a redistributed proximal bundle method for nonconvex optimization proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives this naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting this work proposes an approach based on generating cutting planes models not of the objective function as most bundle methods do but of a local convexification of the objective function the corresponding convexification parameter is calculated on the fly in such a way that the algorithm can inform the user as to what proximal parameters are sufficiently large that the objective function is likely to have well defined proximal points this novel approach shown to be sound from both the objective function and subdifferential modelling perspectives opens the way to create workable nonconvex algorithms based on nonconvex mathcal vu theory both theoretical convergence analysis and some encouraging preliminary numerical experience are provided

solving the minimum sum of squares clustering problem by hyperbolic smoothing and partition into boundary and gravitational regions this article considers the minimum sum of squares clustering mssc problem the mathematical modeling of this problem leads to a min sum min formulation which in addition to its intrinsic bi level nature has the significant characteristic of being strongly nondifferentiable to overcome these difficulties the proposed resolution method called hyperbolic smoothing adopts a smoothing strategy using a special c differentiable class function the final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem this paper introduces the method of partition of the set of observations into two nonoverlapping groups data in frontier and data in gravitational regions the resulting combination of the two methodologies for the mssc problem has interesting properties which drastically simplify the computational tasks

coordination of cluster ensembles via exact methods we present a novel optimization based method for the combination of cluster ensembles for the class of problems with intracluster criteria such as minimum sum of squares clustering mssc we propose a simple and efficient algorithm called examce for this class of problems that is inspired from a set partitioning formulation of the original clustering problem we prove some theoretical properties of the solutions produced by our algorithm and in particular that under general assumptions though the algorithm recombines solution fragments so as to find the solution of a set covering relaxation of the original formulation it is guaranteed to find better solutions than the ones in the ensemble for the mssc problem in particular a prototype implementation of our algorithm found a new better solution than the previously best known for 21 of the test instances of the 40 instance tsplib benchmark data sets used in and and found a worse quality solution than the best known only five times for other published benchmark data sets where the optimal mssc solution is known we match them the algorithm is particularly effective when the number of clusters is large in which case it is able to escape the local minima found by k means type algorithms by recombining the solutions in a set covering context we also establish the stability of the algorithm with extensive computational experiments by showing that multiple runs of examce for the same clustering problem instance produce high quality solutions whose adjusted rand index is consistently above 0 95 finally in experiments utilizing external criteria to compute the validity of clustering examce is capable of producing high quality results that are comparable in quality to those of the best known clustering algorithms

greedy random adaptive memory programming search for the capacitated clustering problem in the capacitated clustering problem ccp a given set of n weighted points is to be partitioned into p clusters such that the total weight of the points in each cluster does not exceed a given cluster capacity the objective is to find a set of p centers that minimises the total scatter of points allocated to these centers in this paper we propose a merger of greedy random adaptive search procedure grasp and adaptive memory programming amp into a new gramps framework for the ccp a learning process is kept in charge of tracking information on the best components in an elite set of gramps solutions the information are strategically combined with problem domain data to restart the construction search phase at early stage of constructions priorities are given to problem domain data and progressively shifted towards generated information as the learning increases gramps is implemented with an efficient local search descent based on a restricted interchange neighbourhood extensive experiments are reported on on a standard set of bench marks from the literature and on a new set of large instances the results show that gramps has an efficient learning mechanism and is competitive with the existing methods in the literature

stochastic turing patterns in the brusselator model a stochastic version of the brusselator model is proposed and studied via the system size expansion the mean field equations are derived and shown to yield to organized turing patterns within a specific parameters region when determining the turing condition for instability we pay particular attention to the role of cross diffusive terms often neglected in the heuristic derivation of reaction diffusion schemes stochastic fluctuations are shown to give rise to spatially ordered solutions sharing the same quantitative characteristic of the mean field based turing scenario in term of excited wavelengths interestingly the region of parameter yielding to the stochastic self organization is wider than that determined via the conventional turing approach suggesting that the condition for spatial order to appear can be less stringent than customarily believed

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

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

fast modified global k means algorithm for incremental cluster construction the k means algorithm and its variations are known to be fast clustering algorithms however they are sensitive to the choice of starting points and are inefficient for solving clustering problems in large datasets recently incremental approaches have been developed to resolve difficulties with the choice of starting points the global k means and the modified global k means algorithms are based on such an approach they iteratively add one cluster center at a time numerical experiments show that these algorithms considerably improve the k means algorithm however they require storing the whole affinity matrix or computing this matrix at each iteration this makes both algorithms time consuming and memory demanding for clustering even moderately large datasets in this paper a new version of the modified global k means algorithm is proposed we introduce an auxiliary cluster function to generate a set of starting points lying in different parts of the dataset we exploit information gathered in previous iterations of the incremental algorithm to eliminate the need of computing or storing the whole affinity matrix and thereby to reduce computational effort and memory usage results of numerical experiments on six standard datasets demonstrate that the new algorithm is more efficient than the global and the modified global k means algorithms

experiments in quadratic 0 1 programming we present computational experience with a cutting plane algorithm for 0 1 quadratic programming without constraints our approach is based on a reduction of this problem to a max cut problem in a graph and on a partial linear description of the cut polytope

new and efficient dca based algorithms for minimum sum of squares clustering the purpose of this paper is to develop new efficient approaches based on dc difference of convex functions programming and dca dc algorithm to perform clustering via minimum sum of squares euclidean distance we consider the two most widely used models for the so called minimum sum of squares clustering mssc in short that are a bilevel programming problem and a mixed integer program firstly the mixed integer formulation of mssc is carefully studied and is reformulated as a continuous optimization problem via a new result on exact penalty technique in dc programming dca is then investigated to the resulting problem secondly we introduce a gaussian kernel version of the bilevel programming formulation of mssc named gkmssc the gkmssc problem is formulated as a dc program for which a simple and efficient dca scheme is developed a regularization technique is investigated for exploiting the nice effect of dc decomposition and a simple procedure for finding good starting points of dca is developed the proposed dca schemes are original and very inexpensive because they amount to computing at each iteration the projection of points onto a simplex and or onto a ball and or onto a box which are all determined in the explicit form numerical results on real word datasets show the efficiency the scalability of dca and its great superiority with respect to k means and kernel k means standard methods for clustering highlightsthe purpose of this paper is to develop new efficient approaches based on dc programming and dca for clustering we consider the two models for the minimum sum of squares clustering mssc a mixed integer program and a bilevel programming formulation the mixed integer program is reformulated as a dc program via an exact penalty technique for which dca is investigated a gaussian kernel version of the bilevel programming formulation is introduced and a simple and efficient dca scheme is developed the two main features of dca say the effect of dc decomposition and of starting points are carefully studied

a new nonsmooth optimization algorithm for minimum sum of squares clustering problems the minimum sum of squares clustering problem is formulated as a problem of nonsmooth nonconvex optimization and an algorithm for solving the former problem based on nonsmooth optimization techniques is developed the issue of applying this algorithm to large data sets is discussed results of numerical experiments have been presented which demonstrate the effectiveness of the proposed algorithm

spectral methods for graph clustering a survey graph clustering is an area in cluster analysis that looks for groups of related vertices in a graph due to its large applicability several graph clustering algorithms have been proposed in the last years a particular class of graph clustering algorithms is known as spectral clustering algorithms these algorithms are mostly based on the eigen decomposition of laplacian matrices of either weighted or unweighted graphs this survey presents different graph clustering formulations most of which based on graph cut and partitioning problems and describes the main spectral clustering algorithms found in literature that solve these problems

new diagonal bundle method for clustering problems in large data sets abstract clustering is one of the most important tasks in data mining recent developments in computer hardware allow us to store in random access memory ram and repeatedly read data sets with hundreds of thousands and even millions of data points this makes it possible to use conventional clustering algorithms in such data sets however these algorithms may need prohibitively large computational time and fail to produce accurate solutions therefore it is important to develop clustering algorithms which are accurate and can provide real time clustering in large data sets this paper introduces one of them using nonsmooth optimization formulation of the clustering problem the objective function is represented as a difference of two convex dc functions then a new diagonal bundle algorithm that explicitly uses this structure is designed and combined with an incremental approach to solve this problem the method is evaluated using real world data sets with both large number of attributes and large number of data points the proposed method is compared with two other clustering algorithms using numerical results

roof duality complementation and persistency in quadratic 0 1 optimization the paper is concerned with the primal problem of maximizing a given quadratic pseudo boolean function four equivalent problems are discussed the primal the complementation the discrete rhys lp and the weighted stability problem of a sam graph each of them has a relaxation the roof dual the quadratic complementation the continuous rhys lp and the fractional weighted stability problem of a sam graph the main result is that the four gaps associated with the four relaxations are equal furthermore a solution to any of these problems leads at once to solutions of the other three equivalent ones the four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem the optimal solutions of the roof dual define best linear majorantsp x off having the following persistency property if theith coefficient inp is positive negative thenx i 1 0 in every optimum of the primal problem several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal on the other hand a class of gap free functions properly including the supermodular ones is exhibited

facets of the clique partitioning polytope a subseta of the edge set of a graphg v e is called a clique partitioning ofg is there is a partition of the node setv into disjoint setsw 1 w k such that eachw i induces a clique i e a complete but not necessarily maximal subgraph ofg and such thata 1 uv u v w i u v given weightsw e for alle e the clique partitioning problem is to find a clique partitioninga ofg such that e a w e is as small as possible this problem known to be hard see wakabayashi 1986 comes up for instance in data analysis and here the underlying graphg is typically a complete graph in this paper we study the clique partitioning polytope of the complete graphk n i e is the convex hull of the incidence vectors of the clique partitionings ofk n we show that triangles 2 chorded odd cycles 2 chorded even wheels and other subgraphs ofk n induce facets of the theoretical results described here have been used to design an empirically efficient cutting plane algorithm with which large real world instances of the clique partitioning problem could be solved these computational results can be found in grotschel and wakabayashi 1989

power spectra methods for a stochastic description of diffusion on deterministically growing domains a central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity generally the mathematical treatments of biological systems have used continuum mean field hypotheses for their constituent parts which ignores any sources of intrinsic stochastic effects in this paper we consider a stochastic space jump process as a description of diffusion i e particles are able to undergo a random walk on a discretized domain by developing analytical fourier methods we are able to probe this probabilistic framework which gives us insight into the patterning potential of diffusive systems further an alternative description of domain growth is introduced with which we are able to rigorously link the mean field and stochastic descriptions finally through combining these ideas it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean field steady state

predator prey cycles from resonant amplification of demographic stochasticity we present the simplest individual level model of predator prey dynamics and show via direct calculation that it exhibits cycling behavior the deterministic analogue of our model recovered when the number of individuals is infinitely large is the volterra system with density dependent prey reproduction which is well known to fail to predict cycles this difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite our results indicate that additional biological mechanisms such as predator satiation may not be necessary to explain observed predator prey cycles in real finite populations

aggregate codifferential method for nonsmooth dc optimization a new algorithm is developed based on the concept of codifferential for minimizing the difference of convex nonsmooth functions since the computation of the whole codifferential is not always possible we use a fixed number of elements from the codifferential to compute the search directions the convergence of the proposed algorithm is proved the efficiency of the algorithm is demonstrated by comparing it with the subgradient the truncated codifferential and the proximal bundle methods using nonsmooth optimization test problems

extended formulations for the a cut problem letg v e be an undirected graph anda v we consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes ina we consider two extended formulations using both node and edge variables an edge variable formulation has previously been considered for this problem chopra and rao 1991 cunningham 1991 we show that the lp relaxations of the extended formulations are stronger than the lp relaxation of the edge variable formulation even with an extra class of valid inequalities added this is interesting because while the lp relaxations of the extended formulations can be solved in polynomial time the lp relaxation of the edge variable formulation cannot we also give a class of valid inequalities for one of the extended formulations computational results using the extended formulations are performed

multiple kernel learning algorithms in recent years several methods have been proposed to combine multiple kernels instead of using a single one these different kernels may correspond to using different notions of similarity or may be using information coming from multiple sources different representations or different feature subsets in trying to organize and highlight the similarities and differences between them we give a taxonomy of and review several multiple kernel learning algorithms we perform experiments on real data sets for better illustration and comparison of existing algorithms we see that though there may not be large differences in terms of accuracy there is difference between them in complexity as given by the number of stored support vectors the sparsity of the solution as given by the number of used kernels and training time complexity we see that overall using multiple kernels instead of a single one is useful and believe that combining kernels in a nonlinear or data dependent way seems more promising than linear combination in fusing information provided by simple linear kernels whereas linear methods are more reasonable when combining complex gaussian kernels

automatic relevance determination in nonnegative matrix factorization with the beta divergence this paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization nmf with the divergence the divergence is a family of cost functions that includes the squared euclidean distance kullback leibler kl and itakura saito is divergences as special cases learning the model order is important as it is necessary to strike the right balance between data fidelity and overfitting we propose a bayesian model based on automatic relevance determination ard in which the columns of the dictionary matrix and the rows of the activation matrix are tied together through a common scale parameter in their prior a family of majorization minimization mm algorithms is proposed for maximum a posteriori map estimation a subset of scale parameters is driven to a small lower bound in the course of inference with the effect of pruning the corresponding spurious components we demonstrate the efficacy and robustness of our algorithms by performing extensive experiments on synthetic data the swimmer dataset a music decomposition example and a stock price prediction task

j means a new local search heuristic for minimum sum of squares clustering abstract a new local search heuristic called j m eans is proposed for solving the minimum sum of squares clustering problem the neighborhood of the current solution is defined by all possible centroid to entity relocations followed by corresponding changes of assignments moves are made in such neighborhoods until a local optimum is reached the new heuristic is compared with two other well known local search heuristics k and h m eans as well as with h m eans an improved version of the latter in which degeneracy is removed moreover another heuristic which fits into the variable neighborhood search metaheuristic framework and uses j m eans in its local search step is proposed too results on standard test problems from the literature are reported it appears that j m eans outperforms the other local search methods quite substantially when many entities and clusters are considered

the hyperbolic smoothing clustering method the minimum sum of squares clustering problem is considered the mathematical modeling of this problem leads to a min sum min formulation which in addition to its intrinsic bi level nature has the significant characteristic of being strongly nondifferentiable to overcome these difficulties the resolution method proposed adopts a smoothing strategy using a special c differentiable class function the final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem the use of this technique called hyperbolic smoothing allows the main difficulties presented by the original problem to be overcome a simplified algorithm containing only the essentials of the method is presented for the purpose of illustrating both the reliability and the efficiency of the method a set of computational experiments was performed making use of traditional test problems described in the literature

modified global k means algorithm for minimum sum of squares clustering problems k means algorithm and its variations are known to be fast clustering algorithms however they are sensitive to the choice of starting points and inefficient for solving clustering problems in large data sets recently a new version of the k means algorithm the global k means algorithm has been developed it is an incremental algorithm that dynamically adds one cluster center at a time and uses each data point as a candidate for the k th cluster center results of numerical experiments show that the global k means algorithm considerably outperforms the k means algorithms in this paper a new version of the global k means algorithm is proposed a starting point for the k th cluster center in this algorithm is computed by minimizing an auxiliary cluster function results of numerical experiments on 14 data sets demonstrate the superiority of the new algorithm however it requires more computational time than the global k means algorithm

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

sparse partial least squares regression for simultaneous dimension reduction and variable selection partial least squares regression has been an alternative to ordinary least squares for handling multicollinearity in several areas of scientific research since the 1960s it has recently gained much attention in the analysis of high dimensional genomic data we show that known asymptotic consistency of the partial least squares estimator for a univariate response does not hold with the very large p and small n paradigm we derive a similar result for a multivariate response regression with partial least squares we then propose a sparse partial least squares formulation which aims simultaneously to achieve good predictive performance and variable selection by producing sparse linear combinations of the original predictors we provide an efficient implementation of sparse partial least squares regression and compare it with well known variable selection and dimension reduction approaches via simulation experiments we illustrate the practical utility of sparse partial least squares regression in a joint analysis of gene expression and genomewide binding data

a least squares algorithm for fitting additive trees to proximity data a least squares algorithm for fitting additive trees to proximity data is described the algorithm uses a penalty function to enforce the four point condition on the estimated path length distances the algorithm is evaluated in a small monte carlo study finally an illustrative application is presented

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

nonsmooth dc programming approach to the minimum sum of squares clustering problems abstract this paper introduces an algorithm for solving the minimum sum of squares clustering problems using their difference of convex representations a non smooth non convex optimization formulation of the clustering problem is used to design the algorithm characterizations of critical points stationary points in the sense of generalized gradients and inf stationary points of the clustering problem are given the proposed algorithm is tested and compared with other clustering algorithms using large real world data sets

formulations and valid inequalities for the node capacitated graph partitioning problem we investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition the objective is to minimize the sum of the costs of the edges between the subsets of the partition this problem has a variety of applications for instance in the design of electronic circuits and devices we present alternative integer programming formulations for this problem and discuss the links between these formulations having chosen to work in the space of edges of the multicut we investigate the convex hull of incidence vectors of feasible multicuts in particular several classes of inequalities are introduced and their strength and robustness are analyzed as various problem parameters change

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

computational study of large scale p median problems given a directed graph g v a the p median problem consists of determining p nodes the median nodes minimizing the total distance from the other nodes of the graph we present a branch and cut and price algorithm yielding provably good solutions for instances with v 3795 key ingredients of the algorithm are a delayed column and row generation technique exploiting the special structure of the formulation to solve the lp relaxation and cutting planes to strengthen the formulation and limit the size of the enumeration tree

robust ecological pattern formation induced by demographic noise we demonstrate that demographic noise can induce persistent spatial pattern formation and temporal oscillations in the levin segel predator prey model for plankton herbivore population dynamics although the model exhibits a turing instability in mean field theory demographic noise greatly enlarges the region of parameter space where pattern formation occurs to distinguish between patterns generated by fluctuations and those present at the mean field level in real ecosystems we calculate the power spectrum in the noise driven case and predict the presence of fat tails not present in the mean field case these results may account for the prevalence of large scale ecological patterns beyond that expected from traditional nonstochastic approaches

kernelized bayesian matrix factorization we extend kernelized matrix factorization with a full bayesian treatment and with an ability to work with multiple side information sources expressed as different kernels kernels have been introduced to integrate side information about the rows and columns which is necessary for making out of matrix predictions we discuss specifically binary output matrices but extensions to realvalued matrices are straightforward we extend the state of the art in two key aspects i a full conjugate probabilistic formulation of the kernelized matrix factorization enables an efficient variational approximation whereas full bayesian treatments are not computationally feasible in the earlier approaches ii multiple side information sources are included treated as different kernels in multiple kernel learning which additionally reveals which side sources are informative we then show that the framework can also be used for supervised and semi supervised multilabel classification and multi output regression by considering samples and outputs as the domains where matrix factorization operates our method outperforms alternatives in predicting drug protein interactions on two data sets on multilabel classification our algorithm obtains the lowest hamming losses on 10 out of 14 data sets compared to five state of the art multilabel classification algorithms we finally show that the proposed approach outperforms alternatives in multi output regression experiments on a yeast cell cycle data set

influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise numerous mathematical models exploring the emergence of complexity within developmental biology incorporate diffusion as the dominant mechanism of transport however self organizing paradigms can exhibit the biologically undesirable property of extensive sensitivity as illustrated by the behavior of the french flag model in response to intrinsic noise and turing s model when subjected to fluctuations in initial conditions domain growth is known to be a stabilizing factor for the latter though the interaction of intrinsic noise and domain growth is underexplored even in the simplest of biophysical settings previously we developed analytical fourier methods and a description of domain growth that allowed us to characterize the effects of deterministic domain growth on stochastically diffusing systems in this paper we extend our analysis to encompass stochastically growing domains this form of growth can be used only to link the meso and macroscopic domains as the box splitting form of growth on the microscopic scale has an ill defined thermodynamic limit the extension is achieved by allowing the simulated particles to undergo random walks on a discretized domain while stochastically controlling the length of each discretized compartment due to the dependence of diffusion on the domain discretization we find that the description of diffusion cannot be uniquely derived we apply these analytical methods to two justified descriptions where it is shown that under certain conditions diffusion is able to support a consistent inhomogeneous state that is far removed from the deterministic equilibrium without additional kinetics finally a logistically growing domain is considered not only does this show that we can deal with nonmonotonic descriptions of stochastic growth but it is also seen that diffusion on a stationary domain produces different effects to diffusion on a domain that is stationary on average

an improved column generation algorithm for minimum sum of squares clustering given a set of entities associated with points in euclidean space minimum sum of squares clustering mssc consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized a column generation algorithm for mssc was given by du merle et al in siam journal scientific computing 21 1485 1505 the bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost we propose a new way to solve this auxiliary problem based on geometric arguments this greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2 300 entities i e more than 10 times as much as previously done

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

uobyqa unconstrained optimization by quadratic approximation uobyqa is a new algorithm for general unconstrained optimization calculations that takes account of the curvature of the objective function f say by forming quadratic models by interpolation therefore because no first derivatives are required each model is defined by n 1 n 2 values of f where n is the number of variables and the interpolation points must have the property that no nonzero quadratic polynomial vanishes at all of them a typical iteration of the algorithm generates a new vector of variables widetilde underline x t say either by minimizing the quadratic model subject to a trust region bound or by a procedure that should improve the accuracy of the model then usually f widetilde underline x t is obtained and one of the interpolation points is replaced by widetilde underline x t therefore the paper addresses the initial positions of the interpolation points the adjustment of trust region radii the calculation of widetilde underline x t in the two cases that have been mentioned and the selection of the point to be replaced further uobyqa works with the lagrange functions of the interpolation equations explicitly so their coefficients are updated when an interpolation point is moved the lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation to f which allows the algorithm to achieve a fast rate of convergence these features are discussed and a summary of the algorithm is given finally a fortran implementation of uobyqa is applied to several choices of f in order to investigate accuracy robustness in the presence of rounding errors the effects of first derivative discontinuities and the amount of work the numerical results are very promising for n 20 but larger values are problematical because the routine work of an iteration is of fourth order in the number of variables

a recursive lagrangian method for clustering problems abstract the problem of partitioning a finite set of objects into subsets or clusters so as to optimize an objective function is treated it is shown that for the particular case where all the objects lie upon a line the clustering problem may be solved exactly by an algorithm utilizing only lagrangian relaxation and dynamic programming by exploiting the approximately unimodal shape of the envelope of optimal objective values a search procedure in the space is devised and this allows the solution in reasonable computation time of the combinatorially large problem several examples are presented and the method is applied to one set of noncollinear data which is first transformed so as to become collinear further promising directions for computational enhancement are noted the important aspects of computer implementation are given

pattern production through a chiral chasing mechanism recent experiments on zebrafish pigmentation suggests that their typical black and white striped skin pattern is made up of a number of interacting chromatophore families specifically two of these cell families have been shown to interact through a nonlocal chasing mechanism which has previously been modeled using integro differential equations we extend this framework to include the experimentally observed fact that the cells often exhibit chiral movement in that the cells chase and run away at angles different to the line connecting their centers this framework is simplified through the use of multiple small limits leading to a coupled set of partial differential equations which are amenable to fourier analysis this analysis results in the production of dispersion relations and necessary conditions for a patterning instability to occur beyond the theoretical development and the production of new pattern planiforms we are able to corroborate the experimental hypothesis that the global pigmentation patterns can be dependent on the chirality of the chromatophores

min cut clustering we describe a decomposition framework and a column generation scheme for solving a min cut clustering problem the subproblem to generate additional columns is itself an np hard mixed integer programming problem we discuss strong valid inequalities for the subproblem and describe some efficient solution strategies computational results on compiler construction problems are reported

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

fuzzy j means a new heuristic for fuzzy clustering abstract a fuzzy clustering problem consists of assigning a set of patterns to a given number of clusters with respect to some criteria such that each of them may belong to more than one cluster with different degrees of membership in order to solve it we first propose a new local search heuristic called fuzzy j means where the neighbourhood is defined by all possible centroid to pattern relocations the integer solution is then moved to a continuous one by an alternate step i e by finding centroids and membership degrees for all patterns and clusters to alleviate the difficulty of being stuck in local minima of poor value this local search is then embedded into the variable neighbourhood search metaheuristic results on five standard test problems from the literature are reported and compared with those obtained with the well known fuzzy c means heuristic it appears that solutions of substantially better quality are obtained with the proposed methods than with this former one

some heuristic methods for solving p median problems with a coverage constraint abstract the aim of this paper is to solve p median problems with an additional coverage constraint these problems arise in location applications when the trade off between distance and coverage is being calculated three kinds of heuristic algorithms are developed first local search procedures are designed both for constructing and improving feasible solutions second a multistart grasp heuristic is developed based on the previous local search methods third by employing lagrangean relaxation methods a very efficient lagrangean heuristic algorithm is designed which extends the well known algorithm of handler and zang for constrained shortest path problems to constrained p median problems finally a comparison of the computational efficiency of the developed methods is made between a variety of problems of different sizes

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

a vns heuristic for escaping local extrema entrapment in normalized cut clustering normalized cut is one of the most popular graph clustering criteria the main approaches proposed for its resolution are spectral clustering methods and a multilevel approach of dhillon et al tpami 29 1944 1957 2007 called graclus their aim is to obtain good solutions in a small amount of time for large instances metaheuristics are general frameworks for stochastic searches often employed in global optimization to improve the solutions obtained by other heuristics variable neighborhood search vns is a metaheuristic which exploits systematically the idea of neighborhood change during the search in this paper we propose a vns heuristic for normalized cut segmentation computational experiments show that in most cases this vns heuristic improves significantly and in moderate time the solutions obtained by the current state of the art algorithms i e graclus and a spectral method proposed by yu and shi iccv 2003

formulating logical implications in combinatorial optimisation abstract when practical problems are formulated as combinatorial optimisation models one must often include logical implications between decisions it is useful to express these implications as linear constraints involving binary variables since linear constraints offer the possibility of using linear programming and branch and bound as an initial solution method often this formulation step of the modelling process seems far from evident to many practitioners students and researchers therefore it is of interest to make simple rules available to clarify and help in this process in this educationally oriented tutorial paper we introduce such a rule lip the logical implication principle it offers an easy and automatic way to translate elementary logical implications involving 0 1 variables into linear constraints based on an extremely simple cutting plane we demonstrate how the rule is extended using complementarity and implicit binary variables and leads to a simple but powerful instrument this is illustrated by several examples of applications in various fields of operational research the paper culminates in the description of a novel set of constraints which fully eliminate all permutation equivalent solutions to numbered clustering problems

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

quasicycles in a spatial predator prey model we use spatial models of simple predator prey interactions to predict that predator and prey numbers oscillate in time and space these oscillations are not seen in the deterministic versions of the models but are due to stochastic fluctuations about the time independent solutions of the deterministic equations which are amplified due to the existence of a resonance we calculate the power spectra of the fluctuations analytically and show that they agree well with results obtained from stochastic simulations this work extends the analysis of these quasicycles from that previously developed for well mixed systems to spatial systems and shows that the ideas and methods used for nonspatial models naturally generalize to the spatial case

nonsmooth nonconvex optimization approach to clusterwise linear regression problems abstract clusterwise regression consists of finding a number of regression functions each approximating a subset of the data in this paper a new approach for solving the clusterwise linear regression problems is proposed based on a nonsmooth nonconvex formulation we present an algorithm for minimizing this nonsmooth nonconvex function this algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function a special procedure is introduced to generate a good starting point for solving global optimization problems at each iteration of the incremental algorithm such an approach allows one to find global or near global solution to the problem when the data sets are sufficiently dense the algorithm is compared with the multistart spath algorithm on several publicly available data sets for regression analysis

an interior point algorithm for minimum sum of squares clustering an exact algorithm is proposed for minimum sum of squares nonhierarchical clustering i e for partitioning a given set of points from a euclidean m space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong this problem is expressed as a constrained hyperbolic program in 0 1 variables the resolution method combines an interior point algorithm i e a weighted analytic center column generation method with branch and bound the auxiliary problem of determining the entering column i e the oracle is an unconstrained hyperbolic program in 0 1 variables with a quadratic numerator and linear denominator it is solved through a sequence of unconstrained quadratic programs in 0 1 variables to accelerate resolution variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region proved minimum sum of squares partitions are determined for the first time for several fairly large data sets from the literature including fisher s 150 iris

on the complexity of clustering with relaxed size constraints in fixed dimension abstract we study the computational complexity of the problem of computing an optimal clustering a 1 a 2 a k of a set of points assuming that every cluster size a i belongs to a given set m of positive integers we present a polynomial time algorithm for solving the problem in dimension 1 i e when the points are simply rational values for an arbitrary set m of size constraints which extends to the l 1 norm an analogous procedure known for the euclidean norm moreover we prove that in dimension 2 assuming euclidean norm the problem is strongly np hard with size constraints m 2 4 this result is extended also to the size constraints m 2 3 both in the case of euclidean and l 1 norm