distributional results for means of normalized random measures with independent increments we consider the problem of determining the distribution of means of random probability measures which are obtained by normalizing increasing additive processes a solution is found by resorting to a well known inversion formula for characteristic functions due to gurland moreover expressions of the posterior distributions of those means in the presence of exchangeable observations are given finally a section is devoted to the illustration of two examples of statistical relevance

superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming this paper establishes the superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming sdp under the assumptions that the semidefinite program has a strictly complementary primal dual optimal solution and that the size of the central path neighborhood tends to zero the interior point algorithm considered here closely resembles the mizuno todd ye predictor corrector method for linear programming which is known to be quadratically convergent it is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearly with order 2 1 2 r the proof relies on a careful analysis of the central path for sdp it is shown that under the strict complementarity assumption the primal dual central path converges to the analytic center of the primal dual optimal solution set and the distance from any point on the central path to this analytic center is bounded by the duality gap

normalized random measures driven by increasing additive processes this paper introduces and studies a new class of nonparametric prior distributions random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes in particular we present results for the distribution of means under both prior and posterior conditions and via the use of strategic latent variables undertake a full bayesian analysis our class of priors includes the well known and widely used mixture of a dirichlet process

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

bayesian poisson process partition calculus with an application to bayesian levy moving averages this article develops and describes how to use results concerning disintegrations of poisson random measures these results are fashioned as simple tools that can be tailor made to address inferential questions arising in a wide range of bayesian nonparametric and spatial statistical models the poisson disintegration method is based on the formal statement of two results concerning a laplace functional change of measure and a poisson palm fubini calculus in terms of random partitions of the integers 1 n the techniques are analogous to but much more general than techniques for the dirichlet process and weighted gamma process developed in ann statist 12 1984 351 3571 and ann inst statist math 41 1989 227 245 in order to illustrate the flexibility of the approach large classes of random probability measures and random hazards or intensities which can be expressed as functionals of poisson random measures are described we describe a unified posterior analysis of classes of discrete random probability which identifies and exploits features common to all these models the analysis circumvents many of the difficult issues involved in bayesian nonparametric calculus including a combinatorial component this allows one to focus on the unique features of each process which are characterized via real valued functions h the applicability of the technique is further illustrated by obtaining explicit posterior expressions for levy cox moving average processes within the general setting of multiplicative intensity models in addition novel computational procedures similar to efficient procedures developed for the dirichlet process are briefly discussed for these models

explicit feature control in structural topology optimization via level set method abstract the present paper aims to address a long standing and challenging problem in structural topology optimization explicit feature control of the optimal topology the basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

a strongly polynomial rounding procedure yielding a maximally complementary solution for p_ kappa linear complementarity problems we deal with linear complementarity problems lcps with p kappa matrices first we establish the convergence rate of the complementary variables along the central path the central path is parameterized by the barrier parameter mu as usual our elementary proof reproduces the known result that the variables on or close to the central path fall apart in three classes in which these variables are cal o 1 cal o mu and cal o sqrt mu respectively the constants hidden in these bounds are expressed in or bounded by the input data all this is preparation for our main result a strongly polynomial rounding procedure given a point with sufficiently small complementarity gap and which is close enough to the central path the rounding procedure produces a maximally complementary solution in at most cal o n 3 arithmetic operations the result implies that interior point methods ipms not only converge to a complementary solution of p kappa lcps but when furnished with our rounding procedure they can also produce a maximally complementary exact solution in polynomial time

hierarchical mixture modeling with normalized inverse gaussian priors in recent years the dirichlet process prior has experienced a great success in the context of bayesian mixture modeling the idea of overcoming discreteness of its realizations by exploiting it in hierarchical models combined with the development of suitable sampling techniques represent one of the reasons of its popularity in this article we propose the normalized inverse gaussian n ig process as an alternative to the dirichlet process to be used in bayesian hierarchical models the n ig prior is constructed via its finite dimensional distributions this prior although sharing the discreteness property of the dirichlet prior is characterized by a more elaborate and sensible clustering which makes use of all the information contained in the data whereas in the dirichlet case the mass assigned to each observation depends solely on the number of times that it occurred for the n ig prior the weight of a single observation depends heavily on the whole number of ties in the sample moreover expressio

analytical properties of the central path at boundary point in linear programming we study the properties of the weighted central paths in linear programming we consider each path as the function of the parameter 0 where the value at d 0 corresponds to the limit point at the boundary of the feasible set we calculate the recursive formulas for the central path derivatives of all orders valid at each 0 we establish the geometric growth of the derivatives and consequently the analyticity of the weighted central path at 0 this paper also provides the analysis of limiting behavior of those projection operators which often appear in the interior point methods the central trajectory is one of the most important concepts in the interior point methods for linear programming the central trajectory or the central path is a distinguished curve that is interior to the feasible set and tends to an optimal point most interior points algorithms follow or are related to the central trajectory although the central path concept is essential for the design and analysis of algorithms relatively few papers study properties of this curve itself for a review on this topic we refer to 3 recently zhao and zhu 16 have studied some analytical properties of the central trajectory in the interior of the feasible set they have established an upper bound for the high order derivatives of the weighted central path and estimated the convergence radius of the corresponding taylor series they have also pointed out the relationship of these properties with the complexity of a path following algorithm where the number of iterations is estimated by a curvature integral see also 15 further guler 3 has analysed some limiting properties of the central trajectory derivatives with respect to the parameter 0 he has established the existence of finite limits for derivatives of all orders as tends to infinity that is the central path stays in the interior and to zero the central path tends to the boundary these results have applications to the construction of polynomial time algorithms which are also locally fast a surprising result of this study is that all derivatives of order 1t end to 0 as 1 which helps to explain why interior point methods for linear programming require so few iterations see 3

on the identification of the optimal partition of second order cone optimization problems this paper discusses the identification of the optimal partition of second order cone optimization soco by giving some condition numbers which only depend on the soco problem itself we derive some bounds on the magnitude of the blocks of variables along the central path and prove that the optimal partition mathcal b mathcal n mathcal r and mathcal t for soco problems can be identified along the central path when the barrier parameter mu is small enough then we generalize the results to a specific neighborhood of the central path

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

means of a dirichlet process and multiple hypergeometric functions the lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a dirichlet process this approach leads to several results which are illustrated here among these are a new and more direct procedure for determining the exact form of the distribution of the mean a correspondence between the distribution of the mean and the parameter of a dirichlet process a characterization of the family of cauchy distributions as the set of the fixed points of this correspondence and an extension of the markov krein identity moreover an expression of the characteristic function of the mean of a dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth lauricella function this expression is then employed to prove that the distribution of the mean of a dirichlet process is symmetric if and only if the parameter of the process is symmetric and to provide a new expression of the moment generating function of the variance of a dirichlet process

adaptive multiscale wavelet galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization the multiscale galerkin formulation of two dimensional elasticity problems is presented for easy interpolation and boundary handlings as well as efficient adaptive analysis two dimensional interpolation wavelets are used as the multiscale trial functions in the galerkin formulation after the validity of the present multiscale adaptive method is verified with some benchmark problems the present wavelet based method is applied to the multiscale topology optimization that progresses design resolution levels dyadically from low to high levels by this application we show the potential of the multiscale method and the possibility of developing a fully integrated analysis and topology design optimization in the multiscale multiresolution setting

on weighted centers for semidefinite programming abstract in this paper we generalize the notion of weighted centers to semidefinite programming our analysis fits in the v space framework which is purely based on the symmetric primal dual transformation and does not make use of barriers existence and scale invariance properties are proven for the weighted centers relations with other primal dual maps are discussed

limiting behavior of the affine scaling continuous trajectories for linear programming problems we consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form by characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems we show that these trajectories tend to an optimal solution which in general depends on the starting point by considering the trajectories that arise from the lagrangian multipliers of the above mentioned logarithmic barrier families of problems we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem we also present results related to asymptotic direction of the affine scaling trajectories we briefly discuss how to apply our results to linear programs formulated in formats different from the standard form finally we extend the results to the primal dual amne scaling algorithm

perturbation analysis of second order cone programming problems we discuss first and second order optimality conditions for nonlinear second order cone programming problems and their relation with semidefinite programming problems for doing this we extend in an abstract setting the notion of optimal partition then we state a characterization of strong regularity in terms of second order optimality conditions this is the first time such a characterization is given for a nonpolyhedral conic problem

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

on consistency of nonparametric normal mixtures for bayesian density estimation the past decade has seen a remarkable development in the area of bayesian nonparametric inference from both theoretical and applied perspectives as for the latter the celebrated dirichlet process has been successfully exploited within bayesian mixture models leading to many interesting applications as for the former some new discrete nonparametric priors have been recently proposed in the literature that have natural use as alternatives to the dirichlet process in a bayesian hierarchical model for density estimation when using such models for concrete applications an investigation of their statistical properties is mandatory of these properties a prominent role is to be assigned to consistency indeed strong consistency of bayesian nonparametric procedures for density estimation has been the focus of a considerable amount of research in particular much attention has been devoted to the normal mixture of dirichlet process in this article we improve on previous contributions by establishing str

on the analyticity properties of infeasible interior point paths for monotone linear complementarity problems infeasible interior point paths x r eta y r eta r 0 0 0 y 0 x iy i r eta i 1 le i le n end array leqno lcp r if the path z r eta x r eta y r eta converges for r downarrow 0 then it converges to a solution of lcp 0 this paper deals with the analyticity properties of z r eta and its derivatives with respect to r near r 0 for solvable monotone complementarity problems lcp 0 it is shown for lcp 0 with a strictly complementary solution that the path z r eta r downarrow0 has an extension to r 0 which is analytic also at r 0 if lcp 0 has no strictly complementary solution then hat z rho eta z rho 2 eta rho sqrt r has an extension to rho 0 that is analytic at rho 0

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

log linear bayesian additive regression trees for multinomial logistic and count regression models we introduce bayesian additive regression trees bart for log linear models including multinomial logistic regression and count regression with zero inflation and overdispersion bart has been app

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

analyticity of the central path at the boundary point in semidefinite programming abstract in this paper we study the limiting behavior of the central path for semidefinite programming sdp we show that the central path is an analytic function of the barrier parameter even at the limit point provided that the semidefinite program has a strictly complementary solution a consequence of this property is that the derivatives of any order of the central path have finite limits as the barrier parameter goes to zero

distribution on warp maps for alignment of open and closed curves alignment of curve data is an integral part of their statistical analysis and can be achieved using model or optimization based approaches the parameter space is usually the set of monotone con

exponential functionals and means of neutral to the right priors summary the mean of a random distribution chosen from a neutral to the right prior can be represented as the exponential functional of an increasing additive process this fact is exploited in order to give sufficient conditions for the existence of the mean of a neutralto the right prior and for the absolute continuity of its probability distribution moreover expressions for its moments of any order are provided for illustrative purposes we consider a generalisation of the neutral to the right prior based on the gamma process and the beta stacy process finally by resorting to the maximum entropy algorithm we obtain an approximation to the probability density function of the mean of a neutral tothe right prior the arguments are easily extended to examine means of posterior quantities the numerical results obtained are compared to those yielded by the application of some well established simulation algorithms

topology optimization in wavelet space we investigate the use of wavelet methods for the design description in topology optimization this is a new setting for topology optimization that does not take the material densities in the elements of the finite element model as the design space this new approach has advantages and limitations which are discussed it is shown how the control of wavelet coefficients can be used to prevent the formation of checkerboard patterns and to obtain some geometry control problems related to overlapping basis functions are analysed and solution methods are proposed examples of optimized designs using the described methods are given copyright 2001 john wiley sons ltd

limiting behavior of weighted central paths in linear programming we study the limiting behavior of the weighted central paths x s 0 in linear programming at both 0 and we establish the existence of a partition b n of the index set 1 n such thatx i ands j as fori b andj n andx n s b converge to weighted analytic centers of certain polytopes for allk 1 we show that thekth order derivativesx k ands k converge when 0 and consequently the derivatives of each order are bounded in the interval 0 we calculate the limiting derivatives explicitly and establish the surprising result that all higher order derivatives k 2 converge to zero when

multi resolution multi scale topology optimization a new paradigm abstract the purpose of this work is to present a new concept multi resolution multi scale topology optimization the key idea of the present strategy is that design optimization should be performed progressively from low to high resolution not at a single resolution level to achieve the multi resolution strategy design optimization is formulated in a wavelet based variable space not in a direct density variable space the major advantages of the multi resolution design optimization include 1 topologically simple and close to the global optimum structures may be obtained without any explicit constraint and 2 the convergence is not sensitive to mathematical programming methods for the efficient numerical implementation of the multi resolution approach the side constraints imposed on the direct density variables are removed by mapping the density variables into intermediate variables these intermediate variables are then wavelet transformed to new design variables it is addressed that the present multi resolution topology optimization can resolve major numerical instability problems such as mesh dependencies and local minima the usefulness of the multi scale nature of the wavelets in the present multi resolution multi scale optimization formulation is also discussed

error bounds for linear matrix inequalities for iterative sequences that converge to the solution set of a linear matrix inequality we show that the distance of the iterates to the solution set is at most o epsilon 2 d the nonnegative integer d is the so called degree of singularity of the linear matrix inequality and epsilon denotes the amount of constraint violation in the iterate for infeasible linear matrix inequalities we show that the minimal norm of epsilon approximate primal solutions is at least 1 o epsilon 1 2 d 1 and the minimal norm of epsilon approximate farkas type dual solutions is at most o 1 epsilon 2 d 1 as an application of these error bounds we show that for any bounded sequence of epsilon approximate solutions to a semidefinite programming problem the distance to the optimal solution set is at most o epsilon 2 k where k is the degree of singularity of the optimal solution set