A note on a quadratic formulation for linear complementarity problems

In this note, we discuss some properties of a quadratic formulation for linear complementarity problems. Projected SOR methods proposed by Mangasarian apply to symmetric matrices only. The quadratic formulation discussed here makes it possible to use these SOR methods for solving nonsymmetric LCPs. SOR schemes based on this formulation preserve sparsity. For proper choice of a free parameter, this quadratic formulation also preserves convexity. The value of the quadratic function for the solution of original LCP is also known.     

Iteratively Reweighted Generalized Least Squares for Estimation and Testing With Correlated Data: An Inference Function Framework

The focus of this article is on fitting regression models and testing of general linear hypotheses for correlated data using quasi-likelihood based techniques. The class of generalized method of moments or GMMs provides an elegant approach for estimating a vector of regression parameters from a set of score functions. Extending the principle of the GMMs, in the generalized estimating equation framework, leads to a quadratic inference function or QIF approach for the analysis of correlated data. We derive an iteratively reweighted generalized least squares or IRGLS algorithm for finding the QIF estimator and establish its convergence properties. A software library implementing the techniques is demonstrated through several datasets.     

A note on shrinkage wavelet estimation in Bayesian analysis

In this short note the closed form of the soft wavelet shrinkage estimator is derived, extending the work of Huang (2002) for the scale mixture of normal distributions.     

A note on three-parameter families and generalized convex functions

The concept of generalized convex functions introduced by Beckenbach [E.F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371] is extended to the two-dimensional case. Using three-parameter families, we define generalized convex (midconvex, M-convex) functions and show some continuity properties of them.     

A unifying approach to solve a class of rank-three programs involving linear and quadratic functions

The aim of this paper is two-fold. First, the so-called ‘optimal level solutions’ method is described in a new unifying framework with the aim to provide an algorithmic scheme able to approach various different classes of problems. Then, the ‘optimal level solutions’ method is used to solve a class of low-rank programmes involving linear and quadratic functions and having a polyhedral feasible region. In particular, the considered class of programmes covers, among all, rank-three d.c., multiplicative and fractional programmes. Some optimality conditions are used to improve the performance of the proposed algorithm.     

A note on unconditionally stable linear multistep methods

It has been shown by Dahlquist [3] that the trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. It is the purpose of this note to show that a slightly different stability requirement permits methods of higher accuracy.The preparation of this paper was sponsored by the Swedish Technical Research Council.     

A note on the Bayesian inference for generalized multivariate gamma distribution

In this paper, we show that the derivation of Lemma 3 of Das and Dey (2010) needs to be corrected by using a logical transformation, instead of the ad-hoc transformation which is partially motivated by its univariate equivalent transformation. The correct derivation is presented by two approaches.     

A note on the generalized symmetries of lattice hierarchies

We show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential-delay equations. All members of these hierarchies of lattice and differential-delay equations commute. It is thus seen that differential-delay hierarchies provide a broader context within which to place lattice hierarchies.     

A short note on solvability of systems of interval linear equations

In this note we formulate necessary and sufficient conditions for strong solvability and feasibility of systems of linear interval equations in terms of absolute value inequalities.     

Statistical inference using a weighted difference-based series approach for partially linear regression models

Partially linear regression models with fixed effects are useful tools for making econometric analyses and normalizing microarray data. Baltagi and Li (2002) [7] proposed a computation friendly difference-based series estimation (DSE) for them. We show that the DSE is not asymptotically efficient in most cases and further propose a weighted difference-based series estimation (WDSE). The weights in it do not involve any unknown parameters. The asymptotic properties of the resulting estimators are established for both balanced and unbalanced cases, and it is shown that they achieve a semiparametric efficient boundary. Additionally, we propose a variable selection procedure for identifying significant covariates in the parametric part of the semiparametric fixed-effects regression model. The method is based on a combination of the nonconcave penalization (Fan and Li, 2001 [13]) and weighted difference-based series estimation techniques. The resulting estimators have the oracle property; that is, they can correctly identify the true model as if the true model (the subset of variables with nonvanishing coefficients) were known in advance. Simulation studies are conducted and an application is given to demonstrate the finite sample performance of the proposed procedures.     

A communication mix for an event planning: a linear quadratic approach

The communication mix is a relevant decision issue for an organization that plans the advertising campaign for a fixed future event. It is assumed that the objectives of the organization are to minimize the cost of the advertising campaign and to drive the final demand as close as possible to a target value. Two different advertising channels are available: the first affects deterministically the consumers’ demand, whereas the second presents some stochastic aspects which are out of decision-maker’s control. Some recent mathematical developments on the stochastic linear quadratic control problem allow to formulate and solve some interesting instances of the problem. A comparative analysis of the efficiency of deterministic and stochastic controls is done and the optimal feedback policies are discussed. The trade-off between efficiency and risk of an advertising channel is essential to understand the features of the optimal solutions.This study was supported by MIUR and University of Padua.     

A note on the singular linear system of the generalized finite element methods

We characterize the kernel of the global stiffness matrix in the singular linear system of the generalized finite element methods (GFEM) which uses the classical finite element (FE) shape functions and local approximation space of harmonic polynomials.     

A note on extrema of linear combinations of elementary symmetric functions

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed.     

A note on quadratic forms of stationary functional time series under mild conditions

We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.     

A note on rational solutions to a Hirota-Satsuma-like equation

With the generalized bilinear operators based on a prime number $p=3$, a Hirota-Satsuma-like equation is proposed. Rational solutions are generated and graphically described by using symbolic computation software Maple.     

A note on a quadratic rational map with two Siegel disks

Suppose f（z） is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim （J（f））〈2.     

A note on asymptotic inference in a class of non-stationary processes

In this note some problems of asymptotic inference in a class of non-stationary stochastic processes are considered. In particular, it is shown that no criterion based on the existence of uniformly most powerful tests over a local neighborhood can be used in this situation.     

A note on a quadratic measure of deviation of density estimates

Summary The results of Rosenblatt on quadratic measure of deviations of density estimates have been generalized to a wider class of weight functions. It is pointed out that the proof of Theorem 1 of Rosenblatt is incorrect. A corrected version of the proof is also provided.     

A note on multi-criteria ABC inventory classification using weighted linear optimization

Ramanathan [R. Ramanathan, ABC inventory classification with multiple-criteria using weighted linear optimization, Computers & Operations Research 33 (2006) 695–700] recently proposed a weighted linear optimization model for multi-criteria ABC inventory classification. Despite its many advantages, Ramanathan’s model (R-model) could lead to a situation where an item with a high value in an unimportant criterion is inappropriately classified as a class A item. In this paper we present an extended version of the R-model for multi-criteria inventory classification. Our model provides a more reasonable and encompassing index since it uses two sets of weights that are most favourable and least favourable for each item. An illustrative example is presented to compare our model and the R-model.