topology of symplectomorphism groups of rational ruled surfaces let m be either s 2 s 2 or the one point blow up cp 2 cp 2 of cp 2 in both cases m carries a family of symplectic forms where 1 determines the cohomology class this paper calculates the rational co homology of the group g of symplectomorphisms of m as well as the rational homotopy type of its classifying space bg it turns out that each group g contains a nite collection kk k 0 of nite dimensional lie subgroups that generate its homotopy we show that these subgroups asymptotically commute i e all the higher whitehead products that they generate vanish as 1 however for each xed there is essentially one nonvanishing product that gives rise to a jumping generator w in h g and to a single relation in the rational cohomology ring h bg an analog of this generator w was also seen by kronheimer in his study of families of symplectic forms on 4 manifolds using seiberg witten theory our methods involve a close study of the space of compatible almost complex structures on m

a convergence analysis of dykstra s algorithm for polyhedral sets let h be a nonempty closed convex set in a hilbert space x determined by the intersection of a finite number of closed half spaces it is well known that given an x 0 in x dykstra s algorithm applied to this collection of closed half spaces generates a sequence of iterates that converge to ph x0 the orthogonal projection of x0 onto h the iterates however do not necessarily lie in h we propose a combined dykstra conjugate gradient method such that given an varepsilon 0 the algorithm computes an x in h with x p h x 0 varepsilon moreover for each iterate xm of dykstra s algorithm we calculate a bound for x m p h x 0 that approaches zero as m tends to infinity applications are made to computing bounds for x m p h x 0 where h is a polyhedral cone numerical results are presented from a sample isotone regression problem

efficient rank reduction of correlation matrices geometric optimisation algorithms are developed that efficiently find the nearest low rank correlation matrix we show in numerical tests that our methods compare favourably with the existing methods in the literature the connection with the lagrange multiplier method is established along with an identification of whether a local minimum is a global minimum an additional benefit of the geometric approach is that any weighted norm can be applied the problem of finding the nearest low rank correlation matrix occurs as part of the calibration of multi factor interest rate market models to correlation

global risk bounds and adaptation in univariate convex regression we consider the problem of nonparametric estimation of a convex regression function phi 0 we study the risk of the least squares estimator lse under the natural squared error loss we show that the risk is always bounded from above by n 4 5 modulo logarithmic factors while being much smaller when phi 0 is well approximable by a piecewise affine convex function with not too many affine pieces in which case the risk is at most 1 n up to logarithmic factors on the other hand when phi 0 has curvature we show that no estimator can have risk smaller than a constant multiple of n 4 5 in a very strong sense by proving a local minimax lower bound we also study the case of model misspecification where we show that the lse exhibits the same global behavior provided the loss is measured from the closest convex projection of the true regression function in the process of deriving our risk bounds we prove new results for the metric entropy of local neighborhoods of the space of univariate convex functions these results which may be of independent interest demonstrate the non uniform nature of the space of univariate convex functions in sharp contrast to classical function spaces based on smoothness constraints

additive conjoint isotonic probabilistic models adisop the isop model or model of twodimensional or bi isotonicity scheiblechner 1995 postulates that the probabilities of ordered response categories increase isotonically in the order of subject ability and item easiness adding a conventional cancellation axiom for the factors of subjects and items gives the adisop model where the c d f s of response categories are functions of an additive item and subject parameter and an ordinal category parameter extending cancellation to the interactions of subjects and categories as well as of items and categories independence axiom of the category factor from the subject and item factor gives the cadisop model completely additive model in which the parallel c d f s are functions of the sum of subject item and category parameters the cadisop model is very close to the unidimensional version of the polytomous rasch model with the logistic item category characteristic s replaced by nonparametric axioms and statistics the axioms representation theorems and algorithms for model fitting of the additive models are presented

computing maximum likelihood estimators of convex density functions we consider the problem of estimating a density function that is known in advance to be convex the maximum likelihood estimator is then the solution of a linearly constrained convex minimization problem this problem turns out to be numerically difficult we show that interior point algorithms perform well on this class of optimization problem though for large samples numerical difficulties are still encountered to eliminate those difficulties we propose a clustering scheme that is reasonable from a statistical point of view we display results for problems with up to 40000 observations we also give a typical picture of the estimated density a piecewise linear function with only very few pieces

multiview stereo and silhouette consistency via convex functionals over convex domains we propose a convex formulation for silhouette and stereo fusion in 3d reconstruction from multiple images the key idea is to show that the reconstruction problem can be cast as one of minimizing a convex functional where the exact silhouette consistency is imposed as convex constraints that restrict the domain of feasible functions as a consequence we can retain the original stereo weighted surface area as a cost functional without heuristic modifications of this energy by balloon terms or other strategies yet still obtain meaningful nonempty reconstructions which are guaranteed to be silhouette consistent we prove that the proposed convex relaxation approach provides solutions that lie within a bound of the optimal solution compared to existing alternatives the proposed method does not depend on initialization and leads to a simpler and more robust numerical scheme for imposing silhouette consistency obtained by projection onto convex sets we show that this projection can be solved exactly using an efficient algorithm we propose a parallel implementation of the resulting convex optimization problem on a graphics card given a photoconsistency map and a set of image silhouettes we are able to compute highly accurate and silhouette consistent reconstructions for challenging real world data sets in particular experimental results demonstrate that the proposed silhouette constraints help to preserve fine scale details of the reconstructed shape computation times depend on the resolution of the input imagery and vary between a few seconds and a couple of minutes for all experiments in this paper

comparison of several treatments with a control using multiple contrasts abstract a problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard we consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control a number of statistical procedures have been proposed for this situation of which the best known is dunnett s 1955 multiple comparison procedure dunnett s test rejects equality of the treatments and the control for a large value of the maximum contrast of the data vector with several vectors that are located symmetrically within the alternative region we study a large class of such tests which includes dunnett s test as a particular case one of these tests which is based on the maximum contrast of the data vector with several orthogonal vectors is very easy to implement and has an uncomplicated good and fairly uniform power function over t

multiple test procedures for identifying the maximum safe dose we consider dose response studies for safety assessment of crop protection compounds and drugs and we offer a hypothesis testing approach for identifying the maximum dose level that is guaranteed to be safe with preassigned confidence the focus is on step down sd multiple test procedures for identifying the maximum safe dose we propose two classes of contrasts among the dose means as test statistics for these procedures pairwise contrasts pc and helmert contrasts hc the first procedure sd2pc consists of a sequence of ordinary t tests and is thus easy to apply but it can have very low power for certain step response functions the second procedure sd1hc does not suffer from this drawback but it requires a weak monotonicity assumption for its mathematical validity the powers of sd2pc and sd1hc are studied via monte carlo simulation we recommend sd2pc for linear response functions and sd1hc for step response functions the procedures are illustrated by applying them to data from an aquatic

selective alternating projections to find the nearest sdd matrix we extend and improve recently proposed algorithms to solve the problem of minimizing the distance from a given matrix to the cone of symmetric and diagonally dominant matrices with positive diagonal sdd we present a variety of criteria to select a subset of the supporting hyperplanes of the faces of sdd and also of the polar cone sdd 0 to then apply dykstra s alternating projection method these selections reduce the number of projections and therefore reduce the required computational work in all our new algorithms the symmetry and the diagonal dominance of the obtained matrix are guaranteed preliminary numerical experiments indicate that some of the selection criteria produce a significant reduction in cpu time

deformations and the rigidity method abstract in d rohrlich false division towers of elliptic curves j algebra 229 1 2000 249 279 d rohrlich a deformation of the tate module j algebra 229 1 2000 280 313 rohrlich proved rigidity for psl 2 z p t for p 5 obtained this group as a galois group over c t using modular function fields and derived from this interesting consequences for galois representations attached to the tate modules of elliptic curves furthermore in an unpublished preprint he established that the corresponding galois representation g c t gal c t alg c t psl 2 z p t is universal here we will turn things around we first provide a general framework for rigid deformations of projective representations of the absolute galois group of a function field in one variable over a separably closed base under natural rather general hypothesis we will determine the corresponding universal deformation ring if the residual representation is geometrically rigid which happens to be the case for many surjective representation to psl 2 f p p 2 which arise from belyi triples then certain universal deformations will be geometrically rigid too this will give new proofs for most of the results of rohrlich our method also applies to thompson tuples we then go on to give two further applications which are based on the example computed by rohrlich over f q t where q is a power of a prime l we construct infinite p adic galois extensions which have finite ramification and whose constant field is finite furthermore for p 5 and p 1 mod 4 we obtain a family of surjective galois representations gal q alg q 1 sl 2 z p 1 where the parameter runs over all p power roots of unity finally we exhibit a general class of rigid universal deformation rings which are finite flat over z p in particular this shows that the above examples of galois representations are not a singular event but a general phenomenon

inadmissibility of studentized tests for normal order restricted models consider the model where xij i 1 k j 1 2 n ni 2 2 are observed here xij are independent n oi a2 hi a2 unknown let xi e xixi ni x x xk 0 61 ak v lk e 1x ij ne1 x2 let a1 be a k m x k matrix of rank k m 2 and test h a10 0 versus k h where k a 0 0 suppose we assume a2 known and consider a constant size a test a 1 2 which is admissible for h versus k h based on x next assume a2 is unknown consider the same test but now as a function of x v1 2 i e studentize the test the resulting test is inadmissible examples are noted

an algorithm for isotonic regression for two or more independent variables algorithms for solving the isotonic regression problem in more than one dimension are difficult to implement because of the large number of lower sets present or because they involve search techniques which require a significant amount of checking and readjustment here a new algorithm for solving this problem based on a simple iterative technique is proposed and shown to converge to the correct solution

norm product belief propagation primal dual message passing for approximate inference inference problems in graphical models can be represented as a constrained optimization of a free energy function in this paper we treat both forms of probabilistic inference estimating marginal probabilities of the joint distribution and finding the most probable assignment through a unified message passing algorithm architecture in particular we generalize the belief propagation bp algorithms of sum product and max product and tree reweighted trw sum and max product algorithms trbp and introduce a new set of convergent algorithms based on convex free energy and linear programming lp relaxation as a zero temperature of a convex free energy the main idea of this work arises from taking a general perspective on the existing bp and trbp algorithms while observing that they all are reductions from the basic optimization formula of f ihi where the function f is an extended valued strictly convex but nonsmooth and the functions hi are extended valued functions not necessarily convex we use tools from convex duality to present the primal dual ascent algorithm which is an extension of the bregman successive projection scheme and is designed to handle optimization of the general type f ihi we then map the fractional free energy variational principle for approximate inference onto the optimization formula above and introduce the norm product message passing algorithm special cases of the norm product include sum product and max product bp algorithms trbp and nmplp algorithms when the fractional free energy is set to be convex convex free energy the norm product is globally convergent for the estimation of marginal probabilities and for approximating the lp relaxation we also introduce another branch of the norm product which arises as the zero temperature of the convex free energy which we refer to as the convex max product the convex max product is convergent unlike max product and aims at solving the lp relaxation

closure procedures for monotone bi factorial dose response designs les essais en doses multiples ont deux buts i demontrer une amelioration de l efficacite pour une combinaison donnee de chacun des composants ii identifier un intervalle de doses efficaces et sures les auteurs s interessent aux deux aspects quoique avec un interet plus marque pour le second avec une procedure de cloture qui garantit un controle fort du risque global d erreur on etudie deux familles differentes d hypotheses nulles pour des dispositifs bi factoriels monotones par rapport a un ordre partiel l une est adaptee a la recherche d une ou plusieurs doses efficaces minimums l autre est suffisamment large pour permettre d identifier les doses efficaces les plus elevees on applique des tests du rapport de vraisemblance et des contrastes multiples appropries a un exemple d essai clinique tire de hung 2000 statistics in medicine 19 2079 2087 le code ecrit en langage r est disponible dans sa totalite sur internet

projection on the intersection of convex sets abstract in this paper we give a solution of the problem of projecting a point onto the intersection of several closed convex sets when a projection on each individual convex set is known the existing solution methods for this problem are sequential in nature here we propose a highly parallelizable method the key idea in our approach is the reformulation of the original problem as a system of semi smooth equations the benefits of the proposed reformulation are twofold a a fast semi smooth newton iterative technique based on clarke s generalized gradients becomes applicable and b the mechanics of the iterative technique is such that an almost decentralized solution method emerges we proved that the corresponding semi smooth newton algorithm converges near the optimal point quadratically these features make the overall method attractive for distributed computing platforms e g sensor networks

ruled 4 manifolds and isotopies of symplectic surfaces we study symplectic surfaces in ruled symplectic 4 manifolds which are disjoint from a given symplectic section as a consequence in any symplectic 4 manifold two homologous symplectic surfaces which are c 0 close must be hamiltonian isotopic

improved tests for comparing treatments against a control and other one sided problems abstract three multivariate one sided testing problems are studied all problems concern a independent normal populations with unknown means and common unknown variance for the first problem the null hypothesis is that all means are zero while the alternative is that each mean is nonnegative and at least one is positive for the second problem the null hypothesis is that all means are equal and the alternative is that all of the first k 1 means are greater than or equal to the mean of the last population with strict inequality for at least one the alternative for the second problem is called the tree top alternative for the third problem the null hypothesis is that all means are equal while the alternative is that the means are ordered say nonincreasing this latter alternative is called the simple order alternative all three of these problems are discussed extensively in the book by robertson wright and dykstra we will assume that random samples of size n are drawn from each population

estimation of multivariate distributions under stochastic ordering abstract let f and g be the cdf s of two p dimensional multivariate distributions such that f is stochastically larger than g a straightforward derivation is given of the generalized maximum likelihood estimators of f and g based on random samples from each population an algorithmic approach to computing these estimators is described and motivating numerical examples are discussed the special case when f and g correspond to multivariate ordinal contingency tables is also presented the relationship of these results to those of robertson and wright 1974 is considered

approximating the level probabilities in order restricted inference the simple tree ordering summary distribution theory for the likelihood ratio statistics for the comparison of several treatments with a control is discussed these test statistics account for prior information that the treatments are at least as effective as the control their null distributions are mixtures of chi squared or beta distributions and the mixing coefficients which are level probabilities are intractable for even a moderate number of treatments and unequal weights which are typically the sample sizes the distribution corresponding to equal weights is considered as an approximation and an approximation based on the pattern of large and small weights is developed both are adequate for moderate variations in the weights but the approximation based on patterns in the weight set is considerably more accurate for larger variations

statistical inference for the difference between the best treatment mean and a control mean in many experiments researchers are interested in comparing several treatment means with a control mean when there are some treatments significantly better than the control it is often of interest to evaluate the difference between the best treatment mean and the control mean and to identify the best treatment in this article we derive lower confidence bounds for the aforementioned difference for the case that treatments are at least as effective as the control and for the case that no restriction is placed on the treatment means and the control mean the evaluation of the lower confidence bound for the difference between the best treatment mean and the control mean is a concave programming problem subject to homogeneous linear inequality constraints we propose two efficient computation algorithms and discuss the connection between our procedures and gupta s subset selection procedure we compare the expected lower confidence bounds of the two procedures with that of dunnett s procedure an applicati

a simple approach to testing homogeneity of order constrained means abstract the likelihood ratio test for equality of partially ordered means first proposed by bartholomew is known to possess power characteristics that are generally superior to those of competing procedures a limitation of this test however is that the null distribution of the test statistic is heavily dependent on both the specific form of the order restriction and the group sample sizes it is not readily available for most orderings especially when the group sample sizes are unequal furthermore computation of the test statistics often involves intricate algorithms we propose a class of procedures based upon classical methods for combining independent tests these procedures are generally applicable and easily implemented a study of the power functions of the competing tests indicates that the power of the test based on fisher s combination method compares reasonably with that of the likelihood ratio test it is concluded that the procedure based on fisher s combination method is an effective

graph theoretic representations for proximity matrices through strongly anti robinson or circular strongly anti robinson matrices there are various optimization strategies for approximating through the minimization of a least squares loss function a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries we extend these approaches to include components in the approximating sum that satisfy what are called the strongly anti robinson sar or circular strongly anti robinson csar restrictions a matrix that is sar or csar is representable by a particular graph theoretic structure where each matrix entry is reproducible from certain minimum path lengths in the graph one published proximity matrix is used extensively to illustrate the types of approximation that ensue when the sar or csar constraints are imposed

computational acceleration of projection algorithms for the linear best approximation problem abstract this is an experimental computational account of projection algorithms for the linear best approximation problem we focus on the sequential and simultaneous versions of dykstra s algorithm and the halpern lions wittmann bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the euclidean space these algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes we compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms for the simultaneous versions we present a new component averaging variant that substantially accelerates their initial behavior for sparse systems

dykstra s algorithm for constrained least squares doubly symmetric matrix problems in this work we apply dykstra s alternating projection algorithm for minimizing email protected where email protected email protected is the frobenius norm and email protected r m x n email protected r m x n and email protected r n x n are doubly symmetric positive definite matrices with entries within prescribed intervals we first solve the constrained least squares matrix problem by using the special structure properties of doubly symmetric matrices and then use the singular value decomposition to transform the original problem into a simpler one that fits nicely with the algorithm originally developed by r escalante m raydan dykstra s algorithm for a constrained least squares matrix problem numer linear algebra appl 3 1996 459 471

projected tests for order restricted alternatives consider the model where x ij i 1 2 k j 1 2 n are independent random variables distributed according to a one parameter exponential family with natural parameter i we test h 0 1 k versus h 1 c h 0 where 1 k and c is a cone determined by a 0 where the rows of a are contrasts with two nonzero elements we offer a method of generating good tests for h 0 versus h 1 the method is to take a good test for h 0 versus h 2 not h 0 and apply the test to projected sample points where the projection is onto c good tests for h 0 versus h 1 are tests that are schur convex good tests for h 0 versus h 1 are tests which are monotone with respect to a cone order we demonstrate that if the test function for h 0 versus h 2 is a constant size schur convex test then the resulting projected test is monotone

an approximate likelihood ratio test for comparing several treatments to a control abstract some parametric tests for comparing several treatments to a control are examined those tests developed earlier tend to rest on a single principle simplicity or power those developed later attempt to fill the gap we propose a new test that represents another such attempt the basic idea is to find a simple statistic that is a good approximation to the likelihood ratio statistic this leads to a power function similar to that of the likelihood ratio test which is optimal among the available competitors like the orthogonal contrast test the new test is built on an orthogonal relationship resulting in a relatively simple null distribution that depends on the sample sizes only through their total thus critical values can be easily computed and a detailed listing results in only a moderately large table computing the statistic is elementary when the sample sizes of the new treatments are equal otherwise some matrix operations are needed a simple procedure is given to estimate the sample si

on the holomorphicity of genus two lefschetz fibrations we prove that any genus 2 lefschetz fibration without reducible fibers and with transitive monodromy is holomorphic the latter condition comprises all cases where the number of singular fibers 10n is not congruent to 0 modulo 40 this proves a conjecture of the authors in siti1 an auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in s 2 bundles over s 2 of relative degree 7 over the base and of symplectic surfaces in cp 2 of degree 17

a unified approach to likelihood inference on stochastic orderings in a nonparametric context abstract for data in a two way contingency table with ordered margins we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities this leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log likelihood ratio tests the usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases in particular for each of these three orderings we determine the transformation required to apply the estimation algorithm we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered as well as testing each stochastic order against an u

approximations to the power of some order restricted tests with slippage alternatives summary in the one way analysis of variance a researcher may believe that the ordering of the means is known and would like to incorporate this information into the test of hypotheses bartholomew developed likelihood ratio tests for such situations and proposed a twomoment chi squared approximation for the minimum power of the test for the case of known variances the accuracy of a closely related approximation is studied and an approximation based on mixtures of noncentral f variables is developed for the case of unknown variances the simple ordering the simple tree ordering and a class of slippage alternatives are considered and an example concerning the choice of sample size needed to provide a specified power at a particular alternative is treated for powers in the range typically of interest the two moment approximation seems adequate

hybrid methods for solving the educational testing problem abstract methods for solving the educational testing problem are considered one approach glunt 1995 is to formulate the problem as a linear convex programming problem in which the constraint is the intersection of three convex sets this method is globally convergent but the rate of convergence is slow however the method does have the capability of determining the correct rank of the solution matrix and this can be done in relatively few iterations if the correct rank of the solution matrix is known it is shown how to formulate the problem as a smooth nonlinear minimization problem for which a rapid convergence can be obtained by l1sqp method 6 this paper studies hybrid methods that attempt to combine the best features of both types of method an important feature concerns the interfacing of the component methods thus it has to be decided which method to use first and when to switch between methods difficulties such as these are addressed in the paper comparative numerical results are also reported

some approximations for the null distribution of the e statistic used in order restricted inference where p l k is the probability that 0 0 takes 1 distinct values and b ab is a beta random variable with parameters a and b kudo sasabuchi choi 1981 pointed out that the likelihood ratio test is applicable even without replication provided that k is adequately large in this case the likelihood ratio statistic is the same as 1l1 with n 1 and its null distribution is given for c 0 by k i 1 1 pr ek c p l k pr b4 1 1 k 1 c pr ek 0 k 13

an algorithm for restricted least squares regression abstract a commonly occurring problem in statistics is that of minimizing a least squares expression subject to side constraints here a simple iterative algorithm is presented and shown to converge to the desired solution several examples are presented including finding the closest concave convex function to a set of points and other general quadratic programming problems the dual problem to the basic problem is also discussed and a solution for it is given in terms of the algorithm finally extensions to expressions other than least squares are given

a semi smooth newton method for a special piecewise linear system with application to positively constrained convex quadratic programming in this paper a special piecewise linear system is studied it is shown that under a mild assumption the semi smooth newton method applied to this system is well defined and the method generates a sequence that converges linearly to a solution besides we also show that the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented as an application we study the convex quadratic programming problem under positive constraints the numerical results suggest that the semi smooth newton method achieves accurate solutions to large scale problems in few iterations

a geometrical approach to iterative isotone regression in the present paper we propose and analyze a novel method for estimating a univariate regression function of bounded variation the underpinning idea is to combine two classical tools in nonparametric statistics namely isotonic regression and the estimation of additive models a geometrical interpretation enables us to link this iterative method with von neumann s algorithm moreover making a connection with the general property of isotonicity of projection onto convex cones we derive another equivalent algorithm and go further in the analysis

a penalized method for multivariate concave least squares with application to productivity analysis abstract we propose a penalized method for the least squares estimator of a multivariate concave regression function this estimator is formulated as a quadratic programing qp problem with o n2 constraints where n is the number of observations computing such an estimator is a very time consuming task and the computational burden rises dramatically as the number of observations increases by introducing a quadratic penalty function we reformulate the concave least squares estimator as a qp with only non negativity constraints this reformulation can be adapted for estimating variants of shape restricted least squares i e the monotonic concave convex least squares the experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem the matlab and r codes for implementing the penalized problems are provided in the paper

order restricted statistical tests on multinomial and poisson parameters the starshaped restriction abstract likelihood ratio statistics for i testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped cf shaked ann statist 1979 and for ii testing the null hypothesis that this parameter vector is starshaped are considered for both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi bar square distribution analogous tests on a collection of poisson means are also found to have asymptotic chi bar square distributions author

accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes abstract the von neumann halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces we propose acceleration schemes making use of two ideas firstly each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection and secondly it is easy to project onto a finite intersection of such hyperplanes we give conditions for which our accelerations converge strongly finally we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem

computing the nearest euclidean distance matrix with low embedding dimensions euclidean distance embedding appears in many high profile applications including wireless sensor network localization where not all pairwise distances among sensors are known or accurate the classical multi dimensional scaling cmds generally works well when the partial or contaminated euclidean distance matrix edm is close to the true edm but otherwise performs poorly a natural step preceding cmds would be to calculate the nearest edm to the known matrix a crucial condition on the desired nearest edm is for it to have a low embedding dimension and this makes the problem nonconvex there exists a large body of publications that deal with this problem some try to solve the problem directly and some are the type of convex relaxations of it in this paper we propose a numerical method that aims to solve this problem directly our method is strongly motivated by the majorized penalty method of gao and sun for low rank positive semi definite matrix optimization problems the basic geometric object in our study is the set of edms having a low embedding dimension we establish a zero duality gap result between the problem and its lagrangian dual problem which also motivates the majorization approach adopted numerical results show that the method works well for the euclidean embedding of network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation

set intersection problems supporting hyperplanes and quadratic programming we study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem for the problem of finding the closest point in the intersection of closed convex sets we propose an algorithm that like dykstra s algorithm converges strongly in a hilbert space moreover this algorithm converges in finitely many iterations when the closed convex sets are cones in mathbb r n r n satisfying an alignment condition next we propose modifications of the alternating projection algorithm and prove its convergence the algorithm converges superlinearly in mathbb r n r n under some nice conditions under a conical condition the convergence can be finite lastly we discuss the case where the intersection of the sets is empty