the need for a flat higher gauge structure to describe a berry phase associated with some resonance phenomena in the presence of a resonance crossing producing splitting of the base manifold for example a circle crossing in a plane we show that the rigorous geometrical structure within which the berry phase arises may be a 2 bundle a structure related to gerbes and to category theory rather than a fiber bundle the bloch wave operator plays an important role in the associated theory

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

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

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

on the adiabatic theorem for nonself adjoint hamiltonians the authors consider the evolution of a two level system driven by a nonself adjoint hamiltonian h in t and treat the adiabatic limit in to 0 while adiabatic theorem like results do not hold true in general for this case they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part least dissipative eigenvalue the theory gives the full asymptotic expansion of the evolution restricted to this subspace the first correction beyond berry s phase is to their best knowledge given explicitly for the first time

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

implicit numerical methods for highly nonlinear neutral stochastic differential equations with time dependent delay abstract this paper represents the continuation of the analysis from papers milosevic 2011 10 and milosevic 2013 11 the main aim of this paper is to establish certain results for the backward euler method for a class of neutral stochastic differential equations with time dependent delay for that purpose the split step backward euler method which represents an extension of the backward euler method is introduced for this class of equations conditions under which the split step backward euler method and thus the backward euler method is well defined are revealed moreover the convergence in probability of the backward euler method is proved under certain nonlinear growth conditions including the one sided lipschitz condition this result is proved using the technique which is based on the application of the continuous time approximation for this reason the discrete forward backward euler method is involved since it allows its continuous version to be well defined from the aspect of measurability the convergence in probability is established for the continuous forward backward euler solution which is essential for proving the same result for both discrete forward backward and backward euler methods additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without the linear growth condition on the drift coefficient of the equation as usual the whole consideration is affected by the presence and properties of the delay function

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

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

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

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

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

a genetic algorithm using hyper quadtrees for low dimensional k means clustering the k means algorithm is widely used for clustering because of its computational efficiency given n points in d dimensional space and the number of desired clusters k k means seeks a set of k cluster centers so as to minimize the sum of the squared euclidean distance between each point and its nearest cluster center however the algorithm is very sensitive to the initial selection of centers and is likely to converge to partitions that are significantly inferior to the global optimum we present a genetic algorithm ga for evolving centers in the k means algorithm that simultaneously identifies good partitions for a range of values around a specified k the set of centers is represented using a hyper quadtree constructed on the data this representation is exploited in our ga to generate an initial population of good centers and to support a novel crossover operation that selectively passes good subsets of neighboring centers from parents to offspring by swapping subtrees experimental results indicate that our ga finds the global optimum for data sets with known optima and finds good solutions for large simulated data sets

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

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

constrained clustering by constraint programming cluster analysis is an important task in data mining with hundreds of different approaches in the literature since the last decade the cluster analysis has been extended to constrained clustering also called semi supervised clustering so as to integrate previous knowledge on data to clustering algorithms in this dissertation we explore constraint programming cp for solving the task of constrained clustering the main principles in cp are 1 users specify declaratively the problem in a constraint satisfaction problem 2 solvers search for solutions by constraint propagation and search relying on cp has two main advantages the declarativity which enables to easily add new constraints and the ability to find an optimal solution satisfying all the constraints when there exists one we propose two models based on cp to address constrained clustering tasks the models are flexible and general and supports instance level constraints and different cluster level constraints it also allows the users to choose among different optimization criteria in order to improve the efficiency different aspects have been studied in the dissertation experiments on various classical datasets show that our models are competitive with other exact approaches we show that our models can easily be embedded in a more general process and we illustrate this on the problem of finding the pareto front of a bi criterion optimization process

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

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

the integration of an interior point cutting plane method within a branch and price algorithm this paper presents a novel integration of interior point cutting plane methods within branch and price algorithms unlike the classical method columns are generated at a central dual solution by applying the analytic centre cutting plane method accpm on the dual of the full master problem first we introduce some modifications to accpm we propose a new procedure to recover primal feasibility after adding cuts and use for the first time a dual newton s method to calculate the new analytic centre after branching second we discuss the integration of accpm within the branch and price algorithm we detail the use of accpm as the search goes deep in the branch and bound tree making full utilization of past information as a warm start we exploit dual information from accpm to generate incumbent feasible solutions and to guide branching finally the overall approach is implemented and tested for the bin packing problem and the capacitated facility location problem with single sourcing we compare against cplex mip 7 5 as well as a classical branch and price 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

a unified continuous optimization framework for center based clustering methods center based partitioning clustering algorithms rely on minimizing an appropriately formulated objective function and different formulations suggest different possible algorithms in this paper we start with the standard nonconvex and nonsmooth formulation of the partitioning clustering problem we demonstrate that within this elementary formulation convex analysis tools and optimization theory provide a unifying language and framework to design analyze and extend hard and soft center based clustering algorithms through a generic algorithm which retains the computational simplicity of the popular k means scheme we show that several well known and more recent center based clustering algorithms which have been derived either heuristically or and have emerged from intuitive analogies in physics statistical techniques and information theoretic perspectives can be recovered as special cases of the proposed analysis and we streamline their relationships

cost optimal constrained correlation clustering via weighted partial maximum satisfiability abstract integration of the fields of constraint solving and data mining and machine learning has recently been identified within the ai community as an important research direction with high potential this work contributes to this direction by providing a first study on the applicability of state of the art boolean optimization procedures to cost optimal correlation clustering under constraints in a general similarity based setting we develop exact formulations of the correlation clustering task as maximum satisfiability maxsat the optimization version of the boolean satisfiability sat problem for obtaining cost optimal clusterings we apply a state of the art maxsat solver for solving the resulting maxsat instances optimally resulting in cost optimal clusterings we experimentally evaluate the maxsat based approaches to cost optimal correlation clustering both on the scalability of our method and the quality of the clusterings obtained furthermore we show how the approach extends to constrained correlation clustering where additional user knowledge is imposed as constraints on the optimal clusterings of interest we show experimentally that added user knowledge allows clustering larger datasets and at the same time tends to decrease the running time of our approach we also investigate the effects of maxsat level preprocessing symmetry breaking and the choice of the maxsat solver on the efficiency of the approach

almost quantum adiabatic dynamics and generalized time dependent wave operators we consider quantum dynamics for which the strict adiabatic approximation fails but which do not escape too far from the adiabatic limit to treat these systems we introduce a generalization of the time dependent wave operator theory which is usually used to treat dynamics which do not escape too far from an initial subspace called the active space our generalization is based on a time dependent adiabatic deformation of the active space the geometric phases associated with the almost adiabatic representation are also derived we use this formalism to study the adiabaticity of a dynamics surrounding an exceptional point of a non hermitian hamiltonian we show that the generalized time dependent wave operator can be used to correct easily the adiabatic approximation which is very imperfect in this situation

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