Allow problems concerning spectral properties of sign pattern matrices: A survey

An n×n sign pattern matrix has entries in {+,-,0}. This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.     

Using Estimated Missing Spatial Data with the 2-Median Model

Spatial operations research problems seek best locations, often points of minimum aggregate weighted distance, requiring georeferenced data as input. Frequently maps of such data are incomplete, with holes in their geographic distributions. Spatial statistical procedures are available to complete these data sets with best estimates of the missing values. Impacts such imputations have on 2-median facility location–allocation solutions are explored. The sampling distribution of the spatial mean and standard distance of these medians are studied. Population density is used as the weight attribute in determining location-allocation solutions because it can be accurately described with a relatively simple spatial statistical model.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

When does the inverse have the same sign pattern as the transpose?

By a sign pattern (matrix) we mean an array whose entries are from the set {+, –, 0}. The sign patterns A for which every real matrix with sign pattern A has the property that its inverse has sign pattern A ^T are characterized. Sign patterns A for which some real matrix with sign pattern A has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices is examined.     

Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

Correlation stress testing is employed in several financial models for determining the value-at-risk (VaR) of a financial institution’s portfolio. The possible lack of mathematical consistence in the target correlation matrix, which must be positive semidefinite, often causes breakdown of these models. The target matrix is obtained by fixing some of the correlations (often contained in blocks of submatrices) in the current correlation matrix while stressing the remaining to a certain level to reflect various stressing scenarios. The combination of fixing and stressing effects often leads to mathematical inconsistence of the target matrix. It is then naturally to find the nearest correlation matrix to the target matrix with the fixed correlations unaltered. However, the number of fixed correlations could be potentially very large, posing a computational challenge to existing methods. In this paper, we propose an unconstrained convex optimization approach by solving one or a sequence of continuously differentiable (but not twice continuously differentiable) convex optimization problems, depending on different stress patterns. This research fully takes advantage of the recently developed theory of strongly semismooth matrix valued functions, which makes fast convergent numerical methods applicable to the underlying unconstrained optimization problem. Promising numerical results on practical data (RiskMetrics database) and randomly generated problems of larger sizes are reported.     

k-Best Solutions under Distance Constraints in Valuated -Matroids

In discrete maximization problems one usually wants to find an optimal solution. However, in several topics like “alignments,” “automatic speech recognition,” and “computer chess” people are interested to find thekbest solutions for somek ≥ 2. We demand that theksolutions obey certain distance constraints to avoid that thekalternatives are too similar. Several results for valuated -matroids are presented, some of them concerning time complexity of algorithms.     

The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: Easy and hard cases

This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge—Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge—Toeplitz QAP: (Pl) The Turbine Problem, i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrixB that lead to a simple solution for the Anti-Monge—Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge—Toeplitz QAP is NP-hard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Dedicated to the memory of Gene LawlerThis research has been supported by the Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Diskrete Optimierung.     

Linear complementarity problems with an invariant number of solutions

This paper studies the class INS of all realn × n matricesM for which the linear complementarity problem (q, M) has exactlyk solutions—k depending only onM—for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. This generalizes the results in Cottle and Stone (1983) which deal with the subclassU in INS wherek equals one.After the first two sections of this paper, which introduce the problem and background material, we move on to examine necessary conditions for a matrixM to be in INS (Section 3) and sufficient conditions under whichM will be in INS (Section 4). Section 5 deals with the possible values whichk may have. Section 6 discusses related results concerning the geometry of linear complementarity problems. Finally, Section 7 deals with some known and new matrix classes which are in INS.     

Convergence of Fixed-Point Continuation Algorithms for Matrix Rank Minimization

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem (Math. Program., doi:, 2009). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported.     

On generating multivariate Poisson data in management science applications

Generating multivariate Poisson random variables is essential in many applications, such as multi echelon supply chain systems, multi‐item/multi‐period pricing models, accident monitoring systems, etc. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix, and therefore are rarely used in management science. Instead, multivariate Poisson data are commonly approximated by either univariate Poisson or multivariate Normal data. However, these approximations are often not adequate in practice. In this paper, we propose a conceptually appealing correction for NORTA (NORmal To Anything) for generating multivariate Poisson data with a flexible correlation structure and rates. NORTA is based on simulating data from a multivariate Normal distribution and converting it into an arbitrary continuous distribution with a specific correlation matrix. We show that our method is both highly accurate and computationally efficient. We also show the managerial advantages of generating multivariate Poisson data over univariate Poisson or multivariate Normal data. Copyright © 2011 John Wiley & Sons, Ltd.     

Matrix-free preconditioning using partial matrix estimation

We consider matrix-free solver environments where information about the underlying matrix is available only through matrix vector computations which do not have access to a fully assembled matrix. We introduce the notion of partial matrix estimation for constructing good algebraic preconditioners used in Krylov iterative methods in such matrix-free environments, and formulate three new graph coloring problems for partial matrix estimation. Numerical experiments utilizing one of these formulations demonstrate the viability of this approach. AMS subject classification (2000) 65F10, 65F50, 49M37, 90C06     

The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples.     

Accelerated Landweber iterations for the solution of ill-posed equations

Summary In this paper, the potentials of so-calledlinear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept.If the problem's right hand side data are contaminated by noise, semiiterative methods may be used asregularization methods. Assuming optimal rate of convergence of the iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy.To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results onfast decreasing polynomials are applied to answer an open question of Brakhage.Numerical examples are given including a comparison to the method of conjugate gradients.This research was sponsored by the Deutsche Forschungsgemeinschaft (DFG).     

On a class of new and practical performance indexes for approximation of fold bifurcations of nonlinear power flow equations

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse point) is one of the key problems in power system operations and planning and such an index indicates the severity of a power system with regard to voltage collapse. There exist many interesting methods and ideas to compute this index. However, some successful methods are not yet mathematically justified while other mathematically sound methods are often proposed directly based on the bifurcation theory and they require the initial stationary state to be too close to the unknown turning point to make the underlying methods practical.This paper first gives a survey of several popular methods for estimating the fold bifurcation point including the continuation methods, bifurcation methods and the test function methods (Seydel's direct solution methods, the tangent vector methods and the reduced Jacobian method) and discuss their relative advantages and problems. Test functions are usually based on scaling of the determinant of the Jacobian matrix and it is generally not clear how to determine the behaviour of such functions. As the underlying nonlinear equations are of a particular type, this allows us to do a new analysis of the determinants of the Jacobian and its submatrices in this paper. Following the analysis, we demonstrate how to construct a class of test functions with a predictable analytical behaviour so that a suitable index can be produced. Finally, examples of two test functions from this class are proposed. For several standard IEEE test systems, promising numerical results have been achieved.     

Time series classification by class-specific Mahalanobis distance measures

To classify time series by nearest neighbors, we need to specify or learn one or several distance measures. We consider variations of the Mahalanobis distance measures which rely on the inverse covariance matrix of the data. Unfortunately??for time series data??the covariance matrix has often low rank. To alleviate this problem we can either use a pseudoinverse, covariance shrinking or limit the matrix to its diagonal. We review these alternatives and benchmark them against competitive methods such as the related Large Margin Nearest Neighbor Classification (LMNN) and the Dynamic Time Warping (DTW) distance. As we expected, we find that the DTW is superior, but the Mahalanobis distance measures are one to two orders of magnitude faster. To get best results with Mahalanobis distance measures, we recommend learning one distance measure per class using either covariance shrinking or the diagonal approach.     

On the eigenvalues of double band matrices

We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form rζ, where r is a nonnegative real number and ζ is a pth root of unity, where p is the period of the matrix, which is computed from the distance between the bands. We also present a problem in the asymptotics of spectra in which such double band matrices are perturbed by banded matrices.     

Optimization of unconstrained functions with sparse hessian matrices-newton-type methods

Newton-type methods for unconstrained optimization problems have been very successful when coupled with a modified Cholesky factorization to take into account the possible lack of positive-definiteness in the Hessian matrix. In this paper we discuss the application of these method to large problems that have a sparse Hessian matrix whose sparsity is known a priori. Quite often it is difficult, if not impossible, to obtain an analytic representation of the Hessian matrix. Determining the Hessian matrix by the standard method of finite-differences is costly in terms of gradient evaluations for large problems. Automatic procedures that reduce the number of gradient evaluations by exploiting sparsity are examined and a new procedure is suggested. Once a sparse approximation to the Hessian matrix has been obtained, there still remains the problem of solving a sparse linear system of equations at each iteration. A modified Cholesky factorization can be used. However, many additional nonzeros (fill-in) may be created in the factors, and storage problems may arise. One way of approaching this problem is to ignore fill-in in a systematic manner. Such technique are calledpartial factorization schemes. Various existing partial factorization are analyzed and three new ones are developed. The above algorithms were tested on a set of problems. The overall conclusions were that these methods perfom well in practice.     

The Steklov Problem in a Half-Plane: the Dependence of Eigenvalues on a Piecewise-Constant Coefficient

The Steklov problem considered in the paper describes free two-dimensional oscillations of an ideal, incompressible, heavy fluid in a half-plane covered by a rigid dock with two symmetric gaps. Equivalent reduction of the problem to two spectral problems for integral operators allows us to find limits for all eigenfrequencies as the distance between the gaps tends to zero or infinity. For the fundamental eigenfrequency and the corresponding eigenfunction, two terms are found in the asymptotic expansion as the distance tends to infinity. It is proved that all eigenvalues are simple for any distance. Bibliography: 15 titles.Dedicated to the centenary of V. A. Steklovs paper [1]__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 162–190.     

When is ‘nearest neighbour’ meaningful: A converse theorem and implications

Beyer et al. gave a sufficient condition for the high dimensional phenomenon known as the concentration of distances. Their work has pinpointed serious problems due to nearest neighbours not being meaningful in high dimensions. Here we establish the converse of their result, in order to answer the question as to when nearest neighbour is still meaningful in arbitrarily high dimensions. We then show for a class of realistic data distributions having non-i.i.d. dimensions, namely the family of linear latent variable models, that the Euclidean distance will not concentrate as long as the amount of ‘relevant’ dimensions grows no slower than the overall data dimensions. This condition is, of course, often not met in practice. After numerically validating our findings, we examine real data situations in two different areas (text-based document collections and gene expression arrays), which suggest that the presence or absence of distance concentration in high dimensional problems plays a role in making the data hard or easy to work with.