8. Subset Selection in Regression (Monographs on Statistics and Applied Probability, no. 40). By A. J. Miller. ISBN 0 412 35380 6. Chapman and Hall, London, 1990. 240 pp. £25.00.

Subset Selection in Regression

Recent work by Kilmer and Martin [Linear Algebra Appl., 435 (2011), pp. 641--658] and Braman [Linear Algebra Appl., 433 (2010), pp. 1241--1253] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including (1) a bilinear operator on the matrices which is nearly an inner product and which leads to definitions for length of matrices, angle between two matrices, and orthogonality of matrices, and (2) the use of t-linear combinations to characterize the range and kernel of a mapping defined by a third-order tensor and the t-product and the quantification of the dimensions of those sets. These theoretical results lead to the study of orthogonal projections as well as an effective Gram--Schmidt process for producing an orthogonal basis of matrices. The theoretical framework also leads us to consider the notion of tensor polynomials and their relation to tensor eigentupl...

Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging

Research in closed-loop supply chains (CLSCs) has grown rapidly over the last 10 years. The authors offer a critical review of influential CLSC research that takes a business economics perspective. Much of the research was inspired by practice-driven thoughtpieces, and this helped to keep research focused on relevant issues. However, CLSC research has several assumptions, such as perfect substitution between new and remanufactured products that risk becoming institutionalized. There is a strong need to carefully examine current industrial practice so that research remains focused on relevant problems. Deeper understanding of consumer perceptions of remanufactured products, product diffusion, and valuation of returned products are needed for the field to continue to add insights into developing sustainable economies.

Production and Operations Management

Introductory Fourier transform spectroscopy

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ
 2n
 →ℜ
 2n
 , u∈ℜ
 n
 is a parameter variable and x∈ℜ
 n
 is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z

 k
 =(u

 k
 
,x

 k
 )} such that the mapping H(·) is continuously differentiable at each z

 k
 and may be non-differentiable at the limiting point of {z

 k
 }. We prove that three most often used Gabriel-More smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising.

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.

/pdf/finding-the-largest-eigenvalue-of-a-nonnegative-tensor-25h11qpxm1.pdf

Finding the Largest Eigenvalue of a Nonnegative Tensor

We introduce M -tensors. This concept extends the concept of M -matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M -tensors must be Z- tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M -tensor must be nonnegative. A symmetric M -tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an M - tensor is its smallest H + -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M -tensor if and only if all its H + -eigenvalues are nonnegative. Some further spectral properties of M -tensors are given. We also introduce strong M -tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M -tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.

/pdf/m-tensors-and-some-applications-meoq7v8gv1.pdf

$M$-Tensors and Some Applications

We investigate the NP-hard absolute value equation (AVE) Ax?|x|=b, where A is an arbitrary n×n real matrix. In this paper, we propose a smoothing Newton method for the AVE. When the singular values of A exceed 1, we show that this proposed method is globally convergent and the convergence rate is quadratic. Preliminary numerical results show that this method is promising.

Guanglu Zhou

Papers

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

Finding the Largest Eigenvalue of a Nonnegative Tensor

$M$-Tensors and Some Applications

A globally and quadratically convergent method for absolute value equations

An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor