a new fuzzy cover approach to clustering this paper presents a new fuzzy cover based clustering algorithm in the proposed algorithm the concept of fuzzy cover and objective function are employed to identify holding points in the dataset and we associate these holding points together to build up the backbones of the final clusters three specific objectives underlie the presentation of the proposed approach in this paper the first is to describe mathematical formulation of the fuzzy covers and the second is to summarize the detailed procedure of constructing fuzzy covers and splicing them into clusters the third goal is to demonstrate that this approach is able to find out reasonable representative patterns in the final clusters we illustrate this approach with four examples in order to verify the clustering effectiveness

cluster differences scaling with a within clusters loss component and a fuzzy successive approximation strategy to avoid local minima cluster differences scaling is a method for partitioning a set of objects into classes and simultaneously finding a low dimensional spatial representation ofk cluster points to model a given square table of dissimilarities amongn stimuli or objects the least squares loss function of cluster differences scaling originally defined only on the residuals of pairs of objects that are allocated to different clusters is extended with a loss component for pairs that are allocated to the same cluster it is shown that this extension makes the method equivalent to multidimensional scaling with cluster constraints on the coordinates a decomposition of the sum of squared dissimilarities into contributions from several sources of variation is described including the appropriate degrees of freedom for each source after developing a convergent algorithm for fitting the cluster differences model it is argued that the individual objects and the cluster locations can be jointly displayed in a configuration obtained as a by product of the optimization finally the paper introduces a fuzzy version of the loss function which can be used in a successive approximation strategy for avoiding local minima a simulation study demonstrates that this strategy significantly outperforms two other well known initialization strategies and that it has a success rate of 92 out of 100 in attaining the global minimum

an fptas for optimizing a class of low rank functions over a polytope we present a fully polynomial time approximation scheme fptas for optimizing a very general class of non linear functions of low rank over a polytope our approximation scheme relies on constructing an approximate pareto optimal front of the linear functions which constitute the given low rank function in contrast to existing results in the literature our approximation scheme does not require the assumption of quasi concavity on the objective function for the special case of quasi concave function minimization we give an alternative fptas which always returns a solution which is an extreme point of the polytope our technique can also be used to obtain an fptas for combinatorial optimization problems with non linear objective functions for example when the objective is a product of a fixed number of linear functions we also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor unless p np we prove this by showing a similar hardness of approximation result for supermodular function minimization a result that may be of independent interest

from the restricted to the full three body problem the three body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three body problem we use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three body problem with one small mass is to the first approximation the product of the restricted three body problem and a harmonic oscillator this allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three body problem for example all the non degenerate periodic solutions generic bifurcations hamiltonian hopf bifurcations bridges and natural centers known to exist in the restricted problem can be continued into the reduced three body problem the classic normalization calculations of deprit and deprit bartholome show that there are two dimensional kam invariant tori near the lagrange point in the restricted problem with the above result this proves that there are three dimensional kam invariant tori near the lagrange point in the reduced three body problem

analysis of parameter selection for gustafson kessel fuzzy clustering using jacobian matrix in fuzzy clustering the fuzzy c means fcm is the most known algorithm several extensions and variations of fcm had been proposed in the literature the first important extension to fcm was proposed by gustafson and kessel gk in the gk fuzzy clustering they considered the effect of different cluster shapes except for spherical shapes by replacing the euclidean distance of the fcm objective function with the mahalanobis distance the gk algorithm has become one of the most frequently used clustering algorithms just like fcm the fuzziness index m is a parameter in which the value will greatly influence the performance of the gk algorithm however there is no theoretical work on the parameter selection for the fuzziness index m of gk in this paper we reveal the relation between the stable fixed points of the gk algorithm and the datasets using jacobian matrix analysis and then provide a theoretical base for selecting the fuzziness index m in the gk algorithm some experimental results verify the effectiveness of our theoretical results

clustering based initialization for non negative matrix factorization non negative matrix factorization nmf is an unsupervised learning algorithm that can extract parts from visual data the goal of this technique is to find intuitive basis such that training examples can be faithfully reconstructed using linear combination of basis images which are restricted to non negative values thus nmf basis images can be understood as localized features that correspond better with intuitive notions of parts of images however there has been few systematic study to explore various methods for initialization of nmf algorithm which is crucial for the performance of nmf algorithm in data analysis in this paper we discuss a structured nmf initialization scheme based on the clustering method comparing with the random initialization in common use our method achieved faster convergence while maintaining the data structure and also obtained good result for the face recognition task furthermore we also proposed to use a normalized aic incorporated with our nmf initialization for rank selection of traditional nmf at the cost of much less computational load while obtaining a good performance in face recognition

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

an fptas for minimizing the product of two non negative linear cost functions we consider a quadratic programming qp problem of the form min x t c x subject to ax b x 0 where c in mathbb r n times n rm rank c 1 and a in mathbb r m times n b in mathbb r m we present an fully polynomial time approximation scheme fptas for this problem by reformulating the qp as a parameterized lp and rounding the optimal solution furthermore our algorithm returns an extreme point solution of the polytope therefore our results apply directly to 0 1 problems for which the convex hull of feasible integer solutions is known such as spanning tree matchings and sub modular flows they also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s t shortest paths and s t min cuts for the above discrete problems the quadratic program models the problem of obtaining an integer solution that minimizes the product of two linear non negative cost functions

a new accelerating method for globally solving a class of nonconvex programming problems abstract in this paper we combine the new global optimization method proposed by jiao h jiao a branch and bound algorithm for globally solving a class of nonconvex programming problems nonlinear anal 70 2 2008 1113 1123 with a suitable deleting technique to propose a new accelerating global optimization algorithm for solving a class of nonconvex programming problems np this technique offers a possibility to cut away a large part of the currently investigated region in which the global optimal solution of np does not exist and can be seen as an accelerating device for the global optimization algorithm of the nonconvex programming problems compared with the method in the above cited reference numerical results show that the computational efficiency is obviously improved by using this new technique in the number of iterations the required list length and the overall execution time of the algorithm

general c means clustering model partitional clustering is an important part of cluster analysis based on various theories numerous clustering algorithms have been developed and new clustering algorithms continue to appear in the literature it is known that occam s razor plays a pivotal role in data based models and partitional clustering is categorized as a data based model however no relation had previously been discovered between occam s razor and partitional clustering as we discuss in this paper the three main contributions of this paper can be summarized as follows 1 according to a novel definition of the mean a unifying generative framework for partitional clustering algorithms called a general c means clustering model gcm is presented and studied and 2 based on the local optimality test of the gcm the connection between occam s razor and partitional clustering is established for the first time as its application a comprehensive review of the existing objective function based clustering algorithms is presented based on gcm 3 under a common assumption about partitional clustering a theoretical guide for devising and implementing clustering algorithm is discovered these conclusions are verified by numerical experimental results

time series k means existing clustering algorithms are weak in extracting smooth subspaces for clustering time series data in this paper we propose a new k means type smooth subspace clustering algorithm named time series k means tskmeans for clustering time series data the proposed tskmeans algorithm can effectively exploit inherent subspace information of a time series data set to enhance clustering performance more specifically the smooth subspaces are represented by weighted time stamps which indicate the relative discriminative power of these time stamps for clustering objects the main contributions of our work include the design of a new objective function to guide the clustering of time series data and the development of novel updating rules for iterative cluster searching with respect to smooth subspaces based on a synthetic data set and five real life data sets our experimental results confirm that the proposed tskmeans algorithm outperforms other state of the art time series clustering algorithms in terms of common performance metrics such as accuracy fscore randindex and normal mutual information

a branch and bound algorithm for globally solving a class of nonconvex programming problems abstract a branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems np for minimizing the problem linear lower bounding functions llbfs of objective function and constraint functions are constructed then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value the proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems and finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm

a generalization of distance functions for fuzzy c means clustering with centroids of arithmetic means fuzzy c means fcm is a widely used fuzzy clustering method which allows an object to belong to two or more clusters with a membership grade between zero and one despite the considerable efforts made by the clustering community the common characteristics of distance functions suitable for fcm remain unclear to fill this crucial void in this paper we first provide a generalized definition of distance functions that fit fcm directly the goal is to provide more flexibility to fcm in the choice of distance functions while preserving the simplicity of fcm by using the centroids of arithmetic means indeed we show that any distance function that fits fcm directly can be derived by a continuously differentiable convex function and thus is an instance of the generalized point to centroid distance p2c d by definition in addition we prove that if the membership grade matrix is nondegenerate any instance of the p2c d fits fcm directly finally extensive experiments have been conducted to demonstrate that the p2c d leads to the global convergence of fcm and that the clustering performances are significantly affected by the choices of distance functions

approximation algorithms for indefinite quadratic programming we consider approximation schemes for indefinite quadratic programming we argue that such an approximation can be found in polynomial time for fixed andt wheret denotes the number of negative eigenvalues of the quadratic term our algorithm is polynomial in 1 for fixedt and exponential int for fixed we next look at the special case of knapsack problems showing that a more efficient polynomial int approximation algorithm exists

a modified k means algorithm for circular invariant clustering several important pattern recognition applications are based on feature vector extraction and vector clustering directional patterns are commonly represented by rotation variant vectors f sub d formed from features uniformly extracted in m directions it is often desirable that pattern recognition algorithms are invariant under pattern rotation this paper introduces a distance measure and a k means based algorithm namely circular k means ck means to cluster vectors containing directional information such as f sub d in a circular shift invariant manner a circular shift of f sub d corresponds to pattern rotation thus the algorithm is rotation invariant an efficient fourier domain representation of the proposed measure is presented to reduce computational complexity a split and merge approach smck means suited to the proposed ck means technique is proposed to reduce the possibility of converging at local minima and to estimate the correct number of clusters experiments performed for textural images illustrate the superior performance of the proposed algorithm for clustering directional vectors f sub d compared to the alternative approach that uses the original k means and rotation invariant feature vectors transformed from f sub d

a practicable contraction approach for the sum of the generalized polynomial ratios problem abstract in this paper a practicable contraction approach is proposed for solving the sum of the generalized polynomial ratios problem p with generalized polynomial constraints due to the intrinsic difficulty of problem p less work has been devoted to solving this problem the proposed approach is based on reducing the original nonconvex problem p as a standard geometric programming gp problem by utilizing simple transformation and contraction strategies the resulting optimization problem can be solved effectively by utilizing the solutions of a series of gp problems the tractability and effectiveness of the proposed successive contraction approach are demonstrated by several numerical examples and the performance comparison of the proposed approach and other methods published is also presented in terms of solution quality

a thresholded fuzzy c means algorithm for semi fuzzy clustering abstract in this paper the problem of achieving semi fuzzy or soft clustering of multidimensional data is discussed a technique based on thresholding the results of the fuzzy c means algorithm is introduced the proposed approach is analysed and contrasted with the soft clustering method see s z selim and m a ismail pattern recognition 17 559 568 showing the merits of the new method separation of clusters in the semi fuzzy clustering context is introduced and the use of the proposed technique to measure the degree of separation is explained

k vectors an alternating minimization algorithm for learning regression functions the k vectors algorithm for learning regression functions proposed here is akin to the well known k means algorithm both algorithms partition the feature space but unlike the k means algorithm the k vectors algorithm aims to reconstruct the response rather than the feature the partitioning rule of the algorithm is based on maximizing the correlation inner product of the feature vector with a set of k vectors and generates polyhedral cells similar to the ones generated by the nearest neighbor rule of the k means algorithm similarly to k means the learning algorithm alternates between two types of steps in the first type of steps k labels are determined via a centroid type rule in the response space which uses a surrogate hinge type loss function to the mean squared error loss function in the second type of steps the k vectors which determine the partition are updated according to a multiclass classification rule in the spirit of support vector machines it is proved that both steps of the algorithm only require solving convex optimization problems and that the algorithm is empirically consistent as the length of the training sequence increases to infinity fixed points of the empirical version of the algorithm tend to fixed points of the population version of the algorithm learnability of the predictor class posit by the algorithm is also established

normal forms and versal deformations of linear hamiltonian systems abstract equivalence classes of time independent linear real hamiltonian systems can be identified up to canonical transformations with the orbits of the adjoint action of the real symplectic group on its lie algebra a new set of representatives also called normal forms for these orbits is given versal deformations of systems in normal form are constructed applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated

soft subspace clustering of categorical data with probabilistic distance categorical data clustering is an important subject in pattern recognition currently subspace clustering of categorical data remains an open problem due to the difficulties in estimating attribute interestingness according to the statistics of categories in clusters in this paper a new algorithm is proposed for clustering categorical data with a novel soft feature selection scheme by which each categorical attribute is automatically assigned a weight that correlates with the smoothed dispersion of the categories in a cluster in the proposed algorithm dissimilarity between categorical data objects is measured using a probabilistic distance function based on kernel density estimation for categorical attributes we also make use of the probabilistic distances to define a cluster validity index for estimating the number of categorical clusters the suitability of the proposal is demonstrated in an empirical study done with some widely used real world data sets and synthetic data sets and the results show its outstanding performance highlightswe define the cluster scatter on object to cluster distances for categorical data we propose a probabilistic distance function using a kernel density estimation method categorical attributes are weighted based on the smoothed dispersion of categories two weighting schemes are offered depending on the attribute types significantly improve clustering performance compared to mode based algorithms

a tabu search based algorithm for the fuzzy clustering problem the fuzzy clustering problem fcp is a mathematical program which is difficult to solve since it is nonconvex which implies possession of many local minima the fuzzy c means heuristic is the widely known approach to this problem but it is guaranteed only to yield local minima in this paper we propose a new approach to this problem which is based on tabu search technique and aims at finding a global solution of fcp we compare the performance of the algorithm with the fuzzy c means algorithm

an efficient algorithm for globally solving generalized linear multiplicative programming generalized linear multiplicative programming problems are difficult problems known to be n p hard and they frequently arise in various areas such as engineering economics finance etc at present most of the global optimization algorithms for this kind of problems are mainly developed based on those kinds of branch and bound algorithms in this study a new algorithm based on the concept of level set is presented for globally solving generalized linear multiplicative programming problems the proposed algorithm is concise and easy to implement so it is quite suitable for engineering applications finally the global convergence is presented and the practical efficiency of the algorithm is demonstrated by the numerical experiments

unsupervised minor prototype detection using an adaptive population partitioning algorithm this paper presents a new partitioning algorithm designated as the adaptive c populations acp clustering algorithm capable of identifying natural subgroups and influential minor prototypes in an unlabeled dataset in contrast to traditional fuzzy c means clustering algorithms which partition the whole dataset equally adaptive clustering algorithms such as that presented in this study identify the natural subgroups in unlabeled datasets in this paper data points within a small dense region located at a relatively large distance from any of the major cluster centers are considered to form a minor prototype the aim of acp is to adaptively separate these isolated minor clusters from the major clusters in the dataset the study commences by introducing the mathematical model of the proposed acp algorithm and demonstrates its convergence to a stable solution the ability of acp to detect minor prototypes is confirmed via its application to the clustering of three different datasets with different sizes and characteristics

global optimization for sum of linear ratios problem with coefficients the global optimization of the sum of linear ratios problem has attracted the interest of researchers and practitioners for a number of years since these types of optimization problems are nonconvex various specialized algorithms have been proposed for globally solving these problems however these algorithms are only for the case that sum of linear ratios problem without coefficients and may be difficult to implement in this paper a branch and bound algorithm is proposed for globally solving the sum of linear ratios problem with coefficients by utilizing an equivalent problem and linearization technique the initial nonconvex programming problem is reduced to a sequence of linear programming problems the proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems numerical results are given to show the feasibility and effectiveness of the proposed algorithm

a simplicial branch and bound duality bounds algorithm for the linear sum of ratios problem in this article we present and validate a simplicial branch and bound duality bounds algorithm for globally solving the linear sum of ratios fractional program the algorithm computes the lower bounds called for during the branch and bound search by solving ordinary linear programming problems these problems are derived by using lagrangian duality theory the algorithm applies to a wide class of linear sum of ratios fractional programs two sample problems are solved and the potential practical and computational advantages of the algorithm are indicated

approximation of linear fractional multiplicative problems in this paper we propose a fully polynomial time approximation scheme for a class of optimization problems where the feasible region is a polyhedral one and the objective function is the sum or product of ratios of linear functions the class includes the well known ones of linear sum of ratios fractional programming and multiplicative programming

on the mathematical and numerical properties of the fuzzy c means algorithm abstract the fuzzy clustering problem is investigated interesting properties of the points generated in the course of applying the fuzzy c means algorithm are revealed using the concept of reduced objective function we investigate seven quantities that could be used for stopping the algorithm and prove relationships among them finally we empirically show that these quantities converge linearly

quantization based clustering algorithm in this paper a quantization based clustering algorithm qbca is proposed to cluster a large number of data points efficiently unlike previous clustering algorithms qbca places more emphasis on the computation time of the algorithm specifically qbca first assigns the data points to a set of histogram bins by a quantization function then it determines the initial centers of the clusters according to this point distribution finally qbca performs clustering at the histogram bin level rather than the data point level we also propose two approaches to improve the performance of qbca further i a shrinking process is performed on the histogram bins to reduce the number of distance computations and ii a hierarchical structure is constructed to perform efficient indexing on the histogram bins finally we analyze the performance of qbca theoretically and experimentally and show that the approach 1 can be easily implemented 2 identifies the clusters effectively and 3 outperforms most of the current state of the art clustering approaches in terms of efficiency

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

a note on the ics algorithm with corrections and theoretical analysis ozdemir and akarun 2001 proposed an intercluster separation ics fuzzy clustering algorithm the ics algorithm is useful in combined quantization and dithering however there are two errors in the update equations for the ics algorithm this correspondence first points out these errors and gives their corrections since the parameters m c and are important factors in the performance of ics we also conduct a theoretical analysis of these ics parameters in order to analyze the parameters in ics we devise a theorem for the calculation of the hessian matrix from the ics objective function we establish the fixed point property of ics based on the decomposition of the hessian matrix and then analyze the effect of the parameters finally we propose a numerical approach in choosing the appropriate parameters m and for ics these experimental results give a better numerical perspective on the effect of parameters in ics and have conclusions consistent with our theoretical analysis

a parameter free community detection method based on centrality and dispersion of nodes in complex networks abstract k means is a simple and efficient clustering algorithm to detect communities in networks however it may suffer from a bad choice of initial seeds also called centers that seriously affect the clustering accuracy and the convergence rate additionally in k means the number of communities should be specified in advance till now it is still an open problem on how to select initial seeds and how to determine the number of communities in this study a new parameter free community detection method named k rank d was proposed first based on the fact that good initial seeds usually have high importance and are dispersedly located in a network we proposed a modified pagerank centrality to evaluate the importance of a node and drew a decision graph to depict the importance and the dispersion of nodes then the initial seeds and the number of communities were selected from the decision graph actively and intuitively as the start parameter of k means experimental results on synthetic and real world networks demonstrate the superior performance of our approach over competing methods for community detection

a method of identifying influential data in fuzzy clustering in multivariate statistical methods it is important to identify influential observations for a reasonable interpretation of the data structure in this paper we propose a method for identifying influential data in the fuzzy c means fcm algorithm to investigate such data we consider a perturbation of the data points and evaluate the effect of a perturbation as a perturbation we consider two cases one is the case in which the direction of a perturbation is specified and the other is the case in which the direction of a perturbation is not specified by computing the change in the clustering result of fcm when given data points are slightly perturbed we can look for data points that greatly affect the result also we confirm an efficacy of the proposed method by numerical examples

the evolution of invariant manifolds in hamiltonian hopf bifurcations we consider the evolution of the stable and unstable manifolds of an equilibrium point of a hamiltonian system of two degrees of freedom which depends on a parameter n the eigenvalues of the linearized system are complex for no0 and pure imaginary for n40 thus for no0 the equilibrium has a two dimensional stable manifold and a two dimensional unstable manifold but for n40 these stable and unstable manifolds are gone if the sign of a certain term in the normal form is positive then for small negative n the stable and unstable manifolds of the system are either identical or must have transverse intersection thus either the system is totally degenerate or the system admits a suspended smale horseshoe as an invariant set r 2002 elsevier science usa all rights reserved

color image segmentation advances and prospects abstract image segmentation is very essential and critical to image processing and pattern recognition this survey provides a summary of color image segmentation techniques available now basically color segmentation approaches are based on monochrome segmentation approaches operating in different color spaces therefore we first discuss the major segmentation approaches for segmenting monochrome images histogram thresholding characteristic feature clustering edge detection region based methods fuzzy techniques neural networks etc then review some major color representation methods and their advantages disadvantages finally summarize the color image segmentation techniques using different color representations the usage of color models for image segmentation is also discussed some novel approaches such as fuzzy method and physics based method are investigated as well

an algorithm for competitive learning in clustering problems abstract competitive learning is one of the algorithms for clustering based on the least sum of squares criterion however competitive learning has a problem of serious decline of learning ability from lack of competition when one unit monopolizes all input vectors this problem must be avoided using an additional algorithm this paper presents an algorithm which makes competitive learning give a good approximate solution for clustering the results using our method are better than previous studies this paper also presents a parameter optimization of competitive learning using anova analysis of variance using anova helps to optimize a parameter condition systematically

a simulated annealing algorithm for the clustering problem abstract in this paper we discuss the solution of the clustering problem usually solved by the k means algorithm the problem is known to have local minimum solutions which are usually what the k means algorithm obtains the simulated annealing approach for solving optimization problems is described and is proposed for solving the clustering problem the parameters of the algorithm are discussed in detail and it is shown that the algorithm converges to a global solution of the clustering problem we also find optimal parameters values for a specific class of data sets and give recommendations on the choice of parameters for general data sets finally advantages and disadvantages of the approach are presented

a global optimization algorithm for linear fractional programming in this paper we present an efficient branch and bound method for general linear fractional problem gfp first by using a transformation technique an equivalent problem ep of gfp is derived then by exploiting structure of ep a linear relaxation programming lrp of ep is obtained to implement the algorithm the main computation involve solving a sequence of linear programming problem which can be solved efficiently the proposed algorithm is convergent to the global maximum through the successive refinement of the solutions of a series of linear programming problems numerical experiments are reported to show the feasibility of our algorithm

versal normal form at the lagrange equilibrium l4 a new normal form called versal for the linearized hamiltonian vector field of the planar restricted three body problem at the lagrange equilibrium point l4 depending smoothly on the mass ratio for all values close to the critical routh s ratio is described then a canonical transformation also depending smoothly on the mass ratio which brings the linear hamiltonian vector field into this versal normal form is explicitly calculated

quadratic programming and combinatorial minimum weight product problems we present a fully polynomial time approximation scheme fptas for minimizing an objective a t x b t x under linear constraints a x d examples of such problems are combinatorial minimum weight product problems such as the following given a graph g v e and two edge weights a b e rightarrow mathbb r find an s t path p that minimizes a p b p the product of its edge weights relative to a and b

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

when do birds of a feather flock together k means proximity and conic programming given a set of data one central goal is to group them into clusters based on some notion of similarity between the individual objects one of the most popular and widely used approaches is k means despite the computational hardness to find its global minimum we study and compare the properties of different convex relaxations by relating them to corresponding proximity conditions an idea originally introduced by kumar and kannan using conic duality theory we present an improved proximity condition under which the peng wei relaxation of k means recovers the underlying clusters exactly our proximity condition improves upon kumar and kannan and is comparable to that of awashti and sheffet where proximity conditions are established for projective k means in addition we provide a necessary proximity condition for the exactness of the peng wei relaxation for the special case of equal cluster sizes we establish a different and completely localized proximity condition under which the amini levina relaxation yields exact clustering thereby having addressed an open problem by awasthi and sheffet in the balanced case our framework is not only deterministic and model free but also comes with a clear geometric meaning which allows for further analysis and generalization moreover it can be conveniently applied to analyzing various data generative models such as the stochastic ball models and gaussian mixture models with this method we improve the current minimum separation bound for the stochastic ball models and achieve the state of the art results of learning gaussian mixture models

versal normal form of the hamiltonian function of the restricted problem of three bodies near l 4 abstract when one studies matrices depending on parameters the transformation into jordan canonical form can become singular when multiple eigenvalues occur the versal normal form for a matrix was introduced by arnol d 1971 in order to overcome this difficulty unfortunately no method was given on how to construct this transformation for a special class of matrices we describe a simple procedure to accomplish this task we use the one to one resonance at the lagrangian triangular equilibrium point in the restricted problem of three bodies as an example to demonstrate our method we then extend the idea of versal normal form to higher order terms in the hamiltonian function after performing this normalization we are able to analyze the behavior of the periodic orbits near the equilibrium point

range division and linearization algorithm for a class of linear ratios optimization problems abstract this article interprets a range division and linearization algorithm for solving a class of fractional programming problems p with the sum or product of linear ratios objective function in the algorithm by dividing the outer space with respect to problem p into many polynomial rectangles the original non convex problem p is transformed and decomposed into a sequence of linear programming problems in order to improve the computational efficiency of the algorithm a new accelerating technique is introduced which deletes a large part of region of the outer space in which there exists no the optimal solution of p moreover the global convergence of the proposed algorithm is outlined and the computational complexity of it is investigated to demonstrate that it is a fully polynomial time approximation scheme finally the numerical computational results illustrate the feasibility and efficiency of the proposed algorithm

a relaxation approach to the fuzzy clustering problem abstract in this paper a new algorithm for fuzzy clustering is presented the proposed algorithm utilizes the idea of relaxation convergence of the proposed algorithm is proved and limits on the relaxation parameter are derived stopping criteria and resulting convergence behaviour of the algorithms are discussed the performance of the new algorithm is compared to the fuzzy c means algorithm by testing both on three published data sets theoretical and empirical results reported in this paper show that the new algorithm is more efficient and leads to significant computational savings

global optimization for sum of generalized fractional functions this paper considers the solution of generalized fractional programming gfp problem which contains various variants such as a sum or product of a finite number of ratios of linear functions polynomial fractional programming generalized geometric programming etc over a polytope for such problems we present an efficient unified method in this method by utilizing a transformation and a two part linearization method a sequence of linear programming relaxations of the initial nonconvex programming problem are derived which are embedded in a branch and bound algorithm numerical results are given to show the feasibility and effectiveness of the proposed algorithm

fuzzy c means optimality of solutions and effective termination of the algorithm abstract in this paper the solutions produced by the fuzzy c means algorithm for a general class of problems are examined and a method to test for the local optimality of such solutions is established an equivalent mathematical program is defined for the c means problem utilizing a generalized norm then the properties of the resulting optimization problem are investigated it is shown that the gradient of the resulting objective function at the solution produced by the c means algorithm in this case takes a special structure which can be used in terminating the algorithm moreover the local optimality of the solution obtained is checked utilizing the hessian of the criterion function the solution is a local minimum point if the hessian matrix at this point is positive semidefinite simple rules are proposed to help in checking the definiteness of the matrix

comments on optimality test for fixed points by kim et al abstract kim et al pattern recognition 21 651 663 1988 claimed that ismail and selim pattern recognition 19 481 1986 and selim and ismail ieee trans pattern anal mach intell pami 8 284 288 1986 contained errors in this note we dismiss this claim

a generalized fuzzy clustering regularization model with optimality tests and model complexity analysis in this paper we propose a generalized fuzzy clustering regularization gfcr model and then study its theoretical properties gfcr unifies several fuzzy clustering algorithms such as fuzzy c means fcm maximum entropy clustering mec fuzzy clustering based on fermi dirac entropy and fuzzy bidirectional associative clustering network etc the proposed gfcr becomes an alternative model of the generalized fcm gfcm that was recently proposed by yu and yang to advance theoretical study we have the following three considerations 1 we give an optimality test to monitor if gfcr converges to a local minimum 2 we relate the gfcr optimality tests to occam s razor principle and then analyze the model complexity for fuzzy clustering algorithms 3 we offer a general theoretical method to evaluate the performance of fuzzy clustering algorithms finally some numerical experiments are used to demonstrate the validity of our theoretical results and complexity analysis

local convergence of the fuzzy c means algorithms abstract much understanding has recently been gained concerning global convergence properties of the fuzzy c means fcm family of clustering algorithms these global convergence properties which hold for all iteration sequences guarantee that every fcm iteration sequence converges at least along a subsequence to a stationary point of an fcm objective function in this paper we prove a local convergence property that is a property pertaining to iteration sequences started near a solution specifically a simple result is proved which shows that whenever an fcm algorithm is started sufficiently near a minimizer of the corresponding objective function then the iteration sequence must converge to that particular minimizer the result guarantees that once captured by the local neighborhood of a minimizer the succeeding iterate sequence will not escape thus infinite oscillation of such a sequence cannot occur the rate of convergence of the sequence to such a point is also discussed

transformation to versal deformations of matrices abstract in the paper versal deformations of matrices are considered the versal deformation is a matrix family inducing an arbitrary multi parameter deformation of a given matrix by an appropriate smooth change of parameters and basis given a deformation of a matrix it is suggested to find transformation functions the change of parameters and the change of basis dependent on parameters in the form of taylor series the general method of construction of recurrent procedures for calculation of coefficients in the taylor expansions is developed and used for the cases of real and complex matrices elements of classical lie and jordan algebras and infinitesimally reversible matrices several examples are given and studied in detail applications of the suggested approach to problems of stability singularity and perturbation theories are discussed

a practicable branch and bound algorithm for sum of linear ratios problem abstract this article presents a practicable algorithm for globally solving sum of linear ratios problem slr the algorithm works by globally solving a bilinear programming problem eq that is equivalent to the problem slr in the algorithm by utilizing convex envelope and concave envelope of bilinear function the initial nonconvex programming problem is reduced to a sequence of linear relaxation programming problems in order to improve the computational efficiency of the algorithm a new accelerating technique is introduced which provides a theoretical possibility to delete a large part of the investigated region in which there exists no global optimal solution of the eq by combining this innovative technique with branch and bound operations a global optimization algorithm is designed for solving the problem slr finally numerical experimental results show the feasibility and efficiency of the proposed algorithm

an outcome space finite algorithm for solving linear multiplicative programming this paper presents an outcome space finite algorithm for solving linear multiplicative programming in each iteration of which a convex quadratic programming is only solved in the paper we give a global optimization condition on a class of multiplicative programming problems and prove that the proposed algorithm is finite terminative and gain a global optimal solution of the former problem when it stops it can be shown by the numerical results that the proposed algorithm is effective and the computational results can be gained in short time

a tabu search based heuristic for clustering abstract this paper considers a clustering problem where a given data set is partitioned into a certain number of natural and homogeneous subsets such that each subset is composed of elements similar to one another but different from those of any other subset for the clustering problem a heuristic algorithm is exploited by combining the tabu search heuristic with two complementary functional procedures called packing and releasing procedures the algorithm is numerically tested for its effectiveness in comparison with reference works including the tabu search algorithm the k means algorithm and the simulated annealing algorithm

a fuzzy k modes algorithm for clustering categorical data this correspondence describes extensions to the fuzzy k means algorithm for clustering categorical data by using a simple matching dissimilarity measure for categorical objects and modes instead of means for clusters a new approach is developed which allows the use of the k means paradigm to efficiently cluster large categorical data sets a fuzzy k modes algorithm is presented and the effectiveness of the algorithm is demonstrated with experimental results

optimality tests for fixed points of the fuzzy c means algorithm abstract the fuzzy c means fcm clustering algorithms are known to converge to either local minima or saddle points of the objective function which defines the fcm method the object of this paper is to derive efficient numerical tests for local extrema of the fcm functional that enable one to identify each candidate as a local minimum or saddle point numerical examples of the theory derived illustrate that the tests proposed cover all possible cases

on the local optimality of the fuzzy isodata clustering algorithm the convergence of the fuzzy isodata clustering algorithm was proved by bezdek 3 two sets of conditions were derived and it was conjectured that they are necessary and sufficient for a local minimum point in this paper we address this conjecture and explore the properties of the underlying optimization problem the notions of reduced objective function and improving and feasible directions are used to examine this conjecture finally based on the derived properties of the problem a new stopping criterion for the fuzzy isodata algorithm is proposed

nondegeneracy and weak global convergence of the lloyd algorithm in mathbb r d the lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer reducing average distortion at every step it constructs a voronoi partition of the domain and replaces each generator with the centroid of the corresponding voronoi cell optimal quantization is obtained in the case of a centroidal voronoi tessellation cvt which is a special voronoi tessellation of a domain omega in mathbb r d having the property that the generators of the voronoi diagram are also the centers of mass with respect to a given density function rho geq 0 of the corresponding voronoi cells the lloyd iteration is currently the most popular and elegant algorithm for computing cvts and optimal quantizers but many questions remain about its convergence especially in d dimensional spaces d 1 in this paper we prove that any limit point of the lloyd iteration in any dimensional spaces is nondegenerate provided that omega is a convex and bounded set and rho belongs to l 1 omega and is positive almost everywhere this ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the lloyd iteration to the set of nondegenerate fixed point quantizers while previously only conjectured the convergence properties of the lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional the results presented in this paper go beyond existing convergence theories for cvts and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities

an interior point method for generalized linear fractional programming we develop an interior point polynomial time algorithm for a generalized linear fractional problem the latter problem can be regarded as a nonpolyhedral extension of the usual linear fractional programming typical example which is of interest for control theory is the minimization of the generalized eigenvalue of a pair of symmetric matrices linearly depending on the decision variables

a tabu search approach to the clustering problem in this paper we consider the problem of clustering m objects into c clusters the objects are represented by points in an n dimensional euclidean space and the objective is to classify these m points into c clusters such that the distance between points within a cluster and its center which is to be found is minimized the problem is a nonconvex program that has many local minima it has been studied by many researchers and the most well known algorithm for solving it is the k means algorithm in this paper we develop a new algorithm for solving this problem based on a tabu search technique preliminary computational experience on the developed algorithm are encouraging and compare favorably with both the k means and the simulated annealing algorithms

cluster analysis for the synthesis of subarrayed monopulse antennas the design of monopulse radar systems requires the synthesis of both a sum pattern and a difference pattern which is quite flexible and hard to be optimized for subarrayed antennas in this study innovative approaches to synthesize the optimal sum and difference patterns independently for subarrayed monopulse arrays are presented by exploiting the relationship of the subarray partition and optimal excitations the synthesis problem is formulated as the clustering of a set induced by the optimal excitations thus the optimal subarray partition is transformed to a clustering problem a clustering method is presented to solve the array synthesis with uniform excitation at element level furthermore an improved approach is proposed to synthesize low sidelobe patterns when compared with several classical approaches various numerical experiments validate the effectiveness and potentiality of the proposed methods

variable branch tree structured vector quantization tree structured vector quantizers tsvq and their variants have recently been proposed all trees used are fixed m ary tree structured such that the training samples in each node must be artificially divided into a fixed number of clusters this paper proposes a variable branch tree structured vector quantizer vbtsvq based on a genetic algorithm which searches for the number of child nodes of each splitting node for optimal coding in vbtsvq moreover one disadvantage of tsvq is that the searched codeword usually differs from the full searched codeword briefly the searched codeword in tsvq sometimes is not the closest codeword to the input vector this paper proposes the multiclassification encoding method to select many classified components to represent each cluster and the codeword encoded in the vbtsvq is usually the same as that of the full search vbtsvq outperforms other tsvqs in the experiments presented here

on solving the sum of ratios problem this paper addresses the development of efficient global search methods for fractional programming problems such problems are in general nonconvex with numerous local extremums and belong to a class of global optimization problems first we reduce a rather general fractional programming problem with d c functions to solving an equation with a vector parameter that satisfies some nonnegativity assumption this theorem allows the justified use of the generalized dinkelbachs approach for solving fractional programming problems with a d c goal function based on solving of some d c minimization problem we developed a global search algorithm for fractional programming problems which was tested on a set of low dimensional test problems taken from the literature as well as on randomly generated problems with up to 200 variables or 200 terms in the sum

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

the local minima problem in hierarchical classes analysis an evaluation of a simulated annealing algorithm and various multistart procedures hierarchical classes models are quasi order retaining boolean decomposition models for n way n mode binary data to fit these models to data rationally started alternating least squares or equivalently alternating least absolute deviations algorithms have been proposed extensive simulation studies showed that these algorithms succeed quite well in recovering the underlying truth but frequently end in a local minimum in this paper we evaluate whether or not this local minimum problem can be mitigated by means of two common strategies for avoiding local minima in combinatorial data analysis simulated annealing sa and use of a multistart procedure in particular we propose a generic sa algorithm for hierarchical classes analysis and three different types of random starts the effectiveness of the sa algorithm and the random starts is evaluated by reanalyzing data sets of previous simulation studies the reported results support the use of the proposed sa algorithm in combination with a random multistart procedure regardless of the properties of the data set under study

modeling differences in the dimensionality of multiblock data by means of clusterwise simultaneous component analysis given multivariate multiblock data e g subjects nested in groups are measured on multiple variables one may be interested in the nature and number of dimensions that underlie the variables and in differences in dimensional structure across data blocks to this end clusterwise simultaneous component analysis sca was proposed which simultaneously clusters blocks with a similar structure and performs an sca per cluster however the number of components was restricted to be the same across clusters which is often unrealistic in this paper this restriction is removed the resulting challenges with respect to model estimation and selection are resolved

optimality test for generalized fcm and its application to parameter selection in cluster analysis the fuzzy c means fcm clustering algorithm is the best known and most widely used method it was proven to converge to either a local minimum or saddle points by bezdek et al wei and mendel produced efficient optimality tests for fcm fixed points recently a weighting exponent selection for fcm was proposed by yu et al inspired by these results we unify several alternative fcm algorithms into one model called the generalized fuzzy c means gfcm this gfcm model presents a wide variation of fcm algorithms and can easily lead to new and interesting clustering algorithms moreover we construct a general optimality test for gfcm fixed points this is applied to theoretically choose the parameters in the gfcm model the experimental results demonstrate the precision of the theoretical analysis