the bunch kaufman factorization of symmetric matrices signature similar to sign regular matrices abstract the bunch kaufman pivoting strategy is a most commonly used method in practice to factor symmetric indefinite matrices however this method in general may destroy the band structure of banded matrices and its growth factor bound of 2 57 n 1 can almost be attained in this paper we investigate the behavior of the method for factorizing symmetric indefinite matrices that are signature similar to sign regular matrices with signature the growth factor bound depending on flips in the signature is derived for factorizing such banded matrices if the bunch kaufman pivoting strategy is modified as suggested by sorensen and van loan then not only do the same growth factor bounds hold but also the bandwidth is nicely preserved

robust clustering by detecting density peaks and assigning points based on fuzzy weighted k nearest neighbors clustering by fast search and find of density peaks referred to as dpc was introduced by alex rodriguez and alessandro laio the dpc algorithm is based on the idea that cluster centers are characterized by having a higher density than their neighbors and by being at a relatively large distance from points with higher densities the power of dpc was demonstrated on several test cases it can intuitively find the number of clusters and can detect and exclude the outliers automatically while recognizing the clusters regardless of their shape and the dimensions of the space containing them however dpc does have some drawbacks to be addressed before it may be widely applied first the local density i of point i is affected by the cutoff distance dc and is computed in different ways depending on the size of datasets which can influence the clustering especially for small real world cases second the assignment strategy for the remaining points after the density peaks that is the cluster centers have been found can create a domino effect whereby once one point is assigned erroneously then there may be many more points subsequently mis assigned this is especially the case in real word datasets where there could exist several clusters of arbitrary shape overlapping each other to overcome these deficiencies a robust clustering algorithm is proposed in this paper to find the density peaks this algorithm computes the local density i of point i relative to its k nearest neighbors for any size dataset independent of the cutoff distance dc and assigns the remaining points to the most probable clusters using two new point assignment strategies the first strategy assigns non outliers by undertaking a breadth first search of the k nearest neighbors of a point starting from cluster centers the second strategy assigns outliers and the points unassigned by the first assignment procedure using the technique of fuzzy weighted k nearest neighbors the proposed clustering algorithm is benchmarked on publicly available synthetic and real world datasets which are commonly used for testing the performance of clustering algorithms the clustering results of the proposed algorithm are compared not only with that of dpc but also with that of several well known clustering algorithms including affinity propagation ap density based spatial clustering of applications with noise dbscan and k means the benchmarks used are clustering accuracy acc adjusted mutual information ami and adjusted rand index ari the experimental results demonstrate that our proposed clustering algorithm can find cluster centers recognize clusters regardless of their shape and dimension of the space in which they are embedded be unaffected by outliers and can often outperform dpc ap dbscan and k means

the angular metric traveling salesman problem motivated by applications in robotics we formulate the problem of minimizing the total angle cost of a tsp tour for a set of points in euclidean space where the angle cost of a tour is the sum of the direction changes at the points we establish the np hardness of both this problem and its relaxation to the cycle cover problem we then consider the issue of designing approximation algorithms for these problems and show that both problems can be approximated to within a ratio of o log n in polynomial time we also consider the problem of simultaneously approximating both the angle and the length measure for a tsp tour in studying the resulting tradeoff we choose to focus on the sum of the two performance ratios and provide tight bounds on the sum finally we consider the extremal value of the angle measure and obtain essentially tight bounds for it in this paper we restrict our attention to the planar setting but all our results are easily extended to higher dimensions

quasi ldu factorization of nonsingular totally nonpositive matrices abstract let a a ij r n n be a nonsingular totally nonpositive matrix in this paper we describe some properties of these matrices when a11 0 and obtain a characterization in terms of the quasi ldu factorization of a where l is a block lower triangular matrix d is a diagonal matrix and u is a unit upper triangular matrix

fast accurate fuzzy clustering through data reduction clustering is a useful approach in image segmentation data mining and other pattern recognition problems for which unlabeled data exist fuzzy clustering using fuzzy c means or variants of it can provide a data partition that is both better and more meaningful than hard clustering approaches the clustering process can be quite slow when there are many objects or patterns to be clustered this paper discusses the algorithm brfcm which is able to reduce the number of distinct patterns which must be clustered without adversely affecting the partition quality the reduction is done by aggregating similar examples and then using a weighted exemplar in the clustering process the reduction in the amount of clustering data allows a partition of the data to be produced faster the algorithm is applied to the problem of segmenting 32 magnetic resonance images into different tissue types and the problem of segmenting 172 infrared images into trees grass and target average speed ups of as much as 59 290 times a traditional implementation of fuzzy c means were obtained using brfcm while producing partitions that are equivalent to those produced by fuzzy c means

an analytical comparison of different formulations of the travelling sales man problem a transformation technique is proposed that permits one to derive the linear description of the imagex of a polyhedronz under an affine linear transformation from the given linear description ofz this result is used to analytically compare various formulations of the asymmetric travelling salesman problem to the standard formulation due to dantzig fulkerson and johnson which are all shown to be weaker formulations in a precise setting we also apply this transformation technique to symmetrize formulations and show in particular that the symmetrization of the standard asymmetric formulation results into the standard one for the symmetric version of the travelling salesman problem

full rank factorization in echelon form of totally nonpositive negative rectangular matrices abstract an n m real matrix a is said to be totally nonpositive negative if every minor is nonpositive negative in this paper we study the full rank factorization in echelon form of a totally nonpositive negative matrix this characterization allows us to significantly reduce the number of minors to be checked in order to decide the total negativity of a matrix

almost strictly sign regular rectangular matrices abstract almost strictly sign regular matrices are sign regular matrices with a special zero pattern and whose nontrivial minors are nonzero in this paper we provide several properties of almost strictly sign regular rectangular matrices of maximal rank and analyze their q r factorization

decompositions of strictly sign regular matrices abstract a matrix is called strictly sign regular if for each k all its k k minors are nonzero and have the same sign we study several decompositions of strictly sign regular matrices including qr decomposition schur decomposition svd and symmetric triangular decomposition several characterizations are provided

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

a geometric proof of total positivity for spline interpolation abstract the total positivity of the spline collocation matrix is the basis of several important results in univariate spline theory this makes it desirable to provide as simple as possible a proof of this total positivity the proofs available in the literature don t qualify since these all rely on certain determinant identities which are not exactly intuitive the authors give a proof that uses nothing more than cramer s rule hard to avoid since total positivity is a statement about determinants and the geometrically obvious fact that a b spline can always be written as a positive combination of b splines on a finer knot sequence the geometric intuition appealed to here stems from the area of computer aided design in which a spline is constructed and manipulated through its b polygon a broken line whose vertices correspond to the b spline coefficients if a knot is added to provide greater potential flexibility the new b polygon is obtained by interpolation to the old this had led lane and riesenfeld to a proof of the variation diminishing property of the spline collocation matrix and is shown here to provide a proof of the total positivity as well

total nonpositivity of nonsingular matrices in this paper nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns these characterizations extend an existing characterization that uses some restrictive hypotheses

characterizations and stable tests for the routh hurwitz conditions and for total positivity given a polynomial of degree n a test of o n2 elementary operations and growth factor 1 is presented in order to check the routh hurwitz conditions this optimal growth factor guarantees that the test presents better stability properties than other known tests we also present a test of o n3 elementary operations and growth factor 1 in order to check if a matrix is strictly totally positive finally totally positive matrices are characterized by their symmetric triangular decompositions

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

lower bounding procedure for the asymmetric quadratic traveling salesman problem abstract in this paper we consider the asymmetric quadratic traveling salesman problem aqtsp given a directed graph and a function that maps every pair of consecutive arcs to a cost the problem consists in finding a cycle that visits every vertex exactly once and such that the sum of the costs is minimal we propose an extended linear programming formulation that has a variable for each cycle in the graph since the number of cycles is exponential in the graph size we propose a column generation approach moreover we apply a particular reformulation linearization technique on a compact representation of the problem and compute lower bounds based on lagrangian relaxation we compare our new bounds with those obtained by some linearization models proposed in the literature computational results on some set of benchmarks used in the literature show that our lower bounding procedures are very promising

on positivity of principal minors of bivariate bezier collocation matrix it is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct thus the corresponding interpolation matrix m is nonsingular schumaker stated the conjecture that all principal submatrices of m are nonsingular too furthermore all of the corresponding determinants the principal minors are conjectured to be positive this result would solve the constrained interpolation problem in this paper the conjecture on minors for polynomial degree 17 and conjecture for some particular configurations of domain points are confirmed

on tightening the relaxations of miller tucker zemlin formulations for asymmetric traveling salesman problems this paper is concerned with applying thereformulation linearization technique rlt to derive tighter relaxations for theasymmetric traveling salesman problem atsp formulation that is based on themiller tucker zemlin mtz subtour elimination constraints the mtz constraints yield a compact representation for the traveling salesman problem tsp and their use is particularly attractive in various routing and scheduling contexts that have an embedded atsp structure however it is well recognized that these constraints yield weak relaxations and with this motivation desrochers and laporte 1991 have lifted the mtz constraints into facets of the underlying atsp polytope we show that a novel application of the rlt process from a nonstandard mtz representation of the atsp reveals a new formulation of this problem that is compact and yet theoretically as well as computationally dominates the lifted mtz formulation of desrochers and laporte this approach is also extended to derive tight formulations for theprecedence constrained asymmetric traveling salesman problem pcatsp based on the mtz subtour elimination constraints additional classes of valid inequalities are also developed for both these versions of the atsp and further ideas for developing tighter representations are suggested for future investigations

linearization strategies for a class of zero one mixed integer programming problems this paper is concerned with a new linearization strategy for a class of zero one mixed integer programming problems that contains quadratic cross product terms between continuous and binary variables and between the binary variables themselves this linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature moreover the proposed technique provides a unifying framework in the sense that all the alternate methods lead to formulations that are accessible through appropriate surrogates of the constraints of the new linearized formulation extensions to various other types of mixed integer nonlinear programming problems are also discussed

a contribution to convergence theory of fuzzy c means and derivatives in this paper we revisit the convergence and optimization properties of fuzzy clustering algorithms in general and the fuzzy c means fcm algorithm in particular our investigation includes probabilistic and a slightly modified implementation of possibilistic memberships which will be discussed under a unified view we give a convergence proof for the axis parallel variant of the algorithm by gustafson and kessel that can be generalized to other algorithms more easily than in the usual approach using reformulated fuzzy clustering algorithms we apply banach s classical contraction principle and establish a relationship between saddle points and attractive fixed points for the special case of fcm we derive a sufficient condition for fixed points to be attractive allowing identification of them as local minima of the objective function excluding the possibility of a saddle point

accurate bidiagonal decomposition of totally positive h bernstein vandermonde matrices and applications abstract an algorithm for the computation of the bidiagonal decomposition of strictly totally positive h bernstein vandermonde matrices is presented these matrices are collocation matrices of h bernstein bases a generalization of the bernstein basis of the space of polynomials of degree less than or equal to n the algorithm is fast its computational complexity is o n 2 and has high relative accuracy the computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices linear system solving eigenvalue computation and inverse computation numerical experiments are also included

characterizations of rectangular totally and strictly totally positive matrices abstract an n m real matrix a is said to be totally positive strictly totally positive if every minor is nonnegative positive in this paper we study characterizations of these classes of matrices by minors by their full rank factorization and by their thin qr factorization

a stable test for strict sign regularity a stable test to check if a given matrix is strictly sign regular is provided among other nice properties we prove that it has an optimal growth factor the test is compared with other alternative tests appearing in the literature and its advantages are shown

a descent method with linear programming subproblems for nondifferentiable convex optimization most of the descent methods developed so far suffer from the computational burden due to a sequence of constrained quadratic subproblems which are needed to obtain a descent direction in this paper we present a class of proximal type descent methods with a new direction finding subproblem especially two of them have a linear programming subproblem instead of a quadratic subproblem computational experience of these two methods has been performed on two well known test problems the results show that these methods are another very promising approach for nondifferentiable convex optimization

elimination technique from extrapolation to totally positive matrices and cagd in this survey we will show some connections between several mathematical problems such as extrapolation linear systems totally positive matrices and computer aided geometric design with elimination techniques as the common tool to deal with all of them

on the extension of some total positivity inequalities abstract some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices among these larger classes of matrices sign consistent sign regular and matrices with signed bidiagonal decomposition are considered the class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses

factorizations of cauchy vandermonde matrices abstract this paper analyzes the factorization of the inverse of a cauchy vandermonde matrix as a product of bidiagonal matrices factorizations of cauchy vandermonde matrices are also considered the results are applied to derive a fast algorithm for solving linear systems whose coefficient matrices are cauchy vandermonde matrices

conditioning and accurate computations with pascal matrices a result on the ill conditioning of pascal matrices is proved however it is shown that bidiagonal factorizations of pascal matrices can be applied to perform computations with high relative accuracy

lower bounds for the quadratic assignment problem based upon a dual formulation a new bounding procedure for the quadratic assignment problem qap is described which extends the hungarian method for the linear assignment problem lap to qaps operating on the four dimensional cost array of the qap objective function the qap is iteratively transformed in a series of equivalent qaps leading to an increasing sequence of lower bounds for the original problem to this end two classes of operations which transform the four dimensional cost array are defined these have the property that the values of the transformed objective function z are the corresponding values of the old objective function z shifted by some amount c in the case that all entries of the transformed cost array are nonnegative then c is a lower bound for the initial qap if moreover there exists a feasible solution u to the qap such that its value in the transformed problem is zero then c is the optimal value of z and u is an optimal solution for the original qap the transformations are iteratively applied until no significant increase in constant c as above is found resulting in the so called dual procedure dp several strategies are listed for appropriately determining c or equivalently transforming the cost array the goal is the modification of the elements in the cost array to obtain new equivalent problems that bring the qap closer to solution in some cases the qap is actually solved though solution is not guaranteed the close relationship between the dp and the linear programming formulation of adams and johnson is presented the dp attempts to solve adams and johnson s clp a continuous relaxation of a linearization of the qap this explains why the dp produces bounds close to the optimum values for clp calculated by johnson in her dissertation and by resende et al in their interior point algorithm for linear programming the benefit of using dp within a branch and bound algorithm is described then two versions of dp are tested on the nugent test instances from size 5 to size 30 as well as several other test instances from qaplib these compare favorably with earlier bounding methods

a factorization of totally nonsingular matrices over a ring with identity abstract we say that a rectangular matrix over a ring with identity is totally nonsingular tns if for all k all its relevant submatrices either having k consecutive rows and the first k columns or k consecutive columns and the first k rows are invertible we prove that a matrix is tns if and only if it admits a certain factorization with bidiagonal type factors and certain invertible entries this approach generalizes the loewner neville factorization usually applied to totally positive matrices

qr decomposition of almost strictly sign regular matrices in this paper l d u factorization of nonsingular sign regular and almost strictly sign regular matrices is analyzed besides nonsingular almost strictly sign regular matrices are characterized through the q r factorization illustrative numerical examples are included

characterizations of jacobi sign regular matrices abstract several characterizations of jacobi nonsingular sign regular matrices are presented moreover a stable test of o n elementary operations is obtained to check if an n n jacobi nonsingular matrix is sign regular

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

almost strictly totally negative matrices a real matrix a a i j 1 i j n is said to be almost strictly totally negative if it is almost strictly sign regular with signature e 1 1 1 which is equivalent to the property that all its nontrivial minors are negative in this paper an algorithmic characterization of nonsingular almost strictly totally negative matrices is presented

convexity of rational curves and total positivity we analyze convexity preserving properties of curves from a geometric point of view we also characterize totally positive systems of functions in terms of geometric convexity preserving properties of the rational curves rational bezier and nonuniform rational b spline curves are included in this setting

a note on matrices with maximal growth factor for neville elimination neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations the growth factor is an indicator of the numerical stability of an algorithm in the literature bounds for the growth factor of neville elimination with some pivoting strategies have appeared in this work we determine all the matrices such that the minimal upper bound of the growth factor of neville elimination with those pivoting strategies is reached

observer biased fuzzy clustering as generated by clustering algorithms clusterings or partitions are hypotheses on data explanation which are better evaluated by experts from the application domain in general clustering algorithms allow a limited usage of domain knowledge about the cluster formation process in this study we propose both a design technique and a new partitioning based clustering algorithm which can be used to assist the data analyst while looking for a set of meaningful clusters i e clusters that actually correspond to the underlying data structure following an observer metaphor according to which the perception of a group of objects depends on the observer position the closer an observer is from an image more details s he perceives we resort to shrinkage to incorporate a regularization term accounting for the observation point within the objective function of an otherwise unbiased clustering algorithm this technique allows our resulting biased algorithm to generate a set of reasonable partitions i e partitions validated by a given cluster validity index corresponding to views of data with different levels of granularity levels of detail in different regions of the data space for the illustration of the design technique we adopted the fuzzy c means fcm algorithm as the unbiased clustering algorithm and include a convergence theorem assuring that changing the point of observation in the corresponding biased algorithm fcm with focal point fcmfp does not jeopardize its convergence experimental studies on both synthetic and real data are included to illustrate the usefulness of the approach in addition and as a convenient side effect of using shrinkage the experimental results suggest that our biased algorithm fcmfp not only seems to scale better than the successive runs of the unbiased one fcm but on the average seems to produce clusters exhibiting higher validity index values as well in addition less sensitivity to initialization was observed for the biased algorithm when compared with the unbiased one

required nonzero patterns for nonsingular sign regular matrices abstract an n n real matrix is called sign regular if for each k 1 k n all its minors of order k have the same nonstrict sign the zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero the patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

sign structure preserves for m banded factorizations of sign regular matrices abstract a square matrix a a ij is called m banded if a ij 0 for all i j m motivated by the results of rainey and habetler j w rainey g j habetler tridiagonalization of completely nonnegative matrices math comp 26 1972 121 128 we provide m banded factorizations of sign regular matrices that preserve their sign structures as applications some eigenvalue interlacing properties and majorization relations are obtained for certain sign regular matrices

inner totally positive matrices abstract suppose a m n is a staircase matrix with row and column staircase sequences 1 2 n 1 2 n a minor a det a with 1 k 1 k is said to be an inner minor of a if i i i i for i 1 2 k a is said to be inner totally positiv e itp if every inner minor of a is positive we prove that a is itp if a 0 for all inner minors with q k n 0 k 1 2 n also a m n is itp iff it is totally non negative and its extreme inner minors are positive we show that an itp matrix may be reduced by similarity transformations to an itp band matrix and may alternatively be filled in by similarity transformations to become a tp matrix in both the reduction and the filling in the matrix is itp at each stage the analysis is applied to some inverse eigenvalue problems for band matrices

a level 2 reformulation linearization technique bound for the quadratic assignment problem abstract this paper studies polyhedral methods for the quadratic assignment problem bounds on the objective value are obtained using mixed 0 1 linear representations that result from a reformulation linearization technique rlt the rlt provides different levels of representations that give increasing strength prior studies have shown that even the weakest level 1 form yields very tight bounds which in turn lead to improved solution methodologies this paper focuses on implementing level 2 we compare level 2 with level 1 and other bounding mechanisms in terms of both overall strength and ease of computation in so doing we extend earlier work on level 1 by implementing a lagrangian relaxation that exploits block diagonal structure present in the constraints the bounds are embedded within an enumerative algorithm to devise an exact solution strategy our computer results are notable exhibiting a dramatic reduction in nodes examined in the enumerative phase and allowing for the exact solution of large instances

similarity to totally positive matrices and accurate computations abstract matrices similar to nonsingular totally positive matrices are analyzed in the particular case of sbd matrices new characterizations are presented for more general classes of matrices methods to assure accurate computations are analyzed

elimination techniques and interpolation a general recurrence interpolation formula due to gasca and lopez carmona j approx th 34 1982 is studied by means of elimination techniques in linear systems the repeated application of this formula is investigated in order to show its equivalence with extrapolation methods

on nonsingular sign regular matrices abstract an m n matrix a is sign regular if for each k 1 k min m n all k k submatrices of a have determinant with the same nonstrict sign the zero pattern of nonsingular sign regular matrices is analyzed it is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature a matrix is totally nonpositive if all its minors are nonpositive a test for recognizing nonsingular totally nonpositive matrices is also provided

reducing the time complexity of the fuzzy c means algorithm in this paper we present an efficient implementation of the fuzzy c means clustering algorithm the original algorithm alternates between estimating centers of the clusters and the fuzzy membership of the data points the size of the membership matrix is on the order of the original data set a prohibitive size if this technique is to be applied to very large data sets with many clusters our implementation eliminates the storage of this data structure by combining the two updates into a single update of the cluster centers this change significantly affects the asymptotic runtime as the new algorithm is linear with respect to the number of clusters while the original is quadratic elimination of the membership matrix also reduces the overhead associated with repeatedly accessing a large data structure empirical evidence is presented to quantify the savings achieved by this new method

combined matrices of almost strictly sign regular matrices abstract the combined matrix is a very useful concept for many applications almost strictly sign regular assr matrices form an important structured class of matrices with two possible zero patterns which are either type i staircase or type ii staircase we prove that under an irreducibility condition the pattern of zero and nonzero entries of an assr matrix is preserved by the corresponding combined matrix without the irreducibility condition it is proved that type i and type ii staircases are still preserved illustrative numerical examples are included

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

rank structure properties of rectangular matrices admitting bidiagonal type factorizations abstract in this paper we investigate the class of rectangular matrices that admit bidiagonal type factorizations by neville elimination without exchanges we provide a complete characterization for a rectangular matrix to be factored as a product of bidiagonal factors and a banded factor in terms of rank structure properties consequently we give a complete characterization of the class of rectangular matrices that have neville factorizations

sign regular matrices and neville elimination abstract a pivoting strategy of o n operations for the neville elimination of n n nonsingular sign regular matrices is introduced among other nice properties it is proved that it preserves sign regularity it is also shown its relationship with scaled partial pivoting strategies for neville elimination

efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem abstract the generalized traveling salesman problem gtsp is a well known combinatorial optimization problem with a host of applications it is an extension of the traveling salesman problem tsp where the set of cities is partitioned into so called clusters and the salesman has to visit every cluster exactly once while the gtsp is a very important combinatorial optimization problem and is well studied in many aspects the local search algorithms used in the literature are mostly basic adaptations of simple tsp heuristics hence a thorough and deep research of the neighborhoods and local search algorithms specific to the gtsp is required we formalize the procedure of adaptation of a tsp neighborhood for the gtsp and classify all other existing and some new gtsp neighborhoods for every neighborhood we provide efficient exploration algorithms that are often significantly faster than the ones known from the literature finally we compare different local search implementations empirically

criteria for sign regularity of sets of matrices abstract we consider matrix intervals with respect to a certain checkerboard partial ordering we show that every real matrix contained in a matrix interval is sign regular if two special matrices taken from that matrix interval are sign regular

full rank factorization in quasi ldu form of totally nonpositive rectangular matrices abstract let a a i j r n m be a totally nonpositive matrix with rank a r min n m and a 11 0 in this paper we obtain a characterization in terms of the full rank factorization in quasi ldu form that is a l d u where l r n r is a block lower echelon matrix u r r m is a unit upper echelon totally positive matrix and d r r r is a diagonal matrix with rank l rank u rank d r we use this quasi ldu decomposition to construct the quasi bidiagonal factorization of a moreover some properties about these matrices are studied

density weighted fuzzy c means clustering in this short paper a unified framework for performing density weighted fuzzy c means fcm clustering of feature and relational datasets is presented the proposed approach consists of reducing the original dataset to a smaller one assigning each selected datum a weight reflecting the number of nearby data clustering the weighted reduced dataset using a weighted version of the feature or relational data fcm algorithm and if desired extending the reduced data results back to the original dataset several methods are given for each of the tasks of data subset selection weight assignment and extension of the weighted clustering results the newly proposed weighted version of the non euclidean relational fcm algorithm is proved to produce the identical results as its feature data analog for a certain type of relational data artificial and real data examples are used to demonstrate and contrast various instances of this general approach

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

determinantal criteria for total positivity a matrix is totally positive respectively strictly totally positive if all its minors are nonnegative respectively positive in this paper we revisit the main determinantal criteria for these matrices and provide two new ones one of them for the total positivity of m n matrices with any n consecutive rows linearly independent and another one to check if an n n totally positive matrix has all minors of order less than k k n positive

a lagrangian based approach for the asymmetric generalized traveling salesman problem this paper presents an optimal approach for the asymmetric generalized traveling salesman problem gtsp the gtsp is defined on a directed graph in which the nodes are grouped into m predefined mutually exclusive and exhaustive sets with the arc set containing no intraset arcs the problem is to find a minimum cost m arc directed cycle which includes exactly one node from each set our approach employs a lagrangian relaxation to compute a lower bound on the total cost of an optimal solution the lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution finally we use an efficient branch and bound procedure which exploits the multiple choice structure of the node sets we present computational results for the optimal approach tested on a series of randomly generated problems the results show success on a range of problems with up to 104 nodes

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

polynomial least squares fitting in the bernstein basis the problem of polynomial least squares fitting in which the usual monomial basis is replaced by the bernstein basis is considered the coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular bernstein vandermonde matrix in order to use the method based on the qr decomposition of a the first stage consists of computing the bidiagonal decomposition of the coefficient matrix a starting from that bidiagonal decomposition an algorithm for obtaining the qr decomposition of a is the applied finally a triangular system is solved by using the bidiagonal decomposition of the r factor of a some numerical experiments showing the behavior of this approach are included

rederivation of the fuzzypossibilistic clustering objective function through bayesian inference unsupervised clustering of a set of datums into homogeneous groups is a primitive operation required in many signal and image processing applications in fact different incarnations and hybrids of fuzzy c means fcm and possibilistic c means pcm have been suggested which address additional requirements such as accepting weighted sets and being robust to the presence of outliers nevertheless arriving at a general framework which is independent of the datum model and the notion of homogeneity of a particular problem class is a challenge however this process has not been followed organically and clustering algorithms are generally based on exogenous objective functions which are heuristically engineered and are believed to lead to the satisfaction of a required behavior these techniques also commonly depend on regularization coefficients which are to be set prudently by the user or through separate processes in contrast in this work we utilize bayesian inference and derive a robustified objective function for a fuzzypossibilistic clustering algorithm by assuming a generic datum model and a generic notion of cluster homogeneity we utilize this model for the purpose of cluster validity assessment as well we emphasize the epistemological importance of the theoretical basis on which the developed methodology rests at the end of this paper we present experimental results to exhibit the utilization of the developed framework in the context of four different problem classes

variation diminishing transformations a direct approach to total positivity and its statistical applications abstract karlin has shown that the variation diminishing property possessed by totally positive distributions allows a unified and straightforward presentation of many basic properties of hypothesis tests his work in fact reveals that total positivity is equivalent to a suitably formulated variation diminishing property this largely expository article begins with the latter concept and consequently gives a more direct account of the theory this approach avoids the extensive mathematical preliminaries previously required and isolates the more important statistical property

almost strictly sign regular matrices and neville elimination with two determinant pivoting in 2007 cortes and pena introduced a pivoting strategy for the neville elimination of nonsingular sign regular matrices and called it two determinant pivoting neville elimination has been very useful for obtaining theoretical and practical properties for totally positive negative matrices and other related types of matrices a real matrix is said to be almost strictly sign regular if all its nontrivial minors of the same order have the same strict sign in this paper some nice properties related with the application of neville elimination with two determinant pivoting strategy to almost strictly sign regular matrices are presented

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

a factorization theorem for banded matrices in this paper we prove a factorization theorem for strictly m banded totally positive matrices we show that such a matrix is a product of m one banded matrices with positive entries a bi infinite matrix a ai i ca i and ai i n ai i o these matrices arise quite freqiently in applications and recently the have been the subject of several independent investigations 2 3 in this paper we prove a factorization theorem for strictly m handed totally positive matrices we show that such a matrix is a product of m one banded matrices with positive entries let us begin by recalling that a matrix a is totally positive if and only if ull its minors are nonnegative

comparing pivoting strategies for almost strictly sign regular matrices abstract in this paper some properties of two determinant pivoting for neville elimination are presented in particular we consider a zero increasing property and we show an optimal normwise growth factor comparisons with other pivoting strategies for neville elimination and with gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed numerical examples are included

a new convergence proof of fuzzy c means in this letter we give a new more direct derivation of the convergence properties of the fuzzy c means fcm algorithm using the equivalence between the original and reduced fcm criterion from the point of view of the reduced criterion the fcm algorithm is simply a steepest descent algorithm with variable steplength we prove that steplength adjustment follows from the majorization principle for steplength by applying the majorization principle we give a straightforward proof of global convergence further convergence properties follow immediately using known results of optimization theory

backward stability with almost strictly sign regular matrices a sign regular matrix is almost strictly sign regular if all its nontrivial minors of the same order have the same strict sign these matrices form a subclass of sign regular matrices matrices whose minors of the same order have the same sign in this paper the backward stability for almost strictly sign regular matrices is studied when neville elimination with two determinant pivoting strategy is applied in addition several numerical experiments are also presented

the symmetric quadratic traveling salesman problem in the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession this structure arises e g if the succession of two edges represents energetic conformations a change of direction or a possible change of transportation means in the symmetric case costs do not depend on the direction of traversal we study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities we establish relations to facets of the boolean quadric polytope exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities but finding a maximally violated inequality among these is np complete for the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications

totally positive bases and progressive iteration approximation in this paper we study the progressive iteration approximation property of a curve tensor product surface generated by blending a given data point set and a set of basis functions the curve tensor product surface has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular thus the b spline and nurbs curve surface have the progressive iteration approximation property and bezier curve surface also has the property if the corresponding collocation matrix is nonsingular

total positivity and toda flow abstract a real matrix a m n is tp totally positive if all its minors are nonnegative ntp if it is non singular and tp stp if it is strictly tp o oscillatory if it is tp and a power a m is stp we consider the toda flow of a symmetric matrix a t and show that if a 0 is one of tp ntp stp or o then a t is tp ntp stp or o respectively

a collection of examples where neville elimination outperforms gaussian elimination neville elimination is an elimination procedure alternative to gaussian elimination it is very useful when dealing with totally positive matrices for which nice stability results are known here we include examples most of them test matrices used in matlab which are not totally positive matrices where neville elimination outperforms gaussian elimination

single point iterative weighted fuzzy c means clustering algorithm for remote sensing image segmentation in this paper a remote sensing image segmentation procedure that utilizes a single point iterative weighted fuzzy c means clustering algorithm is proposed based upon the prior information this method can solve the fuzzy c means algorithm s problem that the clustering quality is greatly affected by the data distributing and the stochastic initializing the centrals of clustering after the probability statistics of original data the weights of data attribute are designed to adjust original samples to the uniform distribution and added in the process of cyclic iteration which could be suitable for the character of fuzzy c means algorithm so as to improve the precision furthermore appropriate initial clustering centers adjacent to the actual final clustering centers can be found by the proposed single point adjustment method which could promote the convergence speed of the overall iterative process and drastically reduce the calculation time otherwise the modified algorithm is updated from multidimensional data analysis to color images clustering moreover with the comparison experiments of the uci data sets public berkeley segmentation dataset and the actual remote sensing data the real validity of proposed algorithm is proved

a tight linearization and an algorithm for zero one quadratic programming problems this paper is concerned with the solution of linearly constrained zero one quadratic programming problems problems of this kind arise in numerous economic location decision and strategic planning situations including capital budgeting facility location quadratic assignment media selection and dynamic set covering a new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods an implicit enumeration algorithm which uses lagrangian relaxation benders cutting planes and local explorations is designed to exploit the strength of this linearization computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm

combined matrices of sign regular matrices abstract the combined matrix of a nonsingular matrix a is the hadamard entry wise product c a a a 1 t since each row and column sum of c a is equal to one the combined matrix is doubly stochastic when it is nonnegative in this work we study the nonnegativity of the combined matrix of sign regular matrices based upon their signature in particular a few coordinates of the signature e of a play a crucial role in determining whether or not c a is nonnegative

nonsingular almost strictly sign regular matrices abstract the class of nonsingular almost strictly totally positive matrices has been characterized m gasca j m pena characterizations and decompositions of almost strictly positive matrices siam j matrix anal appl 28 2006 1 8 in this paper we discuss the class of almost strictly sign regular matrices that includes almost strictly totally positive matrices a characterization is provided for these matrices in terms of their nontrivial minors using consecutive rows and consecutive columns in particular we present a characterization of certain almost strictly sign regular matrices in terms of a very reduced number of boundary almost trivial minors

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

bidiagonal decomposition of rectangular totally positive said ball vandermonde matrices error analysis perturbation theory and applications abstract an algorithm for computing the bidiagonal decomposition of said ball vandermonde matrices in the rectangular case is presented which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is fast and possesses high relative accuracy the error analysis of the algorithm is also addressed along with the perturbation theory for the bidiagonal factorization of totally positive said ball vandermonde matrices some numerical experiments showing the good behavior of the algorithm are included