improvement and comparison of heuristics for solving the uncapacitated multisource weber problem the multisource weber problem is to locate simultaneouslym facilities in the euclidean plane to minimize the total transportation cost for satisfying the demand ofn fixed users each supplied from its closest facility many heuristics have been proposed for this problem as well as a few exact algorithms heuristics are needed to solve quickly large problems and to provide good initial solutions for exact algorithms we compare various heuristics i e alternative location allocation cooper 1964 projection bongartz et al 1994 tabu search brimberg and mladenovic 1996a p median plus weber hansen et al 1996 genetic search and several versions of variable neighbourhood search based on empirical tests that are reported it is found that most traditional and some recent heuristics give poor results when the number of facilities to locate is large and that variable neighbourhood search gives consistently best results on average in moderate computing time

a variable selection heuristic for k means clustering one of the most vexing problems in cluster analysis is the selection and or weighting of variables in order to include those that truly define cluster structure while eliminating those that might mask such structure this paper presents a variable selection heuristic for nonhierarchical k means cluster analysis based on the adjusted rand index for measuring cluster recovery the heuristic was subjected to monte carlo testing across more than 2200 datasets with known cluster structure the results indicate the heuristic is extremely effective at eliminating masking variables a cluster analysis of real world financial services data revealed that using the variable selection heuristic prior to the k means algorithm resulted in greater cluster stability

a tabu search heuristic for deterministic two mode blockmodeling of binary network matrices two mode binary data matrices arise in a variety of social network contexts such as the attendance or non attendance of individuals at events the participation or lack of participation of groups in projects and the votes of judges on cases a popular method for analyzing such data is two mode blockmodeling based on structural equivalence where the goal is to identify partitions for the row and column objects such that the clusters of the row and column objects form blocks that are either complete all 1s or null all 0s to the greatest extent possible multiple restarts of an object relocation heuristic that seeks to minimize the number of inconsistencies i e 1s in null blocks and 0s in complete blocks with ideal block structure is the predominant approach for tackling this problem as an alternative we propose a fast and effective implementation of tabu search computational comparisons across a set of 48 large network matrices revealed that the new tabu search heuristic always provided objective function values that were better than those of the relocation heuristic when the two methods were constrained to the same amount of computation time

statistical analysis of multiple sociometric relations abstract loglinear models are adapted for the analysis of multivariate social networks a set of sociometric relations among a group of actors models that focus on the similarities and differences between the relations and models that concentrate on individual actors are discussed this approach allows for the partitioning of the actors into blocks or subgroups some ideas for combining these models are described and the various models and computational methods are applied to the analysis of data for a corporate interlock network of the 25 largest organizations in minneapolis st paul and for a classic network of 18 monks in a cloister the computational techniques all involve variations on the standard iterative proportional fitting algorithm used extensively in the analysis of multidimensional contingency tables

conformity of two sociometric relations the problem of comparing two sociometric relations or measurements a andb recorded in distinct sociomatrices was originally discussed by katz and powell in the early 1950 s and hubert and baker in the late 1970 s the problem is considered again using a probabilistic model designed specifically for discrete valued network measurements the model allows for the presence of various structural tendencies such as reciprocity and differential popularity a parameter that isolates the tendency for actors to choose other actors on both relations simultaneously is introduced and estimated conditional on the presence of other parameters that reflect additional important network properties the parameter is presented as a symmetric index but is also generalized to the predictive a onb orb ona situation this approach to the problem is illustrated with the same data used by the earlier solutions and the unique nature of the two relations in the data set a received choices b perceived choices as it affects the modeling is discussed significance tests for the parameter and related parameters are described as well as an extension to more than two relations

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

an exponential family of probability distributions for directed graphs abstract directed graph or digraph data arise in many fields especially in contemporary research on structures of social relationships we describe an exponential family of distributions that can be used for analyzing such data a substantive rationale for the general model is presented and several special cases are discussed along with some possible substantive interpretations a computational algorithm based on iterative scaling procedures for use in fitting data is described as are the results of a pilot simulation study an example using previously reported empirical data is worked out in detail an extension to multiple relationship data is discussed briefly

the median tour and maximal covering tour problems formulations and heuristics abstract in this paper the authors introduce two bicriterion routing problems the median tour problem mtp and the maximal covering tour problem mctp in both problems the tour must visit only p of the n nodes on the network in addition both problems have as one of their objectives the minimization of total tour length the second objective in both problems maximizes access to the tour for the nodes not directly on it in the mtp the access objective is the minimization of the total travel distance necessary for demand at the nodes to reach their nearest stop on the tour in the mctp the access objective is to maximize the total demand within some prespecified maximal travel distance from a tour stop it is shown that the mctp may be viewed as a special case of the mtp consequently solution procedures developed for the mtp may also be employed to solve the mctp unfortunately both problems are np hard in addition the number of efficient solutions may grow exponentially with the number of nodes in the particular problem instance consequently a heuristic procedure is developed to generate a good approximation of the efficient frontier not only does this approximation provide the decision marker with insight into the magnitudes of the inherent tradeoffs between the two objectives but it also provides him or her with a set of feasible options to consider computational experience on a realistically scaled problem is provided applications of the problems include the design of mobile service delivery systems e g healthcare delivery in rural areas of developing countries bi modal transportation systems e g overnight mail delivery and distributed computer networks among others

multiobjective blockmodeling for social network analysis to date most methods for direct blockmodeling of social network data have focused on the optimization of a single objective function however there are a variety of social network applications where it is advantageous to consider two or more objectives simultaneously these applications can broadly be placed into two categories 1 simultaneous optimization of multiple criteria for fitting a blockmodel based on a single network matrix and 2 simultaneous optimization of multiple criteria for fitting a blockmodel based on two or more network matrices where the matrices being fit can take the form of multiple indicators for an underlying relationship or multiple matrices for a set of objects measured at two or more different points in time a multiobjective tabu search procedure is proposed for estimating the set of pareto efficient blockmodels this procedure is used in three examples that demonstrate possible applications of the multiobjective blockmodeling paradigm

clustering qualitative data based on binary equivalence relations neighborhood search heuristics for the clique partitioning problem the clique partitioning problem cpp requires the establishment of an equivalence relation for the vertices of a graph such that the sum of the edge costs associated with the relation is minimized the cpp has important applications for the social sciences because it provides a framework for clustering objects measured on a collection of nominal or ordinal attributes in such instances the cpp incorporates edge costs obtained from an aggregation of binary equivalence relations among the attributes we review existing theory and methods for the cpp and propose two versions of a new neighborhood search algorithm for efficient solution the first version ns r uses a relocation algorithm in the search for improved solutions whereas the second ns ts uses an embedded tabu search routine the new algorithms are compared to simulated annealing sa and tabu search ts algorithms from the cpp literature although the heuristics yielded comparable results for some test problems the neighborhood search algorithms generally yielded the best performances for large and difficult instances of the cpp

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

clustering with block mixture models abstract basing cluster analysis on mixture models has become a classical and powerful approach until now this approach which allows to explain some classic clustering criteria such as the well known k means criteria and to propose general criteria has been developed to classify a set of objects measured on a set of variables but for this kind of data if most clustering procedures are designated to construct an optimal partition of objects or sometimes of variables there exist others methods named block clustering methods which consider simultaneously the two sets and organize the data into homogeneous blocks in this work a new mixture model called block mixture model is proposed to take into account this situation this model allows to embed simultaneous clustering of objects and variables in a mixture approach we first consider this probabilistic model in a general context and we develop a new algorithm of simultaneous partitioning based on the cem algorithm then we focus on the case of binary data and we show that our approach allows us to extend a block clustering method which had been proposed in this case simplicity fast convergence and the possibility to process large data sets are the major advantages of the proposed approach

network distance characteristics that affect computational effort in p median location problems abstract in solving location models the effort expended and the quality of the solutions obtained often varies significantly from one problem instance to another our conjecture is that this is not a random occurrence but instead can be correlated with characteristics of the network upon which the model is constructed in this paper we examine a variety of approaches for generating networks for the p median model the types of networks studied include those based on euclidean inter point distances shortest paths developed from a predefined network and randomly generated distance matrices in addition to the process by which the network is constructed we consider the distribution of distances symmetry and the satisfaction of the triangle inequality all of these characteristics influence the effort required to solve the p median model we have found however that the triangle inequality has the most significant impact

a genetic algorithm for the p median problem with pos neg weights let a connected undirected graph g v e be given in the classical p median problem we want to find a set x containing p points in g such that the sum of weighted distances from x to all vertices in v is minimized we consider the semi obnoxious case where every vertex has either a positive or negative weight in this case we have two different objective functions the sum of the minimum weighted distances from x to all vertices and the sum of the weighted minimum distances in this paper we propose a genetic algorithm for both problems computational results are compared with a previously investigated variable neighborhood search algorithm

clustering functional data into groups by using projections we show that in the functional data context by appropriately exploiting the functional nature of the data it is possible to cluster the observations asymptotically perfectly we demonstrate that this level of performance can sometimes be achieved by the k means algorithm as long as the data are projected on a carefully chosen finite dimensional space in general the notion of an ideal cluster is not clearly defined we derive our results in the setting where the data come from two populations whose distributions differ at least in terms of means and where an ideal cluster corresponds to one of these two populations we propose an iterative algorithm to choose the projection functions in a way that optimizes clustering performance where to avoid peculiar solutions we use a weighted least squares criterion we apply our iterative clustering procedure on simulated and real data where we show that it works well

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

stochastic blockmodels for directed graphs abstract holland and leinhardt 1981 proposed the p 1 model for the analysis of binary directed graph data in network studies such a model provides information about the attractiveness and expansiveness of the individual nodes in the network as well as the tendency of a pair of nodes to reciprocate relational ties when the nodes are a priori partitioned into subgroups based on attributes such as race and sex the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups thus creating a situation called blocking in social networks the p 1 model completely ignores this extra piece of information and is therefore unable to explain the block structure blockmodels that are simple extensions of the p 1 model are proposed specifically for such data an iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood the methodology is illustrated in detail on two empirical examples

neighborhood search heuristics for the uncapacitated facility location problem abstract the uncapacitated facility location problem is one of choosing sites among a set of candidates in which facilities can be located so that the demands of a given set of clients are satisfied at minimum costs applications of neighborhood search methods to this problem have not been reported in the literature in this paper we first describe and compare several neighborhood structures used by local search to solve this problem we then describe neighborhood search heuristics based on tabu search and complete local search with memory to solve large instances of the uncapacitated facility location problem our computational experiments show that on medium sized problem instances both these heuristics return solutions with costs within 0 075 of the optimal with execution times that are often several orders of magnitude less than those required by exact algorithms on large sized instances the heuristics generate low cost solutions quite fast and terminate with solutions whose costs are within 0 0345 of each other

solving the p median problem with pos neg weights by variable neighborhood search and some results for special cases the p median problem with positive and negative weights has been introduced by burkard and krarup computing 60 1998 193 in this paper we discuss some special cases of this problem on trees and propose a variable neighborhood search procedure for general networks which is in fact a modification of the one proposed by hansen and mladenovic locat sci 5 1997 207 for the p median we also compare the results with those obtained by a tabu search procedure

transforming contingency tables abstract we present a general theorem on transforming contingency tables and several applications where the transformation technique has allowed us to take advantage of the iterative proportional fitting procedure and has resulted in simple and useful procedures a further advantage of this technique is that it is sometimes possible to recognize closed form estimates in the transformed problem while they would be overlooked in the original setting we shall view the estimation problem as one of minimizing the kullback leibler information distance between two probability mass functions and will roughly follow the notation of csiszar 1976 although we have adopted the information distance point of veiw the duality between maximum likelihood estimation and minimum information estimation see e g darroch and ratcliff 1972 implies that the results of this paper can just as well be interpreted from the maximum likelihood point of view

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

finding cluster centers and sizes via multinomial parameterization the clustering problem consists in dividing a data set into groups of observations that are similar within but different across this paper presents a method for assessment the clusters centers and sizes in a non linear least squares optimization with multinomial parameterization the method is especially useful for large data sets as it operates on the summary statistics only this approach also works for the problem of finding clusters centers and sizes by the covariance matrix when the original data is not available estimation of the clusters centers and sizes can be followed by actual clustering example of application to marketing research problem is discussed

quantum and classical mechanics of q deformed systems quantum and classical mechanics of a system of q deformed bosonic oscillators are considered the q deformed heisenberg weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables the corresponding hilbert space is constructed in this approach the deformation parameter turns out to be a function of the planck constant an oscillator frequency and a parameter with dimension of length the hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q deformed classical theory the phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory the dimensional parameter introduced determines its radius it is argued that the q deformation can be associated with a special non canonical transformation the principle of least action for the classical q deformed system is formulated a representation of uq m in a space of functions on a phase space spanned by commutative coordinates is constructed

a gamma heuristic for the p median problem heuristic concentration hc is a two stage metaheuristic that can be applied to a wide variety of combinatorial problems it is particularly suited to location problems in which the number of facilities is given in advance in such settings the first stage of hc repeatedly applies some random start interchange or other heuristic to produce a number of alternative facility configurations a subset of the best of these alternatives is collected and the union of the facility sites in them is called a concentration set cs among the component elements of the cs are likely to be included those sites which are members of the optimal solution in earlier studies the second stage of hc has consisted of an exact procedure to extract the best possible solution from the cs in this paper we demonstrate for the p median problem the use of two sequentially active heuristics in the second stage of hc that is we offer two additional layers of heuristics to improve solutions which are found in the first stage of hc thus we are describing a variant of the hc metaheuristic consisting of three layers of heuristics which are utilized in sequence we propose for this procedure the name of gamma heuristic

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

an overview of representative problems in location research we present a survey of over 50 representative problems in location research our goal is not to review all variants of different location models or to describe solution results but rather to provide a broad overview of major location problems that have been studied indicating briefly how they are formulated and how they relate to one another we review standard problems such as median center and warehouse location problems as well as less traditional location problems which have emerged in recent years our primary focus is on problems for which operations research type models have been developed most of the problems we review have been formulated as optimization problems

the p median problem a survey of metaheuristic approaches the p median problem is one of the basic models in discrete location theory as with most location problems it is classified as np hard and so heuristic methods are usually used to solve it metaheuristics are frameworks for building heuristics in this survey we examine the p median with the aim of providing an overview on advances in solving it using recent procedures based on metaheuristic rules

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

experiments in projection and clustering by simulated annealing simulated annealing is a stochastic relaxation algorithm which has been used successfully to optimize functions of many variables this paper analyzes the simulated annealing algorithm when applied to the minimization of functions from two common problems encountered in exploratory pattern analysis projection and clustering the projection is a nonlinear mapping of patterns in high dimension to two dimensions the simulated annealing mapping is compared to gradient descent minimization of the same objective function as well as eigenvector projection the simulated annealing clustering is compared to a k means algorithm empirical results show that simulated annealing can produce results as good as those obtained by conventional methods but are impractical for small data sets because of the high computational cost simulated annealing does in the case of the mapping problem yield a better optimization and better retained structure for large data sets containing tight gaussian clusters

optimizing transformations of structural balance in signed networks with potential relationships abstract a signed network includes positive edges negative edges and 0 edges the last of which denote potential relationships however 0 edges are commonly ignored in transformations of unbalanced networks in this paper we take 0 edges into account and solve the optimization problem in a more comprehensive way we transform the unbalanced network by not only changing signs of edges but also changing edges into potential the experimental results show that our method can solve this problem efficiently and that our solutions are cost saving

a branch and bound algorithm for fitting anti robinson structures to symmetric dissimilarity matrices the seriation of proximity matrices is an important problem in combinatorial data analysis and can be conducted using a variety of objective criteria some of the most popular criteria for evaluating an ordering of objects are based on anti robinson forms which reflect the pattern of elements within each row and or column of the reordered matrix when moving away from the main diagonal this paper presents a branch and bound algorithm that can be used to seriate a symmetric dissimilarity matrix by identifying a reordering of rows and columns of the matrix optimizing an anti robinson criterion computational results are provided for several proximity matrices from the literature using four different anti robinson criteria the results suggest that with respect to computational efficiency the branch and bound algorithm is generally competitive with dynamic programming further because it requires much less storage than dynamic programming the branch and bound algorithm can provide guaranteed optimal solutions for matrices that are too large for dynamic programming implementations

a repetitive branch and bound procedure for minimum within cluster sums of squares partitioning minimization of the within cluster sums of squares wcss is one of the most important optimization criteria in cluster analysis although cluster analysis modules in commercial software packages typically use heuristic methods for this criterion optimal approaches can be computationally feasible for problems of modest size this paper presents a new branch and bound algorithm for minimizing wcss algorithmic enhancements include an effective reordering of objects and a repetitive solution approach that precludes the need for splitting the data set while maintaining strong bounds throughout the solution process the new algorithm provided optimal solutions for problems with up to 240 objects and eight well separated clusters poorly separated problems with no inherent cluster structure were optimally solved for up to 60 objects and six clusters the repetitive branch and bound algorithm was also successfully applied to three empirical data sets from the classification literature

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

solution of large scale symmetric travelling salesman problems in this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality our implementation is based on a fast lp solver ibm s mpsx and makes effective use of polyhedral results on the symmetric travelling salesman polytope we describe the important ingredients of our code and give an extensive documentation of its computational performance

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

genetic algorithm based clustering technique abstract a genetic algorithm based clustering technique called ga clustering is proposed in this article the searching capability of genetic algorithms is exploited in order to search for appropriate cluster centres in the feature space such that a similarity metric of the resulting clusters is optimized the chromosomes which are represented as strings of real numbers encode the centres of a fixed number of clusters the superiority of the ga clustering algorithm over the commonly used k means algorithm is extensively demonstrated for four artificial and three real life data sets

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

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

an examination of indexes for determining the number of clusters in binary data sets the problem of choosing the correct number of clusters is as old as cluster analysis itself a number of authors have suggested various indexes to facilitate this crucial decision one of the most extensive comparative studies of indexes was conducted by milligan and cooper 1985 the present piece of work pursues the same goal under different conditions in contrast to milligan and cooper s work the emphasis here is on high dimensional empirical binary data binary artificial data sets are constructed to reflect features typically encountered in real world data situations in the field of marketing research the simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration index results are evaluated and their performance is compared and analyzed

the p q active uncapacitated facility location problem investigation of the solution space and an lp fitting heuristic abstract the p q active uncapacitated facility location problem is the problem of locating p facilities on n possible sites each serving at least q of the m clients at the minimum cost the problem is an extension of the uncapacitated facility location problem ufl where constraints on the number of facilities and their minimum activity have been added a use of this formulation could be the opening of p new schools where each must have at least q pupils p q active is np hard like the ufl in this paper we present a thorough investigation of the p q active ufl and propose a heuristic solution method different geometric and random cost problem instances are considered experiments show that 60 of the problems can be solved to optimality just by solving the corresponding lp relaxation using a simple local search heuristic the remaining geometric problems are solved with an average gap of 0 1 to a lower bound found by lp relaxation an effort is put into isolating problem types that are hard to solve problems with low p p q close to m combined with clustered clients or a low variation in the facility opening cost are most likely to give results worse than average gaps up to 8 are observed in the worst cases

heuristic implementation of dynamic programming for matrix permutation problems in combinatorial data analysis dynamic programming methods for matrix permutation problems in combinatorial data analysis can produce globally optimal solutions for matrices up to size 30 30 but are computationally infeasible for larger matrices because of enormous computer memory requirements branch and bound methods also guarantee globally optimal solutions but computation time considerations generally limit their applicability to matrix sizes no greater than 35 35 accordingly a variety of heuristic methods have been proposed for larger matrices including iterative quadratic assignment tabu search simulated annealing and variable neighborhood search although these heuristics can produce exceptional results they are prone to converge to local optima where the permutation is difficult to dislodge via traditional neighborhood moves e g pairwise interchanges object block relocations object block reversals etc we show that a heuristic implementation of dynamic programming yields an efficient procedure for escaping local optima specifically we propose applying dynamic programming to reasonably sized subsequences of consecutive objects in the locally optimal permutation identified by simulated annealing to further improve the value of the objective function experimental results are provided for three classic matrix permutation problems in the combinatorial data analysis literature a maximizing a dominance index for an asymmetric proximity matrix b least squares unidimensional scaling of a symmetric dissimilarity matrix and c approximating an anti robinson structure for a symmetric dissimilarity matrix

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 efficient tabu search procedure for the p median problem abstract in this paper we present a new solution heuristic for the p median problem the algorithm is based on tabu search principles and uses short term and long term memory as well as strategic oscillation and random tabu list sizes our proposed procedure is compared with two other move heuristics a well known interchange heuristic and a recent hybrid heuristic in computational tests on networks ranging in size up to 500 nodes the new heuristic is found to be superior with respect to the quality of solutions produced

a branch and price algorithm for the capacitated facility location problem the capacitated facility location problem cflp is a well known combinatorial optimization problem with applications in distribution and production planning it consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs a number of solution approaches based on lagrangean relaxation and subgradient optimization has been proposed for this problem subgradient optimization does not provide a primal fractional optimal solution to the corresponding master problem however in order to compute optimal solutions to large or difficult problem instances by means of a branch and bound procedure information about such a primal fractional solution can be advantageous in this paper a stabilized column generation method is therefore employed in order to solve a corresponding master problem exactly the column generation procedure is then employed within a branch and price algorithm for computing optimal solutions to the cflp computational results are reported for a set of larger and difficult problem instances

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

symmetry preserving quantization and self similar potentials a particular one dimensional model of classical mechanics obeying quadratic poisson symmetry algebra is considered after a special quantization the latter algebra is converted into the q weyl algebra describing symmetries of a specific schrodinger operator in this way one reconstructs the simplest self similar potential having infinitely many bound states with eigenvalues forming one geometric series

pseudolikelihood estimation for social networks abstract interest in log linear modeling for social network data has grown steadily since holland and leinhardt 1981 proposed their p 1 model that model was designed for a single binary relationship directed graph representing interactions between individuals it assumed that interactions between pairs of individuals are mutually independent subsequent work has extended the model in various ways including block modeling and the case of dependence between pairs of individuals in empirical work it would often be desirable to fit a wide variety of these models as the differences in predictions or goodness of fit are likely to provide insights into the data this has not been common practice however because estimation for some of the models has been difficult and the maximum likelihood schemes developed for others involve different computer programs not always available in standard packages the focus of this article is on a general estimation technique that maximizes the pseudolikelihood the pro

the many body problem for q oscillators the correct hamiltonian for the n body problem of free q oscillators is found which promotes the symmetries of the standard q 1 oscillator systems to q symmetries the spectrum of the system is found to be rich exhibiting interactions between the levels of the individual oscillators

a distance based point reassignment heuristic for the k hyperplane clustering problem abstract we consider the k hyperplane clustering problem where given a set of m points in r n we have to partition the set into k subsets clusters and determine a hyperplane for each of them so as to minimize the sum of the squares of the euclidean distances between the points and the hyperplane of the corresponding clusters we give a nonconvex mixed integer quadratically constrained quadratic programming formulation for the problem since even very small size instances are challenging for state of the art spatial branch and bound solvers like couenne we propose a heuristic in which many critical points are reassigned at each iteration such points which are likely to be ill assigned in the current solution are identified using a distance based criterion and their number is progressively decreased to zero our algorithm outperforms the best available one proposed by bradley and mangasarian on a set of real world and structured randomly generated instances for the largest instances we obtain an average improvement in the solution quality of 54

quantum algebras in classical mechanics it is shown that the equations of motion do not determine their hamiltonian descriptions uniquely in particular different poisson bracket relations are admissible for a given physical problem the free symmetric top and a charged particle moving on a homogeneous and constant magnetic field for instance may be described by a classical hamiltonian theory with an su 2 q poisson bracket structure

variable neighborhood search heuristics for selecting a subset of variables in principal component analysis the selection of a subset of variables from a pool of candidates is an important problem in several areas of multivariate statistics within the context of principal component analysis pca a number of authors have argued that subset selection is crucial for identifying those variables that are required for correct interpretation of the components in this paper we adapt the variable neighborhood search vns paradigm to develop two heuristics for variable selection in pca the performances of these heuristics were compared to those obtained by a branch and bound algorithm as well as forward stepwise backward stepwise and tabu search heuristics in the first experiment which considered candidate pools of 18 to 30 variables the vns heuristics matched the optimal subset obtained by the branch and bound algorithm more frequently than their competitors in the second experiment which considered candidate pools of 54 to 90 variables the vns heuristics provided better solutions than their competitors for a large percentage of the test problems an application to a real world data set is provided to demonstrate the importance of variable selection in the context of pca

bose realization of a non canonical heisenberg algebra the authors find out the bose realization of a generalized heisenberg algebra in which the bracket of the annihilation and creation operators is proportional to a polynomial function of the number operator the eigenvalues of the corresponding oscillator are derived in a special case they stress also the connection between non canonical commutation relations and q algebras

heuristic algorithms for siting alternative fuel stations using the flow refueling location model abstract this paper presents three heuristic algorithms that solve for the optimal locations for refueling stations for alternative fuels such as hydrogen ethanol biodiesel natural gas or electricity the flow refueling location model frlm locates refueling stations to maximize the flow that can be refueled with a given number of facilities the frlm uses path based demands and because of the limitations imposed by the driving range of vehicles longer paths require combinations of more than one station to refuel round trip travel a mixed integer linear programming milp version of the model has been formulated and published and could be used to obtain an optimal solution however because of the need for combinations of stations to satisfy demands a realistic problem with a moderate size network and a reasonable number of candidate sites would be impractical to generate and solve with milp methods in this research heuristic algorithms specifically the greedy adding greedy adding with substitution and genetic algorithm are developed and applied to solve the frlm problem these algorithms are shown to be effective and efficient in solving complex frlm problems for case study purposes the heuristic algorithms are applied to locate hydrogen refueling stations in the state of florida

matrix reorganization and dynamic programming applications to paired comparisons and unidimensional seriation a recursive dynamic programming strategy is discussed for optimally reorganizing the rows and simultaneously the columns of ann n proximity matrix when the objective function measuring the adequacy of a reorganization has a fairly simple additive structure a number of possible objective functions are mentioned along with several numerical examples using thurstone s paired comparison data on the relative seriousness of crime finally the optimization tasks we propose to attack with dynamic programming are placed in a broader theoretical context of what is typically referred to as the quadratic assignment problem and its extension to cubic assignment

heuristic concentration and tabu search a head to head comparison earlier this year two papers applying the metaheuristics tabu search ts and heuristic concentration hc to the p median problem were published in consecutive volumes of this journal here we apply the method of hc to some of the data sets which were used for computational experience in the paper on ts for these examples which we regard as being of a particularly challenging character hc discovers the superior solution superior to ts in about 95 of the cases and where the optimal solution is known the optimal solution in about 80 of the cases no general conclusion on the relative times for the two approaches can be drawn

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

optimal least squares unidimensional scaling improved branch and bound procedures and comparison to dynamic programming there are two well known methods for obtaining a guaranteed globally optimal solution to the problem of least squares unidimensional scaling of a symmetric dissimilarity matrix a dynamic programming and b branch and bound dynamic programming is generally more efficient than branch and bound but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations we present some new branch and bound procedures that improve computational efficiency and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects experimental tests were conducted to compare the relative performances of the new procedures a previously published branch and bound algorithm and a dynamic programming solution strategy these experiments which included both synthetic and empirical dissimilarity matrices yielded the following findings a the new branch and bound procedures were often drastically more efficient than the previously published branch and bound algorithm b when computationally feasible the dynamic programming approach was more efficient than each of the branch and bound procedures and c the new branch and bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation

block krylov subspace methods for approximating the linear combination of ź functions arising in exponential integrators in this paper we present numerical methods for computing the matrix functions arising in exponential integrators the matrix functions are the linear combination of the form j 0 p j a b j where j is related to exponential function by reducing the corresponding systems of linear odes with polynomial inhomogeneity based on block krylov subspace methods two effective exponentials of low dimension matrix are derived to approximate the objective functions some useful a posteriori error estimates are established the algorithms can be combined with time stepping strategy to satisfy the accuracy requirements under a moderate dimension of the block krylov subspace several numerical experiments are presented to demonstrate the efficiency of the methods

the tunneling method for global optimization in multidimensional scaling this paper focuses on the problem of local minima of the stress function it turns out that unidimensional scaling is particularly prone to local minima whereas full dimensional scaling with euclidean distances has a local minimum that is global for intermediate dimensionality with euclidean distances it depends on the dissimilarities how severe the local minimum problem is for city block distances in any dimensionality many different local minima are found a simulation experiment is presented that indicates under what conditions local minima can be expected we introduce the tunneling method for global minimization and adjust it for multidimensional scaling with general minkowski distances the tunneling method alternates a local search step in which a local minimum is sought with a tunneling step in which a different configuration is sought with the same stress as the previous local minimum in this manner successively better local minima are obtained and experimentation so far shows that the last one is often a global minimum

an algorithm for generating artificial test clusters an algorithm for generating artificial data sets which contain distinct nonoverlapping clusters is presented the algorithm is useful for generating test data sets for monte carlo validation research conducted on clustering methods or statistics the algorithm generates data sets which contain either 1 2 3 4 or 5 clusters by default the data are embedded in either a 4 6 or 8 dimensional space three different patterns for assigning the points to the clusters are provided one pattern assigns the points equally to the clusters while the remaining two schemes produce clusters of unequal sizes finally a number of methods for introducing error in the data have been incorporated in the algorithm

a comparison of heuristic procedures for minimum within cluster sums of squares partitioning perhaps the most common criterion for partitioning a data set is the minimization of the within cluster sums of squared deviation from cluster centroids although optimal solution procedures for within cluster sums of squares wcss partitioning are computationally feasible for small data sets heuristic procedures are required for most practical applications in the behavioral sciences we compared the performances of nine prominent heuristic procedures for wcss partitioning across 324 simulated data sets representative of a broad spectrum of test conditions performance comparisons focused on both percentage deviation from the best found wcss values as well as recovery of true cluster structure a real coded genetic algorithm and variable neighborhood search heuristic were the most effective methods however a straightforward two stage heuristic algorithm hk means also yielded exceptional performance a follow up experiment using 13 empirical data sets from the clustering literature generally supported the results of the experiment using simulated data our findings have important implications for behavioral science researchers whose theoretical conclusions could be adversely affected by poor algorithmic performances

on analysis of projection methods for rational function approximation to the matrix exponential we revisit the analysis of projection methods for rational function approximation to the matrix exponential and present a new result for establishing convergence of the pade plus krylov approximation when a 1

inexact implementation using krylov subspace methods for large scale exponential discriminant analysis with applications to high dimensionality reduction problems abstract exponential discriminant analysis eda is a generalized discriminant analysis method based on matrix exponential it can essentially overcome the intrinsic difficulty of small sample size problem that exists in the classical linear discriminant analysis lda however for data with high dimensionality one has to solve a large matrix exponential eigenproblem in this method and the time complexity is dominated by the computation of exponential of large matrices in this paper we propose two inexact krylov subspace algorithms for solving the large matrix exponential eigenproblem efficiently the contribution of this work is threefold first we consider how to compute matrix exponential vector products efficiently which is the key step in the krylov subspace method second we compare the discriminant analysis criterion of eda and that of lda from a theoretical point of view third we establish the relationship between the accuracy of the approximate eigenvectors and the distance to nearest neighbor classifier and show why the matrix exponential eigenproblem can be solved approximately in practice numerical experiments on some real world databases show the superiority of our new algorithms over their original counterpart for face recognition

suq h 0 2 and suq h 2 the classical and quantum q deformations of the su 2 algebra by means of classical harmonic oscillators and their canonical quantization we show that the q deformation of the su 2 lie algebra can be realized at a classical level in the sense of poisson brackets denoted by suq inf not 0 2 and that the canonical quantization of the system leads to the quantum q deformation of the su 2 algebra denoted by suq 2 this means that the q deformation and the h cross quantization are in general two different concepts

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation

bfv brst analysis of the classical and quantum q deformations of the sl 2 algebra bfv brst charge for quantum algebras is not unique different constructions of it in classical and quantum phase space for the universal enveloping algebra of the q deformed sl 2 are discussed in the quantum framework a positive definite scalar product is used to introduce a co bfv brst charge to study the cohomology problem by means of the techniques of the non deformed case moreover deformation of the phase space without deforming the generators of sl 2 is considered h cross q deformation of the phase space is shown to yield the witten s second deformation for sl 2 to study the bfv brst cohomology problem when both the quantum phase space and the group are deformed a two parameter deformation of sl 2 is proposed and its bfv brst charge is given

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

the quantum mechanical schrodinger picture of a q oscillator a q deformed version of standard quantum mechanics in the coordinate schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so called q discrete derivative as the representative of the momentum operator the chosen q discrete derivative is symmetric with respect to the exchange of q and q 1 under the usually adopted assumptions a q deformed schrodinger equation is derived for a harmonic oscillator the complete set of eigenfunctions can be explicitly constructed as special q functions and the corresponding energy eigenvalues are identical to those obtained by biedenharn in his pioneering work 1989 this q deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions

analysis of projection methods for rational function approximation to the matrix exponential krylov subspace methods for approximating the action of the matrix exponential exp a on a vector v are analyzed with a hermitian and negative semidefinite our approach is based on approximating the exponential with the commonly employed diagonal pade and chebyshev rational functions which yield a system of equations with a polynomial coefficient matrix we derive optimality properties and error bounds for the convergence of a galerkin type approximation and of a computationally feasible and extensively used alternative as complementary results we theoretically justify the use of a popular a posteriori error estimate and we provide upper bounds for the components of the solution vector our theoretical and numerical results show that this methodology may provide an appropriate framework to devise new strategies such as more powerful acceleration schemes

alternative hamiltonian quantization the authors take a standard hamiltonian quadratic in a set of creation and annihilation operators and demonstrate an alternative quantization by quommutators i e a set of rules for transposition of these operators different from the those of the usual canonical commutation relations which yields a new spectrum of a type reminiscent of logarithmic trajectories

meta heuristic approaches with memory and evolution for a multi product production distribution system design problem we consider a multi product two stage production distribution system design problem pdsd where a fixed number of capacitated distribution centers are to be located with respect to capacitated suppliers plants and retail locations customers while minimizing the total costs in the system we present a mixed integer problem formulation that facilitates the development of efficient heuristic procedures we provide meta heuristic procedures including a population based scatter search with path relinking and trajectory based local and tabu search for the solution of the problem we also develop efficient construction heuristics and transshipment heuristics that are incorporated into the heuristic procedures for the solution of subproblems we present extensive computational results that show the high performance of the solution approaches we obtain smaller than 1 0 average optimality gaps with acceptable runtimes even for relatively large problems the computational results also demonstrate the effectiveness of the construction and transshipment heuristics that impact the solution quality and overall runtimes