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

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

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

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

recent results and questions in combinatorial game complexities abstract aim present a systematic development of part of the theory of combinatorial games from the ground up approach computational complexity combinatorial games are completely determined the questions of interest are efficiencies of strategies methodology divide and conquer isolate the various difficulties separating hard from easy games and attack them individually presentation informal examples of games sampled from various levels illustrate the theory with emphasis on formulating and motivating new and old research problems

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

constructions for optimal spl upsi 4 1 optical orthogonal codes four direct constructions three of which are by way of skew starters are given in this correspondence for optimal spl upsi 4 1 optical orthogonal codes ooc s these improve the known existence results concerning optimal spl upsi 4 1 oocs in particular it is shown that an optimal spl upsi 4 1 ooc exists for all positive integers spl upsi spl equiv 6 mod 12 or spl upsi spl equiv 24 mod 48 it is also shown that an optimal 12 spl upsi 4 1 ooc exists for any positive integer spl upsi whose prime factors are all congruent to 1 modulo 4

application of general semi infinite programming to lapidary cutting problems we consider a volume maximization problem arising in gemstone cutting industry the problem is formulated as a general semi infinite program gsip and solved using an interior point method developed by stein o stein bi level strategies in semi infinite programming kluwer academic publishers boston 2003 it is shown that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling clustering techniques are used to reduce the number of container constraints which is necessary to make the subproblems practically tractable an iterative process consisting of gsip optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem some numerical results based on real world data are also presented

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

clustering by a fuzzy metric in this paper we consider the clustering problem in which a distance between any two elements is given by a fuzzy metric our clustering criterion is to minimize the total of the average within cluster sum of the fuzzy metrics and an optimal clustering is concretely sought by applying an approach by cuts of fuzzy numbers also a numerical example is given to illustrate our theoretical results

optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4 an optical orthogonal signature pattern code oospc finds application in transmitting 2 d images through multicore fiber in code division multiple access cdma communication systems observing a one to one correspondence between an oospc and a certain combinatorial subject called a packing design we present a construction of optimal oospcs with weight 4 and maximum collision parameter 2 which generalizes a well known kohler construction of optimal optical orthogonal codes ooc with weight 4 and maximum collision parameter 2 using this new construction enables one to obtain infinitely many optimal oospcs whose existence was previously unknown we prove that for a multiple n of 4 there exists no optimal oospc of size 6 n with weight 4 and maximum collision parameter 2 together with a report which shows a gap between optimal oocs and optimal oospcs when 6 and n are not coprime we also present a recursive construction of oospcs which are asymptotically optimal with respect to the johnson bound as a by product we obtain an asymptotically optimal m n 4 2 oospc for all positive integers m and n

combinatorial constructions of optimal optical orthogonal codes with weight 4 a v k spl lambda optical orthogonal code c is a family of 0 1 sequences of length v and weight k satisfying the following correlation properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les spl lambda for any x x sub 0 x sub 1 x sub v 1 spl isin c and any integer i spl ne 0 mod v 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les spl lambda for any x x sub 0 x sub 1 x sub v 1 spl isin c y y sub 0 y sub 1 y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are taken modulo v a v k spl lambda optical orthogonal code ooc with spl lfloor 1 k spl lfloor v 1 k 2 spl lfloor v 2 k 2 spl lfloor spl middot spl middot spl middot spl lfloor v spl lambda k spl lambda spl rfloor m spl rfloor spl rfloor spl rfloor codewords is said to be optimal oocs are essential for success of fiber optic code division multiple access cdma communication systems the use of an optimal ooc enables the largest possible number of asynchronous users to transmit information efficiently and reliably in this paper various combinatorial constructions for optimal v 4 1 oocs such as those via skew starters and weil s theorem on character sums are given for v spl equiv 0 mod 12 these improve the known existence results on optimal oocs in particular it is shown that an optimal v 4 1 ooc exists for any positive integer v spl equiv 0 mod 24

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

clustering by sorting potential values cspv a novel potential based clustering method a novel clustering method called clustering by sorting potential values cspv is proposed the clustering is done in an efficient tree growing fashion based on both the distances and the hypothetical potential values produced from the distribution of all the data points the method is simple but is shown to be very effective in identifying different kinds of clusters it outperforms four popular clustering methods in most of our experiments and is the only one that works for all the six studied data sets moreover it is designed as a generic method which can be easily applied to different clustering problems

column generation algorithms for exact modularity maximization in networks finding modules or clusters in networks currently attracts much attention in several domains the most studied criterion for doing so due to newman and girvan phys rev e 69 026113 2004 is modularity maximization many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality there are few exact algorithms prominent among which is a paper by xu et al eur phys j b 60 231 2007 modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of grotschel and wakabayashi math program 45 59 1989 applied we propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming performance of the four resulting algorithms is compared on problems from the literature instances with up to 512 entities are solved exactly moreover the computing time of previously solved problems are reduced substantially

complexity of the cluster deletion problem on subclasses of chordal graphs we consider the following vertex partition problem on graphs given a graph with real nonnegative edge weights partition the vertices into clusters in this case cliques to minimize the total weight of edges out of the clusters this optimization problem is known to be an np complete problem even for unweighted graphs and has been studied extensively in the scope of fixed parameter tractability fpt where it is commonly known as the cluster deletion problem many of the recently developed fpt algorithms rely on being able to solve cluster deletion in polynomial time on restricted graph structures in this paper the complexity of the cluster deletion problem is investigated for the family of chordal graphs it is shown that this problem is np complete for edge weighted split graphs edge weighted interval graphs and edge unweighted chordal graphs we also prove that the cluster deletion is an np complete problem for edge weighted cographs some polynomial time solvable cases are also identified in particular cluster deletion for unweighted split graphs unweighted proper interval graphs and weighted block graphs

two dimensional optical orthogonal codes and semicyclic group divisible designs a nxm k two dimensional optical orthogonal code 2 d ooc c is a family of nxm 0 1 arrays of constant weight k such that i 1n j 0m 1a i j b i j m for any arrays a b in c and any integer except when a b and 0 mod m where m denotes addition modulo m such codes are of current practical interest as they enable optical communication at lower chip rate to simplify practical implementation the am oppw at most one pulse per wavelength restriction is often appended to a 2 d ooc an am oppw 2 d ooc is optimal if its size is the largest possible in this paper the notion of a perfect am oppw 2 d ooc is proposed which is an optimal nxm k am oppw 2 d ooc with cardinality m n n 1 n k k 1 k a link between optimal nxm k am oppw 2 d oocs and block designs is developed some new constructions for such optimal codes are described by means of semicyclic group divisible designs several new infinite families of perfect nxm k 1 am oppw 2 d oocs with k 2 3 4 are thus produced

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

general constructions of optimal variable weight optical orthogonal codes variable weight optical orthogonal code ooc was introduced by yang for multimedia optical cdma systems with multiple quality of service qos requirements in this paper four general constructions for optimal cyclic packings and optimal variable weight oocs are presented many new infinite classes of optimal v w 1 q oocs are obtained new infinite classes of optimal v w 1 q oocs for w 4 are easily obtained by the constructions

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

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

determination of the exact value for psi m k k 1 impulse radio sequences irss were first introduced by chu and colbourn in ieee trans inf theory vol 50 pp 2402 2407 2004 let m k denote the maximal possible number of sequences in an m k irs in this paper a fundamental equivalence between impulse radio sequences and strictly cyclic packing having the impulse position property is established with their relationship the exact value of m k k 1 is finally determined for any positive integers m and k

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

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

optical orthogonal codes design analysis and applications an optical orthogonal code ooc is a family of 0 1 sequences with good auto and cross correlation properties i e the autocorrelation of each sequence exhibits the thumbtack shape and the cross correlation between any two sequences remains low throughout the use of optical orthogonal codes enables a large number of asynchronous users to transmit information efficiently and reliably the thumbtack shaped autocorrelation facilitates the detection of the desired signal and low profile cross correlation reduces interference from unwanted signals theoretical upper and lower bounds on the maximum possible size of oocs are derived methods for the design and analysis of oocs using tools from projective geometry the greedy algorithm iterative constructions algebraic coding theory block design and various other combinational disciplines are discussed

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

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

optical orthogonal codes their bounds and new optimal constructions a v k spl lambda sub a spl lambda sub c optical orthogonal code ooc c is a family of 0 1 sequences of length v and weight k satisfying the following two correlation properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les spl lambda sub a for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c and any integer i not equivalent 0 mod v and 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les spl lambda sub b for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c y y sub 0 y sub 1 spl middot spl middot spl middot y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are taken modulo v the study of optical orthogonal codes is motivated by an application in optical code division multiple access communication systems in this paper upper bounds on the size of an optical orthogonal code are discussed several new constructions for optimal optical orthogonal codes with weight k spl ges 4 and correlation constraints spl lambda sub a spl lambda sub c 1 are described by means of optimal cyclic packings many new infinite series of such optimal optical orthogonal codes are thus produced

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

variable neighborhood search for minimum sum of squares clustering on networks abstract euclidean minimum sum of squares clustering amounts to finding p prototypes by minimizing the sum of the squared euclidean distances from a set of points to their closest prototype in recent years related clustering problems have been extensively analyzed under the assumption that the space is a network and not any more the euclidean space this allows one to properly address community detection problems of significant relevance in diverse phenomena in biological technological and social systems however the problem of minimizing the sum of squared distances on networks have not yet been addressed two versions of the problem are possible either the p prototypes are sought among the set of nodes of the network or also points along edges are taken into account as possible prototypes while the first problem is transformed into a classical discrete p median problem the latter is new in the literature and solved in this paper with the variable neighborhood search heuristic the solutions of the two problems are compared in a series of test examples

dynamic cluster formation using level set methods density based clustering has the advantages for 1 allowing arbitrary shape of cluster and 2 not requiring the number of clusters as input however when clusters touch each other both the cluster centers and cluster boundaries as the peaks and valleys of the density distribution become fuzzy and difficult to determine we introduce the notion of cluster intensity function cif which captures the important characteristics of clusters when clusters are well separated cifs are similar to density functions but when clusters become closed to each other cifs still clearly reveal cluster centers cluster boundaries and degree of membership of each data point to the cluster that it belongs clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation which are often oscillatory or oversmoothed these problems of kernel density estimation are resolved using level set methods and related techniques comparisons with two existing density based methods valley seeking and dbscan are presented which illustrate the advantages of our approach

optimal partitioning of a data set based on the p median model although the k means algorithm for minimizing the within cluster sums of squared deviations from cluster centroids is perhaps the most common method for applied cluster analyses a variety of other criteria are available the p median model is an especially well studied clustering problem that requires the selection of p objects to serve as cluster centers the objective is to choose the cluster centers such that the sum of the euclidean distances or some other dissimilarity measure of objects assigned to each center is minimized using 12 data sets from the literature we demonstrate that a three stage procedure consisting of a greedy heuristic lagrangian relaxation and a branch and bound algorithm can produce globally optimal solutions for p median problems of nontrivial size several hundred objects five or more variables and up to 10 clusters we also report the results of an application of the p median model to an empirical data set from the telecommunications industry

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

visualizing data as objects by dc difference of convex optimization in this paper we address the problem of visualizing in a bounded region a set of individuals which has attached a dissimilarity measure and a statistical value as convex objects this problem which extends the standard multidimensional scaling analysis is written as a global optimization problem whose objective is the difference of two convex functions dc suitable dc decompositions allow us to use the difference of convex algorithm dca in a very efficient way our algorithmic approach is used to visualize two real world datasets

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

approximating k means type clustering via semidefinite programming one of the fundamental clustering problems is to assign n points into k clusters based on minimal sum of squared distances mssc which is known to be np hard in this paper by using matrix arguments we first model mssc as a so called 0 1 semidefinite programming sdp problem we show that our 0 1 sdp model provides a unified framework for several clustering approaches such as normalized k cut and spectral clustering moreover the 0 1 sdp model allows us to solve the underlying problem approximately via the linear programming and sdp relaxations second we consider the issue of how to extract a feasible solution of the original 0 1 sdp model from the optimal solution of the relaxed sdp problem by using principal component analysis we develop a rounding procedure to construct a feasible partitioning from a solution of the relaxed problem in our rounding procedure we need to solve a k means clustering problem in re k 1 which can be done in o n k 2 2k 2 time in case of biclustering the running time of our rounding procedure can be reduced to o n log n we show that our algorithm provides a 2 approximate solution to the original problem promising numerical results for biclustering based on our new method are reported

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

clustering objects on subsets of attributes with discussion read before the royal statistical society at a meeting organized by the research section on wednesday may 5th 2004 professor j t kent in the chair summary a new procedure is proposed for clustering attribute value data when used in conjunction with conventional distance based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously the relevant attribute subsets for each individual cluster can be different and partially or completely overlap with those of other clusters enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed applications in different domains including gene expression arrays are presented