bounded above by the number b g of induced nonisomorphic connected subgraphs of g furthermore c g b g if the perron roots of the adjacency matrices of these subgraphs are all distinct finally the maximum number of complementary eigenvalues for the adjacency matrices of graphs is shown to grow faster than any polynomial on the number of vertices

spectral analysis of coupled linear complementarity problems abstract this note deals with the so called cone constrained bivariate eigenvalue problem the equilibrium model under consideration is a system of linear complementarity problems p x ax by x p q y cx dy y q involving two closed convex cones and their corresponding duals we study the set of pairs r 2 for which this system has a nontrivial solution x y r n m we discuss also the link between the cone constrained version and the unconstrained one

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

on the estimation of intracluster correlation for time to event outcomes in cluster randomized trials cluster randomized trials crts involve the random assignment of intact social units rather than independent subjects to intervention groups time to event outcomes often are endpoints in crts analyses of such data need to account for the correlation among cluster members the intracluster correlation coefficient icc is used to assess the similarity among binary and continuous outcomes that belong to the same cluster however estimating the icc in crts with time to event outcomes is a challenge because of the presence of censored observations the literature suggests that the icc may be estimated using either censoring indicators or observed event times a simulation study explores the effect of administrative censoring on estimating the icc results show that icc estimators derived from censoring indicators or observed event times are negatively biased analytic work further supports these results observed event times are preferred to estimate the icc under minimum frequency of administrative censoring to our knowledge the existing literature provides no practical guidance on the estimation of icc when substantial amount of administrative censoring is present the results from this study corroborate the need for further methodological research on estimating the icc for correlated time to event outcomes copyright 2016 john wiley sons ltd

a note on the kappa statistic for clustered dichotomous data the kappa statistic is widely used to assess the agreement between two raters motivated by a simulation based cluster bootstrap method to calculate the variance of the kappa statistic for clustered physician patients dichotomous data we investigate its special correlation structure and develop a new simple and efficient data generation algorithm for the clustered physician patients dichotomous data based on the delta method and its special covariance structure we propose a semi parametric variance estimator for the kappa statistic an extensive monte carlo simulation study is performed to evaluate the performance of the new proposal and five existing methods with respect to the empirical coverage probability root mean square error and average width of the 95 confidence interval for the kappa statistic the variance estimator ignoring the dependence within a cluster is generally inappropriate and the variance estimators from the new proposal bootstrap based methods and the sampling based delta method perform reasonably well for at least a moderately large number of clusters e g the number of clusters k 50 the new proposal and sampling based delta method provide convenient tools for efficient computations and non simulation based alternatives to the existing bootstrap based methods moreover the new proposal has acceptable performance even when the number of clusters is as small as k 25 to illustrate the practical application of all the methods one psychiatric research data and two simulated clustered physician patients dichotomous data are analyzed

likelihood inference for exchangeable binary data with varying cluster sizes ce papier etudie l estimation au maximum de vraisemblance dans des modeles satures et insatures pour des donnees binaires correlees echangeables lorsqu on dispose d un echantillon de clusters independants de tailles variables nous discutons plusieurs parametrisations de ces modeles et nous proposons l utilisation de l algorithme em pour obtenir des estimations du maximum de vraisemblance la methodologie est illustree par des applications a une etude d agregation familiale de maladie et a la construction d un essai randomise par groupe pour la prevention du cancer

interval estimation for a difference between intraclass kappa statistics summary model based inference procedures for the kappa statistic have developed rapidly over the last decade however no method has yet been developed for constructing a confidence interval about a difference between independent kappa statistics that is valid in samples of small to moderate size in this article we propose and evaluate two such methods based on an idea proposed by newcombe 1998 statistics in medicine 17 873 890 for constructing a confidence interval for a difference between independent proportions the methods are shown to provide very satisfactory results in sample sizes as small as 25 subjects per group sample size requirements that achieve a prespecified expected width for a confidence interval about a difference of kappa statistic are also presented

kappa coefficients in medical research kappa coefficients are measures of correlation between categorical variables often used as reliability or validity coefficients we recapitulate development and definitions of the k categories by m ratings kappas k x m discuss what they are well or ill designed to do and summarize where kappas now stand with regard to their application in medical research the 2 x m m 2 intraclass kappa seems the ideal measure of binary reliability a 2 x 2 weighted kappa is an excellent choice though not a unique one as a validity measure for both the intraclass and weighted kappas we address continuing problems with kappas there are serious problems with using the k x m intraclass k 2 or the various k x m weighted kappas for k 2 or m 2 in any context either because they convey incomplete and possibly misleading information or because other approaches are preferable to their use we illustrate the use of the recommended kappas with applications in medical research

criteria for copositive matrices of order four abstract it is proved that the copositivity of a symmetric matrix of order n is equivalent to the copositivity of two symmetric matrices of order n 1 if the matrix has a row whose off diagonal elements are all nonpositive this result is used for deriving criteria for copositive matrices of order 3 and 4

blow up of smooth solutions of the korteweg de vries equation abstract the paper is devoted both to some initial boundary value problems and to the cauchy problem for the kdv equation blow up results are proved for these problems by using the nonlinear capacity approach 9

correlated binomial variates properties of estimator of intraclass correlation and its effect on sample size calculation in group randomized studies the sample size calculations are complicated by within group worksite community etc correlation we compare by simulation the moment method and the more standard anova method of estimating the intraclass correlation we find the former is less biased for a small to moderate number of clusters but the difference disappears when the appropriate degree of freedom is used for the anova estimator we propose a simulation approach for sample size determination and illustrate it with an example

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

testing the definiteness of matrices on polyhedral cones abstract the principal pivoting scheme for quadratic programming is used to derive finite criteria for testing whether a real symmetric matirx is nonnegative definite nonnegative definite plus or positive definite on a convex polyhedral cone the paper is an extension of a recent one where the author gave criteria for different classes of copositive matrices

on the finiteness of the cone spectrum of certain linear transformations on euclidean jordan algebras abstract let l be a linear transformation on a finite dimensional real hilbert space h and k be a closed convex cone with dual k in h the cone spectrum of l relative to k is the set of all real for which the linear complementarity problem x k y l x x k and x y 0 admits a nonzero solution x in the setting of a euclidean jordan algebra h and the corresponding symmetric cone k we discuss the finiteness of the cone spectrum for z transformations and quadratic representations on h

sample size considerations for stratified cluster randomization design with binary outcomes and varying cluster size stratified cluster randomization trials crts have been frequently employed in clinical and healthcare research comparing with simple randomized crts stratified crts reduce the imbalance of baseline prognostic factors among different intervention groups due to the popularity there has been a growing interest in methodological development on sample size estimation and power analysis for stratified crts however existing work mostly assumes equal cluster size within each stratum and uses multilevel models clusters are often naturally formed with random sizes in crts with varying cluster size commonly used ad hoc approaches ignore the variability in cluster size which may underestimate overestimate the required number of clusters for each group per stratum and lead to underpowered overpowered clinical trials we propose closed form sample size formulas for estimating the required total number of subjects and for estimating the number of clusters for each group per stratum based on cochran mantel haenszel statistic for stratified cluster randomization design with binary outcomes accounting for both clustering and varying cluster size we investigate the impact of various design parameters on the relative change in the required number of clusters for each group per stratum due to varying cluster size simulation studies are conducted to evaluate the finite sample performance of the proposed sample size method a real application example of a pragmatic stratified crt of a triad of chronic kidney disease diabetes and hypertension is presented for illustration

profile likelihood based confidence interval of the intraclass correlation for binary outcome data sampled from clusters the intraclass correlation in binary outcome data sampled from clusters is an important and versatile measure in many biological and biomedical investigations properties of the different estimators of the intraclass correlation based on the parametric semi parametric and nonparametric approaches have been studied extensively mainly in terms of bias and efficiency see for example ridout et al biometrics 1999 55 137 148 paul et al journal of statistical computation and simulation 2003 73 507 523 and lee statistical modelling 2004 4 113 126 but little attention has been paid to extending these results to the problem of the confidence intervals in this article we generalize the results of the four point estimators by constructing asymptotic confidence intervals obtaining closed form asymptotic and sandwich variance expressions of those four point estimators it appears from simulation results that the asymptotic confidence intervals based on these four estimators have serious under coverage to remedy this we introduce the fisher s z transformation approach on the intraclass correlation coefficient the profile likelihood approach based on the beta binomial model and the hybrid profile variance approach based on the quadratic estimating equation for constructing the confidence intervals of the intraclass correlation for binary outcome data as assessed by monte carlo simulations these confidence interval approaches show significant improvement in the coverage probabilities moreover the profile likelihood approach performs quite well by providing coverage levels close to nominal over a wide range of parameter combinations we provide applications to biological data to illustrate the methods copyright 2012 john wiley sons ltd

propensity score methods for estimating relative risks in cluster randomized trials with low incidence binary outcomes and selection bias despite randomization selection bias may occur in cluster randomized trials classical multivariable regression usually allows for adjusting treatment effect estimates with unbalanced covariates however for binary outcomes with low incidence such a method may fail because of separation problems this simulation study focused on the performance of propensity score ps based methods to estimate relative risks from cluster randomized trials with binary outcomes with low incidence the results suggested that among the different approaches used multivariable regression direct adjustment on ps inverse weighting on ps and stratification on ps only direct adjustment on the ps fully corrected the bias and moreover had the best statistical properties

criteria for copositive matrices using simplices and barycentric coordinates abstract we present criteria for verifying the copositivity of an n n matrix given that all its principal submatrices of order n 1 are copositive for n 4 5 necessary and sufficient conditions for copositivity are given based on the sign distribution of the off diagonal elements of a single row here it is only assumed that one principal submatrix is copositive

a test for copositive matrices abstract the paper presents necessary and sufficient conditions that a symmetric matrix be copositive or strictly copositive the conditions are given in terms of the eigenvalues and eigenvectors of the principal submatrices of the given matrix

the difference between 5 5 doubly nonnegative and completely positive matrices abstract the convex cone of n n completely positive cp matrices and its dual cone of copositive matrices arise in several areas of applied mathematics including optimization every cp matrix is doubly nonnegative dnn i e positive semidefinite and component wise nonnegative and it is known that for n 4 only every dnn matrix is cp in this paper we investigate the difference between 5 5 dnn and cp matrices defining a bad matrix to be one which is dnn but not cp we i design a finite procedure to decompose any n n dnn matrix into the sum of a cp matrix and a bad matrix which itself cannot be further decomposed ii show that every bad 5 5 dnn matrix is the sum of a cp matrix and a single bad extreme matrix and iii demonstrate how to separate bad extreme matrices from the cone of 5 5 cp matrices

copositivity and constrained fractional quadratic problems we provide completely positive and copositive optimization formulations for the constrained fractional quadratic problem cfqp and standard fractional quadratic problem stfqp based on these formulations semidefinite programming relaxations are derived for finding good lower bounds to these fractional programs which can be used in a global optimization branch and bound approach applications of the cfqp and stfqp related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

finite criteria for conditional definiteness of quadratic forms abstract a real symmetric n n matrix q is a conditionally positivesemidefinite where a is a given m n matrix if x qx 0 whenever ax 0 and is a conditionally positive definite if strict inequality holds except when x 0 when a is the identity matrix these notions reduce to the well studied notions of copositivity and strict copositivity respectively this paper presents finite criteria involving only the solution of sets of linear equations constructed from the matrices q a for testing both types of conditional definiteness these criteria generalize known facts about copositive matrices and when q is invertible and all row submatrices of a have maximal rank can be very elegantly stated in terms of schur complements of the matrix aq 1 a

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

complementarity forms of theorems of lyapunov and stein and related results abstract the well known lyapunov s theorem in matrix theory continuous dynamical systems asserts that a complex square matrix a is positive stable i e all eigenvalues lie in the open right half plane if and only if there exists a positive definite matrix x such that ax xa is positive definite in this paper we prove a complementarity form of this theorem a is positive stable if and only if for any hermitian matrix q there exists a positive semidefinite matrix x such that ax xa q is positive semidefinite and x ax xa q 0 by considering cone complementarity problems corresponding to linear transformations of the form i s we show that a complex matrix a has all eigenvalues in the open unit disk of the complex plane if and only if for every hermitian matrix q there exists a positive semidefinite matrix x such that x axa q is positive semidefinite and x x axa q 0 by specializing q to i we deduce the well known stein s theorem in discrete linear dynamical systems a has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix x such that x axa is positive definite

on eigenvalues induced by a cone constraint abstract let a be an n n real matrix and k r n be a closed convex cone the spectrum of a relative to k denoted by a k is the set of all r for which the linear complementarity problem x k ax x k x ax x 0 admitsa nonzero solution x r n the notation k stands for the positive dual cone of k the purpose of this work is to study the main properties of the mapping k and discuss some structural differences existing between the polyhedral case i e k is finitely generated and the nonpolyhedral case

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

solving inverse cone constrained eigenvalue problems we compare various algorithms for constructing a matrix of order n whose pareto spectrum contains a prescribed set lambda lambda 1 ldots lambda p of reals in order to avoid overdetermination one assumes that p does not exceed n 2 the inverse pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations we also address the issue of computing lorentz spectra and solving inverse lorentz eigenvalue problems

a comparison of generalized linear mixed model procedures with estimating equations for variance and covariance parameter estimation in longitudinal studies and group randomized trials response data in longitudinal studies and group randomized trials are gathered on units that belong to clusters within which data are usually positively correlated therefore estimates and confidence intervals for intraclass correlation or variance components are helpful when designing a longitudinal study or group randomized trial data simulated from both study designs are used to investigate the estimation of variance and covariance parameters from the following procedures for continuous outcomes restricted maximum likelihood reml and estimating equations ee for binary outcomes restricted pseudo likelihood repl and estimating equations ee we evaluate these procedures to see which provide valid and precise estimates as well as correct standard errors for the intraclass correlation coefficient or variance components reml seems the better choice for estimating terms related to correlation for models with normal outcomes especially in group randomized trial situations results for reml and ee are mixed when outcomes are continuous and non normal with binary outcomes neither repl nor ee provides satisfactory estimation or inference in longitudinal study situations while repl is preferable for group randomized trials

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

new results on pareto spectra abstract a pareto eigenvalue of a matrix a of order n is a scalar r for which the complementarity problem 0 x a x x 0 admits a nonzero solution x r n pareto eigenvalues are also known as complementarity eigenvalues they have found applications in graph theory cone constrained dynamical systems and mathematical modeling in general in this paper we continue our study on theoretical properties of pareto spectra special attention is paid to classification of pareto eigenvalues and cardinality issues

perron frobenius property of copositive matrices and a block copositivity criterion abstract haynsworth and hoffman proved in 1969 that the spectral radius of a symmetric copositive matrix is an eigenvalue of this matrix this note investigates conditions which guarantee that an eigenvector corresponding to this dominant eigenvalue has no negative coordinates i e whether the perron frobenius property holds also a block copositivity criterion using the schur complement is specified which may be helpful to reduce dimension in copositivity checks and which generalizes results proposed by andersson et al in 1995 and johnson and reams in 2005 apparently the latter five researchers were unaware of the more general results by the author precedingly published in 1987 and 1996 respectively

z transformations on proper and symmetric cones z transformations motivated by the similarities between the properties of z matrices on r n and lyapunov and stein transformations on the semidefinite cone mathcal s n we introduce and study z transformations on proper cones we show that many properties of z matrices extend to z transformations we describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations finally we study the equivalence of q and p properties of z transformations on symmetric cones in particular we prove such an equivalence on the lorentz cone

complementarity eigenvalue analysis of connected graphs abstract this work concerns the spectral analysis of connected graphs from a non traditional point of view instead of the usual eigenvalues of the adjacency matrix a g of a graph g under consideration we compute and analyze the complementarity eigenvalues of a g the complementarity eigenvalues of a general square matrix are defined in terms of a certain complementarity system relative to the componentwise ordering the complementarity eigenvalues of a g form the so called complementarity spectrum of g in general the structure of a connected graph is better discriminated in terms of its complementarity spectrum than in terms of its usual spectrum this observation is one of the leading motivation behind our work

estimating the number of clusters for the analysis of correlated binary response variables from unbalanced data dominating the literature on sample size determination are papers concerned with observations collected from independent responses we discuss in this paper under the common correlation model for binary responses the issue of estimating the number of groups when the sampling units are clusters of variable sizes

some inertia theorems in euclidean jordan algebras abstract this paper deals with some inertia theorems in euclidean jordan algebras first based on the continuity of eigenvalues we give an alternate proof of kaneyuki s generalization of sylvester s law of inertia in simple euclidean jordan algebras as a consequence we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite second we present ostrowski schneider type inertia results in euclidean jordan algebras in particular we relate the inertias of objects a and x in a euclidean jordan algebra when l a x 0 or s a x 0 where l a and s a denote lyapunov and stein transformations respectively

inference procedures for assessing interobserver agreement among multiple raters summary we propose a new procedure for constructing inferences about a measure of interobserver agreement in studies involving a binary outcome and multiple raters the proposed procedure based on a chi square goodness of fit test as applied to the correlated binomial model bahadur 1961 in studies in item analysis and prediction 158 176 is an extension of the goodness of fit procedure developed by donner and eliasziw 1992 statistics in medicine11 1511 1519 for the case of two raters the new procedure is shown to provide confidence interval coverage levels that are close to nominal over a wide range of parameter combinations the procedure also provides a sample size formula that may be used to determine the required number of subjects and raters for such studies

an interval estimate for the interclass correlation in beta binomial sampling the beta binomial distribution has been frequently applied to model intraclass correlation among dichotomous responses in cluster sampling to avoid using sophisticated numerical procedures to calculate maximum likelihood estimates many authors have proposed different moment estimators most of them focused their discussion however on point estimation only in this paper we have developed a closed form interval estimate for the intraclass correlation and have shown it has asymptotically correct coverage we have further used a monte carlo simulation to evaluate the performance of the proposed interval estimation in a variety of situations we have found that when the number of clusters is moderate 50 the proposed interval estimation performs reasonably well except for the situation where the intraclass correlation is large and the underlying mean response probability is small even when the number of clusters is as small as 20 the proposed interval estimation may still be useful in the situation where the intraclass correlation is small we have also extended the proposed interval estimate to accommodate stratified sampling in which each stratum has a different mean response probability but has a common intraclass correlation

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

conjugate cone characterization of positive definite and semidefinite matrices abstract positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in rn these results are obtained by generalizing moreau s polar decomposition to a conjugate decomposition some typical results are the matrix a is positive definite if and only if for some closed convex cone k a is positive definite on k and a at 1 exists and is semidefinite on the polar cone k the matrix a is positive semidefinite if and only if for some closed convex cone k such that either k is polyhedral or a at k is closed a is positive semidefinite on both k and the conjugate cone ka s xt a at s 0 x k and a at x 0 for all x in k such that xtax 0

a block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem in this paper we address the solution of the symmetric eigenvalue complementarity problem eicp by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex the spectral projected gradient spg method has been recommended to this optimization problem when the dimension of the symmetric eicp is large and the accuracy of the solution is not a very important issue we suggest a new algorithm which combines elements from the spg method and the block active set method where the latter was originally designed for box constrained quadratic programs in the new algorithm the projection onto the unit simplex in the spg method is replaced by the much cheaper projection onto a box this can be of particular advantage for large and sparse symmetric eicps global convergence to a solution of the symmetric eicp is established computational experience with medium and large symmetric eicps is reported to illustrate the efficacy and efficiency of the new algorithm

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

2 x 2 kappa coefficients measures of agreement or association summary two general but different contexts in which kappa might be used are defined agreement and association two models one for agreement and one for utility of association are defined yielding different kappa coefficients and different sampling theory asymptotic results are derived for both models small sample evaluations are presented for the model for agreement

bounds for the spectral radius of a graph when nodes are removed abstract we present a new type of lower bound for the spectral radius of a graph in which m nodes are removed as a corollary cioab s theorem 4 which states that the maximum normalized principal eigenvector component in any graph never exceeds 1 2 with equality for the star appears as a special case of our more general result

an alternative theorem for quadratic forms and extensions abstract this note concerns an alternative theorem for quadratic forms established recently by y yuan we explore some directions in which yuan s result can be extended and we indicate also some important situations where an alternative theorem a la yuan does not hold

on pairs of vectors achieving the maximal angle of a convex cone in this paper we explore the concept of antipodality relative to a closed convex cone the problem under consideration is that of finding a pair of unit vectors in k achieving the maximal angle of the cone we mention also a few words on the attainability of critical angles by way of application of the general theory we briefly discuss the problem of estimating the radius of pointedness of a cone

tensor eigenvalue complementarity problems this paper studies tensor eigenvalue complementarity problems basic properties of standard and complementarity tensor eigenvalues are discussed we formulate tensor eigenvalue complementarity problems as constrained polynomial optimization when one tensor is strictly copositive the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity when no tensor is strictly copositive we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process the complementarity eigenvalues can be computed sequentially the formulated polynomial optimization can be solved by lasserre s hierarchy of semidefinite relaxations we show that it has finite convergence for generic tensors numerical experiments are presented to show the efficiency of proposed methods

on copositive matrices and strong ellipticity for isotropic elastic materials on etablit les conditions necessaires et suffisantes pour une forte ellipticite des equations d un materiau elastique non lineaire isotrope compressible a l equilibre on utilise un theoreme de representation de matrices copositives du a cottle habetler et lemke 1970

confidence interval estimation of the intraclass correlation coefficient for binary outcome data we obtain closed form asymptotic variance formulae for three point estimators of the intraclass correlation coefficient that may be applied to binary outcome data arising in clusters of variable size our results include as special cases those that have previously appeared in the literature fleiss and cuzick 1979 applied psychological measurement 3 537 542 bloch and kraemer 1989 biometrics 45 269 287 altaye donner and klar 2001 biometrics 57 584 588 simulation results indicate that confidence intervals based on the estimator proposed by fleiss and cuzick provide coverage levels close to nominal over a wide range of parameter combinations two examples are presented

searching for critical angles in a convex cone the concept of antipodality relative to a closed convex cone has been explored in detail in a recent work of ours the antipodality problem consists of finding a pair of unit vectors in k achieving the maximal angle of the cone our attention now is focused not just in the maximal angle but in the angular spectrum of the cone by definition the angular spectrum of a cone is the set of angles satisfying the stationarity or criticality condition associated to the maximization problem involved in the determination of the maximal angle in the case of a polyhedral cone the angular spectrum turns out to be a finite set among other results we obtain an upper bound for the cardinality of this set we also discuss the link between the critical angles of a cone k and the critical angles of its dual cone

copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints abstract a symmetric matrix c is called copositive if the quadratic form x cx is nonnegative for all nonnegative values of the variables x 1 x 2 x n x a known sufficient condition for a quadratic form x qx to be positive unless x 0 subject to the linear inequality constraints ax 0 is that there should exist a copositive matrix c such that q a ca is positive definite the main result of this paper establishes the necessity of this condition for x qx to be merely nonnegative subject to ax 0 the situation is less straightforward the necessity of the existence of a copositive matrix c such that q a ca is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of a and counterexamples are given to show that in general no such matrix may exist even when slater s constraint qualification holds our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x qx subject to ax 0 in which specific symmetric matrices constructed by rational operations from a and q and depending upon a single real parameter v must be tested for positive definiteness or strict copositivity for large values of v this technique is illustrated by several examples

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

spectral theory of copositive matrices abstract let a r n n we provide a block characterization of copositive matrices with the assumption that one of the principal blocks is positive definite haynsworth and hoffman showed that if r is the largest eigenvalue of a copositive matrix then r for all other eigenvalues of a we continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues moreover if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues then it is possible to increase the nonnegative eigenvalues to form a copositive matrix a without changing the eigenvectors we also show that if a copositive matrix has just one positive eigenvalue and n 1 nonpositive eigenvalues then a has a nonnegative eigenvector corresponding to a nonnegative eigenvalue

critical angles in polyhedral convex cones numerical and statistical considerations this work concerns the numerical computation of critical angles in polyhedral convex cones the set of proper critical angles is evaluated explicitly by solving a series of generalized eigenvalue problems involving the generators of the cone the local maximal angles are identified by using a necessary condition for local maximality the expected numbers of critical angles in random polyhedral convex cones are estimated experimentally

regularity versus degeneracy in dynamics games and optimization a unified approach to different aspects in this paper links are established between optimality conditions for quadratic optimization problems qualitative properties in the nonlinear selection replicator dynamics and central solution concepts of evolutionary game theory with particular emphasis on several regularity conditions that are desirable in any of the three fields mentioned above as strictness conditions for locally optimal solutions as hyperbolicity conditions for fixed points and as quasi strictness conditions for game equilibria

on spectral maps induced by convex cones abstract let l e be the vector space of linear endomorphisms on a euclidean space e and k be a closed convex cone in e the spectral map k l e 2 r induced by k is defined as follows to each linear endomorphism a on e it associates the set k a of all r for which the complementarity problem k x a x x k has a nonzero solution x e here k is the dual cone of k and stands for orthogonality the set k a is known as the k spectrum of a the purpose of this work is twofold firstly we wish to compare algebraic properties of k and geometric properties of k secondly we wish to identify classes of linear transformations f l e l e that leave the k spectrum invariant i e such that k f a k a for all a l e the set of linear transformations preserving the k spectrum depends of course on the cone k under consideration

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

using copositivity for global optimality criteria in concave quadratic programming problems in this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems using this condition it follows that from the perspective of worst case complexity of concave quadratic problems the difference between local and global optimality conditions is not as large as in general as an essential ingredient we here use thee subdifferential calculus via an approach of hiriart urruty and lemarechal 1990

inverse eigenvalue problems for linear complementarity systems abstract traditionally an inverse eigenvalue problem is about reconstructing a matrix from a given spectral data in this work we study the set of real matrices a of order n such that the linear complementarity system x 0 ax x 0 x ax x 0 holds for prescribed pairs x 1 1 x p p the analysis of this new type of inverse eigenvalue problem differs substantially from the classical one