a composite likelihood approach to binary spatial data abstract conventional geostatistics addresses the problem of estimation and prediction for continuous observations but in many practical applications in public health environmental remediation or ecological research the most commonly available data are in the form of counts e g number of cases or indicator variables denoting above or below threshold values also in many situations it is less expensive to obtain an imprecise categorical observation than to obtain precise measurements of the variable of interest such as a contaminant this article proposes a computationally simple method for estimation and prediction using binary or indicator data in space the proposed method is based on pairwise likelihood contributions and the large sample properties of the estimators are obtained in a straightforward manner we illustrate the methodology through application to indicator data related to gypsy moth defoliation in massachusetts

multivariate product shot noise cox point process models we introduce a new multivariate product shot noise cox process which is useful for modeling multi species spatial point patterns with clustering intra specific interactions and neutral negative or positive inter specific interactions the auto and cross pair correlation functions of the process can be obtained in closed analytical forms and approximate simulation of the process is straightforward we use the proposed process to model interactions within and among five tree species in the barro colorado island plot

a nonparametric test for spatial isotropy using subsampling a common requirement for spatial modeling is the development of an appropriate correlation structure although the assumption of isotropy is often made for this structure it is not always appropriate a conventional practice when checking for isotropy is to informally assess plots of direction specific sample semi variograms although a useful diagnostic these graphical techniques are difficult to assess and open to interpretation formal alternatives to graphical diagnostics are valuable but have been applied to a limited class of models in this article we propose a formal approach to test for isotropy that is both objective and valid for a wide class of models this approach which is based on the asymptotic joint normality of the sample variogram can be used to compare sample variograms in multiple directions an l2 consistent subsampling estimator for the asymptotic covariance matrix of the sample variogram is derived and used to construct a test statistic a subsampling approach and a limiting

a combined estimating function approach for fitting stationary point process models a composite likelihood technique based on pairwise contributions provides a computationally simple but potentially inefficient approach for fitting spatial point process models we propose a new estimation procedure that improves the efficiency our approach combines estimating functions derived from pairwise composite likelihood estimation and estimating functions that account for correlations among the pairwise contributions our method can be used to fit a variety of parametric spatial point process models and can yield more efficient estimators for the clustering parameters than pairwise composite likelihood estimation we demonstrate the efficacy of our proposed method through a simulation study and an application to the longleaf pine data

superlinear convergence of an infeasible predictor corrector path following interior point algorithm for a semidefinite linear complementarity problem using the helmberg kojima monteiro direction an interior point method ipm defines a search direction at each interior point of a region these search directions form a direction field which in turn gives rise to a system of ordinary differential equations odes the solutions of the system of odes can be viewed as underlying paths in the interior of the region in c k sim and g zhao math program ser a 110 2007 pp 475 499 these off central paths are shown to be well defined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem sdlcp the study of these paths provides a way to understand how iterates generated by an interior point algorithm behave in this paper we give a sufficient condition using these off central paths that guarantees superlinear convergence of a predictor corrector path following interior point algorithm for sdlcp using the helmberg kojima monteiro hkm direction this sufficient condition is implied by a currently known sufficient condition for superlinear convergence using this sufficient condition we show that for any linear semidefinite feasibility problem superlinear convergence using the interior point algorithm with the hkm direction can be achieved for a suitable starting point we work under the assumption of strict complementarity

improved complexity using higher order correctors for primal dual dikin affine scaling in this paper we show that the primal dual dikin affine scaling algorithm for linear programming of jansen roos and terlaky enhances an asymptotical o sqrt n l complexity by using corrector steps we also show that the result remains valid when the method is applied to positive semi definite linear complementarity problems

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

pairwise likelihood methods for inference in image models summary we consider parameter estimation for certain random set models for image data for the spatial models considered here a full likelihood based approach to this problem is often difficult since the likelihood generally cannot be computed for images of typical size this has motivated the consideration of various less computationally demanding parameter estimation techniques we discuss one method based on likelihoods for small subsets of the data consideration of the basic technique within the framework of the theory of estimating functions leads to a weighted version of the basic approach results of a simulation study are reported for some variants of the boolean model and excursion sets of random fields the methods are illustrated for some data on the spatial incidence of heather

an estimating function approach to inference for inhomogeneous neyman scott processes this article is concerned with inference for a certain class of inhomogeneous neyman scott point processes depending on spatial covariates regression parameter estimates obtained from a simple estimating function are shown to be asymptotically normal when the mother intensity for the neyman scott process tends to infinity clustering parameter estimates are obtained using minimum contrast estimation based on the k function the approach is motivated and illustrated by applications to point pattern data from a tropical rain forest plot

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

on the choice of parameters for power series interior point algorithms in linear programming in this paper we study higher order interior point algorithms especially power series algorithms for solving linear programming problems since higher order differentials are not parameter invariant it is important to choose a suitable parameter for a power series algorithm we propose a parameter transformation to obtain a good choice of parameter called ak parameter for general truncated powerseries approximations we give a method to find ak parameter this method is applied to two powerseries interior point algorithms which are built on a primal dual algorithm and a dual algorithm respectively computational results indicate that these higher order power series algorithms accelerate convergence compared to first order algorithms by reducing the number of iterations also they demonstrate the efficiency of thek parameter transformation to amend an unsuitable parameter in power series algorithms

free nilpotent and nilpotent quadratic lie algebras abstract in this paper we introduce an equivalence between the category of the t nilpotent quadratic lie algebras with d generators and the category of some symmetric invariant bilinear forms over the t nilpotent free lie algebra with d generators taking into account this equivalence t nilpotent quadratic lie algebras with d generators are classified up to isometric isomorphisms and over any field of characteristic zero in the following cases d 2 and t 5 d 3 and t 3

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

logarithmic sumt limits in convex programming the limits of a class of primal and dual solution trajectories associated with the sequential unconstrained minimization technique sumt are investigated for convex programming problems with non unique optima logarithmic barrier terms are assumed for linear programming problems such limits of both primal and dual trajectories are strongly optimal strictly complementary and can be characterized as analytic centers of loosely speaking optimality regions examples are given which show that those results do not hold in general for convex programming problems if the latter are weakly analytic bank et al 3 primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability that class of programming problems contains faithfully convex linear and convex quadratic programming problems as strict subsets in the differential case dual trajectory limits can be characterized similarly albeit under different conditions one of which suffices for strict complementarity

underlying paths in interior point methods for the monotone semidefinite linear complementarity problem an interior point method defines a search direction at each interior point of the feasible region the search directions at all interior points together form a direction field which gives rise to a system of ordinary differential equations odes given an initial point in the interior of the feasible region the unique solution of the ode system is a curve passing through the point with tangents parallel to the search directions along the curve we call such curves off central paths we study off central paths for the monotone semidefinite linear complementarity problem sdlcp we show that each off central path is a well defined analytic curve with parameter ranging over 0 and any accumulation point of the off central path is a solution to sdlcp through a simple example we show that the off central paths are not analytic as a function of sqrt mu and have first derivatives which are unbounded as a function of at 0 in general on the other hand for the same example we can find a subset of off central paths which are analytic at 0 these nice paths are characterized by some algebraic equations

examples of ill behaved central paths in convex optimization this paper presents some examples of ill behaved central paths in convex optimization some contain infinitely many fixed length central segments others manifest oscillations with infinite variation these central paths can be encountered even for infinitely differentiable data

linear programming and the newton barrier flow in this note we report a simple characteristic of linear programming central trajectories which has a surprising consequence specifically we show that given a bounded polyhedral setp with nonempty interior the logarithmic barrier function with no objective component induces a vector field of negative newton directions which flows from the center ofp along central trajectories to solutions of every possible linear program onp

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

karmarkar s linear programming algorithm and newton s method this paper describes a full dimensional version of karmarkar s linear programming algorithm theprojective scaling algorithm which is defined for any linear program in n having a bounded full dimensional polytope of feasible solutions if such a linear program hasm inequality constraints then it is equivalent under an injective affine mappingj n m to karmarkar s original algorithm for a linear program in m havingm n equality constraints andm inequality constraints karmarkar s original algorithm minimizes a potential functiong x and the projective scaling algorithm is equivalent to that version of karmarkar s algorithm whose step size minimizes the potential function in the step direction

optimizing over three dimensional subspaces in an interior point method for linear programming abstract interior point algorithms for solving linear programming problems are considered a three dimensional method is developed that at each iteration solves a subproblem based on minimizing the cost function on low dimensional cross sections of the feasible region the generators for the three dimensional subproblem include the dual affine search direction and two higher order search directions one of the higher order directions is a third order correction to the newton recentering direction and the other is a correction to the dual affine direction that is motivated by the use of rank one updates of the second derivative information numerical results are presented for this method that indicate a nearly 20 reduction in cpu time compared to our best dual affine implementation

a weighted estimating equation approach for inhomogeneous spatial point processes we introduce a new estimation method for parametric intensity function models of inhomogeneous spatial point processes based on weighted estimating equations the weights can incorporate information on both inhomogeneity and dependence of the process simulations show that significant efficiency gains can be achieved for non poisson processes compared to the poisson maximum likelihood estimator an application to tropical forest data illustrates the use of the proposed method

a survey of the implications of the behavior of the central path for the duality theory of linear programming the literature in the field of interior point methods for linear programming has been almost exclusively algorithmic oriented very few contributions have been made towards the theory of linear programming itself in particular none of them offer a simple self contained introduction to the theory of linear programming and linear inequalities the purpose of this paper is to show that the interior point methodology can be used to introduce the field of linear programming starting from scratch and using only elementary results from calculus and linear algebra we prove that for every value of the barrier parameter the logarithmic barrier function for the primal dual problem has a unique minimizer and that the path of these minimizers the central path converges to a strictly complementary pair of optimal solutions these results were proved more than a decade ago with advanced mathematical arguments our proofs are new they are also simpler and often more natural than the ones currently known they p

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

minimum seeking properties of analog neural networks with multilinear objective functions in this paper we study the problem of minimizing a multilinear objective function over the discrete set

solving combinatorial optimization problems using karmakar s algorithm we describe a cutting plane algorithm for solving combinatorial optimization problems the primal projective standard form variant of karmarkar s algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem

global convergence property of the affine scaling methods for primal degenerate linear programming problems in this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy the behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence for dual nondegenerate lp problems the result can be regarded as a counterpart to dikin s global convergence result on the affine scaling method assuming primal nondegeneracy

on the continuous trajectories for a potential reduction algorithm for linear programming for a possibly degenerate linear program minx ctx ax b x 0 a is an m n real matrix b rm and c rn whose optimal value is 0 we study the limiting behavior of the trajectories of the family of vector fields i qx i xpax xc q i1ctx1 for all values of q n where x is the diagonal matrix associated with x and pax is the projection operator onto the null space of ax a polynomial algorithm based on the directions i qx has been presented by gonzaga 6 when q n or q n n we show that all trajectories of i qx converge to the unique center of the optimal face of the given linear program when this face consists of a unique vertex it is shown that any trajectory of i qx approaches this vertex along the same direction when the optimal face consists of more than one point we show that there is a threshold value i 0 such that for q i most of the trajectories of i qx converge to the center tangentially to the optimal face and that the direction of approach of a trajectory of i qx depends on the initial condition for q i q i the trajectories of i qx converge to the center along a unique direction along several directions which depend on the initial condition forming a positive angle with the optimal face

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

containing and shrinking ellipsoids in the path following algorithm we describe a new potential function and a sequence of ellipsoids in the path following algorithm for convex quadratic programming each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors furthermore the volumes of the ellipsoids shrink at the ratio 2 omega sqrt n in comparison to 2 1 in karmarkar s algorithm and 2 1 n in the ellipsoid method we also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming

affine scaling for linear programs with free variables the affine scaling modification of karmarkar s algorithm is extended to solve problems with free variables this extended primal algorithm is used to prove two important results first the geometrically elegant feasibility algorithm proposed by chandru and kochar is the same algorithm as the one obtained by appending a single column of residuals to the constraint matrix second the dual algorithm as first described by adler et al is the same as the extended primal algorithm applied to the dual

aberrant crypt foci and semiparametric modeling of correlated binary data motivated by the spatial modeling of aberrant crypt foci acf in colon carcinogenesis we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent gaussian spatial process that accounts for short range dependencies the mean is modeled in a general way using regression splines the mean function can be viewed as a fixed effect and is estimated with a penalty for regularization with the latent process viewed as another random effect the model becomes a generalized linear mixed model in our motivating data set and other applications the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation reml so pairwise likelihood a special case of composite likelihood is used instead we develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications including but not limited to the problem that motivated this work the splines have penalty parameters that must converge to zero asymptotically we derive theory for this along with a data driven method for selecting the penalty parameter a method that is shown in simulations to improve greatly upon standard devices such as likelihood crossvalidation finally we apply the methods to the data from our experiment acf we discover an unexpected location for peak formation of acf

moment estimation for statistics from marked point processes in spatial statistics the data typically consist of measurements of some quantity at irregularly scattered locations in other words the data form a realization of a marked point process in this paper we formulate subsampling estimators of the moments of general statistics computed from marked point process data and we establish their l 2 consistency the variance estimator in particular can be used for the construction of confidence intervals for estimated parameters a practical data based method for choosing a subsampling parameter is given and illustrated on a data set finite sample simulation examples are also presented

barrier operators and associated gradient like dynamical systems for constrained minimization problems we study some continuous dynamical systems associated with constrained optimization problems for that purpose we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient like dynamical systems called a driven descent method a dm prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method continuous gradient projection methods and newton type methods as well as continuous interior descent methods such as lotka volterra type differential equations and riemannian gradient methods related discrete iterative methods such as proximal interior point algorithms based on bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics we prove global existence and strong viability results of the corresponding trajectories of a dm for a smooth objective function when the objective function is convex we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems a dm in particular we derive a general criterion ensuring the global convergence of the trajectory of a dm to a minimizer of a convex function over a closed convex set this result is then applied to several dynamics built upon specific elliptic barrier operators throughout the paper our results are illustrated with many examples

a polynomial arc search interior point algorithm for convex quadratic programming abstract arc search is developed for linear programming in 24 25 the algorithms search for optimizers along an ellipse that is an approximation of the central path in this paper the arc search method is applied to primal dual path following interior point method for convex quadratic programming a simple algorithm with iteration complexity o n log 1 is devised several improvements on the simple algorithm which improve computational efficiency increase step length and further reduce duality gap in every iteration are then proposed and implemented it is intuitively clear that the iteration with these improvements will reduce the duality gap more than the iteration of the simple algorithm without the improvements though it is hard to show how much these improvements reduce the complexity bound the proposed algorithm is implemented in matlab and tested on quadratic programming problems originating from 13 the result is compared to the one obtained by loqo in 22 the proposed algorithm uses fewer iterations in all these problems and the number of total iterations is 27 fewer than the one obtained by loqo this preliminary result shows that the proposed algorithm is promising

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

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

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations we present an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations wherem is the number of constraints andn is the number of variables each operation is performed to a precision of o l bits l is bounded by the number of bits in the input the worst case running time of the algorithm is better than that of karmarkar s algorithm by a factor of sqrt m n

the nonlinear geometry of linear programming iii projective legendre transform coordinates and hilbert geometry this paper studies projective scaling trajectories which are the tra jectories obtained by following the infinitesimal version of karmarkar s linear programming algorithm a nonlinear change of variables projective legendre transform coordinates is introduced to study these trajectories the projective legendre transform mapping has a coordinate free geometric interpretation in terms of the notion of centering by a projective transformation let h be a set of linear programming constraints aj x bj 1 s j s m on rn such that its polytope of feasible solutions ph is bounded and contains 0 in its interior the projective legendre transform mapping i h is given by m a where ph x l b j 1 aj x i x ph x h m ph x x here ph x is the legendre transform coordinate mapping introduced in part ii l h x is a one to one and onto mapping of the interior of the feasible solu tion polytope int ph to the interior of its polar polytope int p the set of projective scaling trajectories with objective function c x co are mappcd under l h to the set of straight line segments in int p passing through the boundary point clco of p as a consequence the projective scaling tra jectories for all objective functions can be interpreted as the complete set of geodesics actually distinguished chords of a projectively invariant metric ge ometry on int ph which is isometric to hilbert geometry on the interior of the polar polytope p

long step strategies in interior point primal dual methods in this paper we analyze from a unique point of view the behavior of path following and primal dual potential reduction methods on nonlinear conic problems we demonstrate that most interior point methods with o root n ln 1 epsilon efficiency estimate can be considered as different strategies of minimizing a convex primal dual potential function in an extended primal dual space their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path it is shown that the neighborhood of this path is a region of the fastest decrease of the potential therefore the long step path following methods are in a sense the best potential reduction strategies we present three examples of such long step strategies we prove also an efficiency estimate for a pure primal dual potential reduction method which can be considered as an implementation of a penalty strategy based on a functional proximity measure using the convex primal dual potential we prove efficiency estimates for karmarkar type and dikin type methods as applied to a homogeneous reformulation of the initial primal dual problem

a new vector field method for eigen decomposition of symmetric matrices abstract we study a new approach to eigen decomposition of symmetric matrices called the vector field method in contrast to the methods currently in use the vector field method computes an eigenvector by computing a discrete approximation to the integral curve of a special tangent vector field on the unit sphere the optimization problems embedded in each iteration of the vector field algorithms admit closed form solutions making the vector field approach promisingly efficient besides establishing the general convergence results about the vector field method and the complexity bounds for computing approximations to eigenvectors we also computationally compare the performance of a family of algorithms called the recursive vector field algorithms with that of the power method the vector field method like the power method offers a significant advantage in that it can exploit sparseness to speed up computation even further the convergence analysis that we present also suggests that one could speed up the vector field algorithms through preprocessing that contracts the spectral radius of the given matrix

feasible region contraction interior point algorithm abstract in this paper we propose a new interior point algorithm for linear programming by combining the affine scaling algorithm and the primal path following algorithm we define a series of subproblems whose feasible region contracts toward the set of optimal solutions in each iteration we take a step in the steepest descent direction in a transformed space to reduce the objective value our preliminary computational results are very favorable

gradient projection network analog solver for linearly constrained nonlinear programming an analog approach is presented for solving nonlinear programming problems with linear constraint conditions the present method is based on transformation of variables with exponential functions which enables every trajectory to pass through an interior of feasible regions along a gradient direction projected onto the feasible space convergence of its trajectory to the solution of optimization problems is guaranteed and it is shown that the present scheme is an extension of the affine scaling method for linear programming to nonlinear programs under a slight modification of riemannian metric an analog electronic circuit is also presented that implements the proposed scheme in real time

a class of primal affine scaling algorithms abstract we obtain a class of primal affine scaling algorithms which generalize some known algorithms this class depending on a r parameter is constructed through a family of metrics generated by r power r 1 of the diagonal iterate vector matrix we prove the so called weak convergence of the primal class for nondegenerate linearly constrained convex programming we observe the computational performance of the class of primal affine scaling algorithms accomplishing tests with linear programs from the netlib library and with some quadratic programming problems described in the maros and meszaros repository

a note on pseudolikelihood constructed from marginal densities for likelihood based inference involving distributions in which high dimensional dependencies are present it may be useful to use approximate likelihoods based for example on the univariate or bivariate marginal distributions the asymptotic properties of formal maximum likelihood estimators in such cases are outlined in particular applications in which only a single q x i vector of observations is observed are examined conditions under which consistent estimators of parameters result from the approximate likelihood using only pairwise joint distributions are studied some examples are analysed in detail

models for free nilpotent lie algebras if a lie algebra g can be generated by m of its elements e1 em and if any other lie algebra generated by m other elements f1 fm is a homomorphic image of g under the map ei fi we say that it is the free lie algebra on m generators the free nilpotent lie algebra gm r on m generators of rank r is the quotient of the free lie algebra by the ideal gr 1 generated as follows g1 g and gk gk 1 g let n denote dimension of gm r free lie algebras are those which have as few relations as possible only those which are a conseqence of the anti commutativity of the bracket and of the the jacobi identity free nilpotent lie algebras add the relations that any iterated lie bracket of more than r elements vanishes for further details see 18 or 13 in this paper we are interested in explicit computations of the lie algebras gm r it is well known 19 that that there is a representation of gm r on upper triangular n by n matrices the problem is that many computations are difficult using this representation in this paper we present an algorithm that yields vector fields e1 em defined in r with the property that they generate a lie algebra isomorphic to gm r see 6 for another approach to this problem in section 3 we restrict to two generators and give three consequences of this algorithm first the form of the vector fields is such that flows of the control system t e1 u t e2 x t may be computed explicitly research supported in part by postdoctoral research fellowships from the national science foundation

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

on the voronoi estimator for the intensity of an inhomogeneous planar poisson process summary the voronoi estimator may be defined for any location as the inverse of the area of the corresponding voronoi cell we investigate the statistical properties of this estimator for the intensity of an inhomogeneous poisson process and demonstrate it is approximately unbiased with a gamma sampling distribution we also introduce the centroidal voronoi estimator a simple extension based on spatial regularization of the point pattern simulations show the voronoi estimator has remarkably low bias while the centroidal voronoi estimator has slightly more bias but much less variability the performance is compared to kernel estimators using two simulated data sets and a dataset consisting of earthquakes within the continental united states

learning in games via reinforcement and regularization we investigate a class of reinforcement learning dynamics where players adjust their strategies based on their actions cumulative payoffs over time specifically by playing mixed strategies that maximize their expected cumulative payoff minus a regularization term a widely studied example is exponential reinforcement learning a process induced by an entropic regularization term which leads mixed strategies to evolve according to the replicator dynamics however in contrast to the class of regularization functions used to define smooth best responses in models of stochastic fictitious play the functions used in this paper need not be infinitely steep at the boundary of the simplex in fact dropping this requirement gives rise to an important dichotomy between steep and nonsteep cases in this general framework we extend several properties of exponential learning including the elimination of dominated strategies the asymptotic stability of strict nash equilibria and the convergence of time averaged trajectories in zero sum games with an interior nash equilibrium

global convergence of the affine scaling methods for degenerate linear programming problems in this paper we show the global convergence of the affine scaling methods without assuming any condition on degeneracy the behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence of the affine scaling methods

asymptotic behavior of the central path for a special class of degenerate sdp problems this paper studies the asymptotic behavior of the central path x s y as 0 for a class of degenerate semidefinite programming sdp problems namely those that do not have strictly complementary primal dual optimal solutions and whose degenerate diagonal blocks of the central path are assumed to satisfy we establish the convergence of the central path towards a primal dual optimal solution which is characterized as being the unique optimal solution of a certain log barrier problem a characterization of the class of sdp problems which satisfy our assumptions are also provided it is shown that the re parametrization t 0 x t4 s t4 y t4 of the central path is analytic at t 0 the limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is finally we apply our results to the convex quadratically constrained convex programming cqccp problem and characterize the class of cqccp problems which can be formulated as sdps satisfying the assumptions of this paper in particular we show that cqccp problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

symmetric invariant non degenerate bilinear form on a lie algebra the killing form is a powerful tool in the study of semi simple lie algebras unfortunately it is non degenerate only in the semi simple case therefore one has to look more generally for a form which has all the useful properties of the killing form bilinearity symmetry invariance nondegeneracy such a form will be a scalar multiple of the killing form in the central simple case but in the general case it may not even exist just take the solvable lie algebra of dimension 2 in this work we study what we call regular quadratic lie algebra g cp i e lie algebra g equipped with a symmetric invariant non degenerate bilinear form cp if d is a derivation of g belonging to the orthogonal lie algebra o cp then following the ideas of v g kac we define a regular quadratic lie algebra of dimension 2 dim g denoted g pi which we call the extension of g cp by d proposition 2 3 the interesting point of this extension is the fact that any indecomposable regular quadratic lie algebra with nontrivial center is of the form g pi proposition 2 9 which allows an inductive classification moreover a regular quadratic solvable lie algebra has a non zero center lemma 2 8 therefore the solvable case is completely covered proposition 2 11 classified all extensions of a given g cp

window subsampling of estimating functions with application to regression models abstract we propose a subsampling method for estimating the asymptotic standard error of a statistic n that is the solution to an estimating equation 1 n n j 1 uj y j xj 0 where the data yj may be temporally or spatially correlated and the estimating function may depend on covariates xj a key statistic that we consider in detail is a generalized linear model regression coefficient computed under the assumption of independence the availability of a consistent variance estimator allows semiparametric regression approaches for clustered and longitudinal data to be used with time series and spatial data the methods that we develop extend the subsampling ideas of carlstein sherman and garcia soidan and hall to estimating functions our approach provides an attractive alternative to the jackknife method of lele particularly for large datasets because we do not require parameter reestimation

hamiltonian structure of dynamical systems which solve linear programming problems abstract a new class of completely integrable hamiltonian systems which solve linear programming problems is introduced the relations with reduced toda flows double bracket equations interior point methods are established an interpretation in terms of total actions on kahler manifolds is given a complete description of the phase portrait including the explicit description of action angle variables is provided

error bounds and limiting behavior of weighted paths associated with the sdp map x 1 2 sx 1 2 this paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming sdp obtained from centrality equations of the form x1 2sx1 2 nu w where w is a fixed positive definite matrix and nu 0 is a parameter under the assumption that the problem has a strictly complementary primal dual optimal solution it is shown that a weighted central path as a function of sqrt nu can be extended analytically beyond 0 and hence that the path converges as nu downarrow 0 characterization of the limit points of the path and its normalized first order derivatives are also provided as a consequence it is shown that a weighted central path can have two types of behavior it converges either as theta nu or as theta sqrt nu depending on whether the matrix w on a certain scaled space is block diagonal or not respectively we also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal dual optimal solutions finally in light of the results of this paper we give a characterization of a sufficient condition proposed by potra and sheng which guarantees the superlinear convergence of a class of primal dual interior point sdp algorithms

convergence analysis of the projective scaling algorithm based on a long step homogeneous affine scaling algorithm in this paper we give a new convergence analysis of a projective scaling algorithm we consider a long step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem this algorithm takes a fixed fraction 2 3 of the way towards the boundary of the nonnegative orthant at each iteration the iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection which generates the same search direction as the original projective scaling algorithm at each iterate the recent convergence results for the long step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm specifically we will show i polynomiality of the algorithm with complexities of o nl and o n2l iterations for 2 3 and 2 3 respectively ii global covnergence of the algorithm when the optimal face is unbounded iii convergence of the primal iterates to a relative interior point of the optimal face iv convergence of the dual estimates to the analytic center of the dual optimal face and v convergence of the reduction rate of the objective function value to 1

a composite likelihood approach in fitting spatial point process models we propose a new likelihood based approach in fitting spatial point process models a composite likelihood is first formed by adding some pairwise composite likelihood functions that are defined in terms of the second order intensity function of the underlying process and then used for estimating the unknown parameters the estimation procedure is computationally simple and yields consistent and asymptotically normal estimators under some mild conditions we demonstrate through a simulation study and applications to two real data examples that the proposed approach may lead to improved estimations compared with the commonly used minimum contrast estimation approach

polynomial time algorithms for linear programming based only on primal scaling and projected gradients of a potential function this paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function the paper contains extensions and analysis of two polynomial time algorithms for linear programming we first present an extension of gonzaga s o nl iteration algorithm that computes dual variables and does not assume a known optimal objective function value this algorithm uses only affine scaling and is based on computing the projected gradient of the potential function q ln x t s sum limits j 1 n ln x j wherex is the vector of primal variables ands is the vector of dual slack variables and q n sqrt n the algorithm takes either a primal step or recomputes dual variables at each iteration we next present an alternate form of ye s o sqrt n l iteration algorithm that is an extension of the first algorithm of the paper but uses the potential function q ln x t s sum limits j 1 n ln x j sum limits j 1 n ln s j where q n sqrt n we use this alternate form of ye s algorithm to show that ye s algorithm is optimal with respect to the choice of the parameterq in the following sense suppose thatq n n t wheret 0 then the algorithm will solve the linear program in o n r l iterations wherer max t 1 t thus the value oft that minimizes the complexity bound ist 1 2 yielding ye s o sqrt n l iteration bound

improving the rate of convergence of interior point methods for linear programming this paper proposes a procedure for improving the rate of convergence of interior point methods for linear programming if x k is the sequence generated by an interior point method the procedure derives an auxiliary sequence bar x k under the suitable assumptions it is shown that the sequence bar x k converges superlinearly faster to the solution than x k application of the procedure to the projective and afflne scaling algorithms is discussed and some computational illustration is provided

a quadratically convergent method for linear programming abstract a new method to solve linear programming problems is introduced this method follows a path defined by a system of o d e and for nondegenerate problem is quadratically convergent

continuous dynamical systems that realize discrete optimization on the hypercube we study the problem of finding a local minimum of a multilinear function e over the discrete set 0 1 n the search is achieved by a gradient like system in 0 1 n with cost function e under mild restrictions on the metric the stable attractors of the gradient like system are shown to produce solutions of the problem even when they are not in the vicinity of the discrete set 0 1 n moreover the gradient like system connects with interior point methods for linear programming and with the analog neural network studied by vidyasagar vid95 in the same context ams subject classifications 34d20 90c51 key words gradient like systems multilinear functions 0 1 programming interior point methods lyapunov stability

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

the cylindrical k function and poisson line cluster point processes the analysis of point patterns with linear structures is of interest in many applications to detect anisotropy in such patterns in particular in the case of a columnar structure we introduce a functional summary statistic the cylindrical k function which is a directional k function whose structuring element is a cylinder we further introduce a class of anisotropic cox point processes called poisson line cluster point processes the points of such a process are random displacements of poisson point processes defined on the lines of a poisson line process parameter estimation for this model based on moment methods or bayesian inference is discussed in the case where the underlying poisson line process is latent to illustrate the proposed methods we analyse two and three dimensional point pattern datasets the three dimensional dataset is of particular interest as it relates to the minicolumn hypothesis in neuroscience which claims that pyramidal and other brain cells have a columnar arrangement perpendicular to the surface of the brain

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems

free nilpotent lie algebras admitting ad invariant metrics abstract in this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent lie algebra to be endowed with an ad invariant metric for such nilpotent lie algebras admitting an ad invariant metric the corresponding automorphisms groups are studied

a class of polynomial variable metric algorithms for linear optimization in the paper the behaviour of interior point algorithms is analyzed by using a variable metric method approach a class of polynomial variable metric algorithms is given achieving o n l iterations for solving a canonical form linear optimization problem with respect to a wide class of riemannian metrics wheren is the number of dimensions and a fixed value it is shown that the vector fields of several interior point algorithms for linear optimization is the negative riemannian gradient vector field of a linear a potential or a logarithmic barrier function for suitable riemannian metrics

extending the applicability of newton s method on lie groups we use newton s method to approximate a zero of a mapping from a lie group into its lie algebra under the same computational cost as before we show the semilocal convergence of newton s method with the following advantages over earlier works 55 weaker sufficient convergence conditions tighter error bounds on the distances involved and at least as precise information on the location of the solution numerical examples are also given for solving equations in cases not covered before