primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

associative and jordan algebras and polynomial time interior point algorithms for symmetric cones we present a general framework whereby analysis of interior point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras in particular such analyses are extendible to the cone of positive semidefinite hermitian matrices with complex and quaternion entries and to the lorentz cone we prove the case of the lorentz cone by using the embedding of its associated jordan algebra in the clifford algebra as an example of such extensions we take monterio s polynomial time complexity analysis of the family of similarly scaled directions introduced by monteiro and zhang 1998 and generalize it to cone lp over all representable symmetric cones

bayes estimation from a markov renewal process a procedure for bayes nonparametric estimation from a markov renewal process is developed it is based on a conjugate class of a priori distributions on the parameter space of semi markov transition distributions the class is characterized by a dirichlet family of distributions for random markov matrices and a beta family of levy processes for random cumulative hazard functions the main result is the derivation of the posterior law from an observation of the markov renewal process over a period of time

on inconsistent bayes estimates of location on montre que dans certaines situations relativement naturelles les estimateurs de position bayesiens sont inconsistants

a potential reduction newton method for constrained equations extending our previous work t wang r d c monteiro and j s pang math programming 74 1996 pp 159 195 this paper presents a general potential reduction newton method for solving a constrained system of nonlinear equations a major convergence result for the method is established specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed strengthened convergence results are established in the context of these specializations

nonparametric inference for a class of semi markov processes with censored observations a class of semi markov models those which have proportional hazards and which are forward going if state can be reached from i then i cannot be reached from j are shown to fit into the multiplicative intensity model of counting processes after suitable random time changes standard large sample results for counting processes following this multiplicative model can therefore be used to make inferences on the above class of semi markov models including the case where observations may be censored large sample results for a fourstate model used in clinical trials are presented

a bayesian nonparametric approach to reliability abstract it is suggested that problems in a reliability context may be handled by a bayesian non parametric approach a stochastic process is defined whose sample paths may be assumed to be either increasing hazard rates or decreasing hazard rates by properly choosing the parameter functions of the process the posterior distribution of the hazard rates are derived for both exact and censored data bayes estimates of hazard rates c d f s densities and means are found under squared error type loss functions some simulation is done and estimates graphed to better understand the estimators finally estimates of the c d f from some data in a paper by kaplan and meier are constructed author

on similarity invariants of matrix commutators and jordan products abstract denote by x y the additive commutator xy yx of two square matrices x y over a field f in a previous paper the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of a x 1 x 2 x k when a is a fixed matrix and x 1 x k vary were studied moreover given any expression g x 1 x k obtained from distinct noncommuting variables x 1 x k by applying recursively the lie product and without using the same variable twice the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of g x 1 x k when one of the variables x 1 x k takes a fixed value in f n n and the others vary were studied the purpose of the present paper is to show that analogous results can be obtained when additive commutators are replaced with multiplicative commutators or jordan products

a note on the existence of the alizadeh haeberly overton direction for semidefinite programming this note establishes a new sufficient condition for the existence and uniqueness of the alizadeh haeberly overton direction for semidefinite programming

a superlinearly convergent primal dual infeasible interior point algorithm for semidefinite programming a primal dual infeasible interior point path following algorithm is proposed for solving semidefinite programming sdp problems if the problem has a solution then the algorithm is globally convergent if the starting point is feasible or close to being feasible the algorithm finds an optimal solution in at most o sqrt n l iterations where n is the size of the problem and l is the logarithm of the ratio of the initial error and the tolerance if the starting point is large enough then the algorithm terminates in at most o nl steps either by finding a solution or by determining that the primal dual problem has no solution of norm less than a given number moreover we propose a sufficient condition for the superlinear convergence of the algorithm in addition we give two special cases of sdp for which the algorithm is quadratically convergent

first and second order methods for semidefinite programming in this paper we survey the most recent methods that have been developed for the solution of semidefinite programs we first concentrate on the methods that have been primarily motivated by the interior point ip algorithms for linear programming putting special emphasis in the class of primal dual path following algorithms we also survey methods that have been developed for solving large scale sdp problems these include first order nonlinear programming nlp methods and more specialized path following ip methods which use the preconditioned conjugate gradient or residual scheme to compute the newton direction and the notion of matrix completion to exploit data sparsity

on the number of invariant polynomials of the matrix xax 1 b abstract let f be a field if a is an n n matrix over f we denote by i a the number of invariant polynomials of a different from 1 we shall prove that if a b are n n matrices over f and t 1 n then i a i b n t if and only if there exists a nonsingular matrix x over f such that i xax 1 b t except in a few cases

spectrally complete pairs of matrices abstract we say that a pair a b of n n matrices over a field f is spectrally complete if for every n tuple 1 n of elements of f satisfying i tr a tr b there is an a similar to a such that a b has eigenvalues 1 n in this paper we identify all the spectrally complete pairs of matrices when f has at least three elements

bnp seq bayesian nonparametric differential expression analysis of sequencing count data abstractwe perform differential expression analysis of high throughput sequencing count data under a bayesian nonparametric framework removing sophisticated ad hoc pre processing steps commonly required in existing algorithms we propose to use the gamma beta negative binomial process which takes into account different sequencing depths using sample specific negative binomial probability dispersion parameters to detect differentially expressed genes by comparing the posterior distributions of gene specific negative binomial dispersion probability parameters these model parameters are inferred by borrowing statistical strength across both the genes and samples extensive experiments on both simulated and real world rna sequencing count data show that the proposed differential expression analysis algorithms clearly outperform previously proposed ones in terms of the areas under both the receiver operating characteristic and precision recall curves supplementary materials for this article are avai

polynomial convergence of a new family of primal dual algorithms for semidefinite programming this paper establishes the polynomial convergence of a new class of primal dual interior point path following feasible algorithms for semidefinite programming sdp whose search directions are obtained by applying newton s method to the symmetric central path equation p x p t 1 2 p t s p 1 p x p t 1 2 mu i 0 where p is a nonsingular matrix specifically we show that the short step path following algorithm based on the frobenius norm neighborhood and the semilong step path following algorithm based on the operator 2 norm neighborhood have o sqrt n l and o nl iteration complexity bounds respectively when p i this yields the first polynomially convergent semilong step algorithm based on a pure newton direction restricting the scaling matrix p at each iteration to a certain subset of nonsingular matrices we are able to establish an o n3 2l iteration complexity for the long step path following method the resulting subclass of search directions contains both the nesterov todd direction and the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction

strengthened existence and uniqueness conditions for search directions in semidefinite programming abstract primal dual interior point p d i p methods for semidefinite programming sdp are generally based on solving a system of matrix equations for a newton type search direction for a symmetrization of the optimality conditions these search directions followed the derivation of similar p d i p methods for linear programming lp among these a computationally interesting search direction is the aho direction however in contrast to the lp case existence and uniqueness of the aho search direction is not guaranteed under the standard nondegeneracy assumptions two different sufficient conditions are known that guarantee the existence and uniqueness independent of the specific linear constraints the first is given by shida shindoh kojima and is based on the semidefiniteness of the symmetrization of the product sx at the current iterate the second is a centrality condition given first by monteiro zanjacomo and then improved by monteiro todd in this paper we revisit and strengthen both of the above mentioned sufficient conditions we include characterizations for existence and uniqueness in the matrix equations arising from the linearization of the optimality conditions as well we present new results on the relationship between the kronecker product and the symmetric kronecker product that arise from these matrix equations we conclude with a proof that the existence and uniqueness of the aho direction is a generic property for every sdp problem and extend the results to the general monteiro zhang family of search directions

posterior simulation and bayes factors in panel count data models abstract this paper is concerned with the problems of posterior simulation and model choice for poisson panel data models with multiple random effects efficient algorithms based on markov chain monte carlo methods for sampling the posterior distribution are developed a new parameterization of the random effects and fixed effects is proposed and compared with a parameterization in common use and computation of marginal likelihoods and bayes factors via chib s 1995 method is also considered the methods are illustrated with two real data applications involving large samples and multiple random effects

the number of invariant polynomials of a matrix with prescribed off diagonal blocks abstract let a be an n n matrix over an arbitrary field f of the form a a 1 2 a 1 2 a 2 1 a 2 2 where a1 1 fp p and a2 2 fq q and p q n we characterize the possible number of nontrivial invariant polynomials of a when the submatrices a1 2 and a2 1 are prescribed

combinatorial clustering and the beta negative binomial process we develop a bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual we introduce a combinatorial stochastic process known as the negative binomial process rm nbp as an infinite dimensional prior appropriate for such problems we show that the rm nbp is conjugate to the beta process and we characterize the posterior distribution under the beta negative binomial process rm bnbp and hierarchical models based on the rm bnbp the rm hbnbp we study the asymptotic properties of the rm bnbp and develop a three parameter extension of the rm bnbp that exhibits power law behavior we derive mcmc algorithms for posterior inference under the rm hbnbp and we present experiments using these algorithms in the domains of image segmentation object recognition and document analysis

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

nonparametric bayes estimators based on beta processes in models for life history data the article studies the problem of finding bayes estimators for cumulative hazard rates and related quantities w r t prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments a particular class of prior processes termed beta processes is introduced and is shown to constitute a conjugate class an important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated for example time inhomogeneous markov chains we find posterior distributions and derive bayes estimators in such models and present a semiparametric bayesian analysis of the cox regression model

a conditional likelihood approach to residual maximum likelihood estimation in generalized linear models summary residual maximum likelihood reml estimation is often preferred to maximum likelihood estimation as a method of estimating covariance parameters in linear models because it takes account of the loss of degrees of freedom in estimating the mean and produces unbiased estimating equations for the variance parameters in this paper it is shown that reml has an exact conditional likelihood interpretation where the conditioning is on an appropriate sufficient statistic to remove dependence on the nuisance parameters this interpretation clarifies the motivation for reml and generalizes directly to non normal models in which there is a low dimensional sufficient statistic for the fitted values the conditional likelihood is shown to be well defined and to satisfy the properties of a likelihood function even though this is not generally true when conditioning on statistics which depend on parameters of interest using the conditional likelihood representation the concept of reml is extended to generalized linear models with varying dispersion and canonical link explicit calculation of the conditional likelihood is given for the one way lay out a saddlepoint approximation for the conditional likelihood is also derived

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

polynomiality of primal dual algorithms for semidefinite linear complementarity problems based on the kojima shindoh hara family of directions kojima shindoh and hara proposed a family of search directions for the semidefinite linear complementarity problem sdlcp and established polynomial convergence of a feasible short step path following algorithm based on a particular direction of their family the question of whether polynomiality could be established for any direction of their family thus remained an open problem this paper answers this question in the affirmative by establishing the polynomiality of primal dual interior point algorithms for sdlcp based on any direction of the kojima shindoh and hara family of search directions we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of sdlcp

primal dual affine scaling algorithms fail for semidefinite programming in this paper we give an example of a semidefinite programming problem in which primal dual affine scaling algorithms using the hrvw ksh m mt and aho directions fail we prove that each of these algorithms can generate a sequence converging to a non optimal solution and that for the aho direction even its associated continuous trajectory can converge to a non optimal point in contrast with these directions we show that the primal dual affine scaling algorithm using the nt direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution both primal and dual problems have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and are nondegenerate everywhere

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

on the eigenvalues of jordan products abstract this paper studies the possible eigenvalues of the jordan product xa ax when a is fixed and x varies

general interior point maps and existence of weighted paths for nonlinear semidefinite complementarity problems extending the previous work of monteiro and pang 1998 this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com plementarity problems over the cone of symmetric positive semidefinite matrices instead of focusing our attention on a specific map as was done in the approach of monteiro and pang 1998 this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of monteiro and pang 1998 to this general map each fundamental map leads to a family of weighted continuous trajectories which include the central trajectory as a special case hence for complementarity problems over the cone of symmetric positive semidefinite matrices the notion of weighted central path depends on the fundamental map used to represent the central path

a multigrid approach to sdp relaxations of sparse polynomial optimization problems we propose a multigrid approach for the global optimization of polynomial optimization problems with sparse support the problems we consider arise from the discretization of infinite dimensional optimization problems such as pde optimization problems boundary value problems and some global optimization applications in many of these applications the level of discretization can be used to obtain a hierarchy of optimization models that capture the underlying infinite dimensional problem at different degrees of fidelity this approach inspired by multigrid methods has been successfully used for decades to solve large systems of linear equations however multigrid methods are difficult to apply to semidefinite programming sdp relaxations of polynomial optimization problems the main difficulty is that the information between grids is lost when the original problem is approximated via an sdp relaxation despite the loss of information we develop a multigrid approach and propose prolongation operator

on the number of invariant factors of matrix products we prove an inequality relating the number of nontrivial invariant factors of n n matrices a and b with those of ab and get some results on the cases of equality in particular we characterize the similarity classes a and b with all eigenvalues in the base field such that ab is nilpotent for some a a and b b

priors for random count matrices derived from a family of negative binomial processes abstractwe define a family of probability distributions for random count matrices with a potentially unbounded number of rows and columns the three distributions we consider are derived from the gamma poisson gamma negative binomial and beta negative binomial processes which we refer to generically as a family of negative binomial processes because the models lead to closed form update equations within the context of a gibbs sampler they are natural candidates for nonparametric bayesian priors over count matrices a key aspect of our analysis is the recognition that although the random count matrices within the family are defined by a row wise construction their columns can be shown to be independent and identically distributed iid this fact is used to derive explicit formulas for drawing all the columns at once moreover by analyzing these matrices combinatorial structure we describe how to sequentially construct a column iid random count matrix one row at a time and derive the predictive d

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

medlda maximum margin supervised topic models a supervised topic model can use side information such as ratings or labels associated with documents or images to discover more predictive low dimensional topical representations of the data however existing supervised topic models predominantly employ likelihood driven objective functions for learning and inference leaving the popular and potentially powerful max margin principle unexploited for seeking predictive representations of data and more discriminative topic bases for the corpus in this paper we propose the maximum entropy discrimination latent dirichlet allocation medlda model which integrates the mechanism behind the max margin prediction models e g svms with the mechanism behind the hierarchical bayesian topic models e g lda under a unified constrained optimization framework and yields latent topical representations that are more discriminative and more suitable for prediction tasks such as document classification or regression the principle underlying the medlda formalism is quite general and can be applied for jointly max margin and maximum likelihood learning of directed or undirected topic models when supervising side information is available efficient variational methods for posterior inference and parameter estimation are derived and extensive empirical studies on several real data sets are also provided our experimental results demonstrate qualitatively and quantitatively that medlda could 1 discover sparse and highly discriminative topical representations 2 achieve state of the art prediction performance and 3 be more efficient than existing supervised topic models especially for classification

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

on the invariant polynomials of jordan products let f be an algebraically closed field this paper describes the possible numbers of nonconstant invariant polynomials of the jordan product xa ax when a is fixed and x varies

negative binomial process count and mixture modeling the seemingly disjoint problems of count and mixture modeling are united under the negative binomial nb process a gamma process is employed to model the rate measure of a poisson process whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an nb process for count modeling a draw from the nb process consists of a poisson distributed finite number of distinct atoms each of which is associated with a logarithmic distributed number of data samples we reveal relationships between various count and mixture modeling distributions and construct a poisson logarithmic bivariate distribution that connects the nb and chinese restaurant table distributions fundamental properties of the models are developed and we derive efficient bayesian inference it is shown that with augmentation and normalization the nb process and gamma nb process can be reduced to the dirichlet process and hierarchical dirichlet process respectively these relationships highlight theoretical structural and computational advantages of the nb process a variety of nb processes including the beta geometric beta nb marked beta nb marked gamma nb and zero inflated nb processes with distinct sharing mechanisms are also constructed these models are applied to topic modeling with connections made to existing algorithms under poisson factor analysis example results show the importance of inferring both the nb dispersion and probability parameters

primal dual interior point methods for semidefinite programming in finite precision recently a number of primal dual interior point methods for semidefinite programming have been developed to reduce the number of floating point operations each iteration of these methods typically performs block gaussian elimination with block pivots that are close to singular near the optimal solution as a result these methods often exhibit complex numerical properties in practice we consider numerical issues related to some of these methods our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well conditioned but are unstable otherwise with this result we explain why one particular method the one introduced by alizadeh haeberly and overton is in general more stable than others we also explain why the so called least squares variation introduced for some of these methods does not yield more numerical accuracy in general finally we present results from our numerical experiments to support our analysis

on the number of invariant polynomials of partially prescribed matrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 a 1 2 a 2 2 are prescribed

the number of invariant polynomials of a matrix with prescribed complementary principal submatrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 and a 2 2 are prescribed

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

frequency of frequencies distributions and size dependent exchangeable random partitions abstractmotivated by the fundamental problem of modeling the frequency of frequencies fof distribution this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions a cluster structure naturally arising from a completely random measure mixed poisson process allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size a distinct and motivated feature that makes it more flexible than a partition structure this allows it to model an entire fof distribution whose structural properties change as the population size varies a fof vector can be simulated by drawing an infinite number of poisson random variables or by a stick breaking construction with a finite random number of steps a generalized negative binomial process model is proposed to generate a cluster structure where in the prior the number of clusters is f

search directions in the sdp and the monotone sdlcp generalization and inexact computation we propose a family of directions that generalizes many directions proposed so far in feasible interior point methods for the sdp semidefinite programming and for the monotone sdlcp semidefinite linear complementarity problem we derive the family from the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction by relaxing its centrality equation into a centrality inequality using this family we present a predictor corrector infeasible interior point method to provide a theoretical basis for inexact computation of directions in primal dual interior point methods for the sdp

semidefinite programming in the space of partial positive semidefinite matrices we build upon the work of fukuda et al siam j optim 11 2001 pp 647 674 and nakata et al math program 95 2003 pp 303 327 in which the theory of partial positive semidefinite matrices was applied to the semidefinite programming sdp problem as a technique for exploiting sparsity in the data in contrast to their work which improved an existing algorithm based on a standard search direction we present a primal dual path following algorithm that is based on a new search direction which roughly speaking is defined completely within the space of partial symmetric matrices we show that the proposed algorithm computes a primal dual solution to the sdp problem having duality gap less than a fraction varepsilon 0 of the initial duality gap in cal o n log varepsilon 1 iterations where n is the size of the matrices involved moreover we present computational results showing that the algorithm possesses several advantages over other existing implementations

matrices with prescribed submatrices and number of invariant polynomials abstract we give necessary and sufficient conditions for the existence of a matrix a b over a field where a is a square matrix such that a b has a prescribed submatrix and the pencil xi a b has a prescribed number of invariant factors different from 1 as a corollary we give a necessary and sufficient condition for the existence of a completely controllable pair a b such that a b has a prescribed submatrix

quadratic convergence of a nonsmooth newton type method for semidefinite programs without strict complementarity we consider a newton type method for the solution of semidefinite programs this newton type method is based on a semismooth reformulation of the semidefinite program as a nonsmooth system of equations we establish local quadratic convergence of this method under a linear independence assumption and a weak nondegeneracy condition that implies uniqueness of the optimal solution but does not imply strict complementarity

bayesian nonparametric statistical inference for poisson point processes a random measure is said to be selected by a weighted gamma prior probability if the values it assigns to disjoint sets are independent gamma random variables with positive multipliers if the intensity measure of a nonhomogeneous poisson point process is selected by a weighted gamma prior probability and if a sample is drawn from the poisson point process having this intensity measure then the posterior random intensity measure given the observations is also selected by a weighted gamma prior probability if the measure space is euclidean and if the true intensity measure is continuous and finite the centered posterior process rescaled by the square root of the sample size will converge weakly in skorohod topology to a wiener process subject to a change of time scale

the number of kronecker indices of square pencils of a special kind abstract in this paper we describe the possible numbers of kronecker indices of the pencils xa b where a and b run over two prescribed similarity classes a and b respectively

a class of projective transformations for linear programming a class of projective transformations under which potential functions are invariant for linear programming is described as a result a new projective algorithm converging in o l sqrt n iterations is developed the algorithm does not require centering conditions and its convergence speed is improved by a factor sqrt n over karmarkar s projective algorithm and by a constant over the recent affine potential reduction algorithm

a feasible primaldual interior point method for linear semidefinite programming in this paper we consider a feasible primaldual interior point method for linear semidefinite programming problem sdp based on alizadehhaeberlyoverton aho direction monteiro 1997 firstly and by a new and simple technique we establish the existence and uniqueness of optimal solution of the perturbed problem sdp and its convergence to optimal solution of sdp next we present new different alternatives to calculate the displacement step after we establish the convergence of the obtained algorithm and we show that its complexity is o nln 1 x0 s0 finally we present some numerical simulations which show the effectiveness of the algorithm developed in this work

an interior point perspective on sensitivity analysis in semidefinite programming we study the asymptotic behavior of the interior point bounds arising from the work of yildirim and todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds we introduce a weaker notion of nondegeneracy and discuss its implications for perturbations of the right hand side vector or the cost matrix we show that the interior point bounds evaluated on the central path using the monteiro zhang family of search directions converge as the duality gap tends to zero to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions furthermore our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner we also show that the same convergence results carry over to iterates lying in an appropriate very narrow central path neighborhood if the nesterov todd direction is used to evaluate the interior point bounds we extend our results to the case of simultaneous perturbations of the right hand side vector and the cost matrix we also provide examples illustrating that our assumptions in general cannot be weakened

a predictor corrector interior point algorithm for the semidefinite linear complementarity problem using the alizadeh haeberly overton search direction this paper proposes a globally convergent predictor corrector infeasible interior point algorithm for the monotone semidefinite linear complementarity problem using the alizadeh haeberly overton search direction and shows its quadratic local convergence under the strict complementarity condition

on similarity invariants of matrix commutators abstract we study the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of a x 1 x 2 x k when a is a fixed matrix and x 1 x k vary then we generalize these results in the following way let g x 1 x k be any expression obtained from distinct noncommuting variables x 1 x k by applying recursively the lie product and without using the same variable twice we study the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of g x 1 x k when one of the variables x 1 x k takes a fixed value in f n n and the others vary

on the number of invariant polynomials of matrix commutators we study the possible numbers of noneonstant invariant polynomials of the matrix commutator xa ax when x varies elsevier science inc 1997

on the number of invariant polynomials of the product of matrices with prescribed similarity classes abstract we study the possibilities for the number of nontrivial invariant polynomials of the product of two nonsingular matrices with prescribed similarity classes over an algebraically closed field