By utilizing a dual complementarity condition, we propose an iterative method for solving the NPhard absolute value equation (AVE): Ax−|x| = b, where A is an n×n square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving a succession of linear programs. The algorithm was tested on 500 consecutively generated random solvable instances of the AVE w...

We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston [Thu].

We consider the linear complementarity problem (LCP): Mz + q ≥ 0, z ≥ 0, z′(Mz + q) = 0 as an absolute value equation (AVE): (M + I)z + q = |(M − I)z + q|, where M is an n× n square matrix and I is the identity matrix. We propose a concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two. The algorithm was tested on 500 consecutively generat...

We investigate the NP-hard absolute value equation (AVE) Au |u| = b, where A is an arbitrary square matrix. In this paper, we present a smoothing method for the AVE. First, we replace the absolute value function by a smooth one, called aggregate function. With this smoothing technique, we formulate the non-smooth AVE as a smooth nonlinear equations, furthermore, an unconstrained differentiable ...

This article provides a review of research involving least absolute value (LAV) regression. The review is concentrated primarily on research publisbed since Ihe sur\'ey article by Dielman (Dielman, T. E. (1984). lx"a.sl absolute value estimation in regression mtxlels; An annotated bibliography. Communications ill Statistics Theory and Methoih. 4. 513-541.) and includes articles on LAV estimatio...

Evaluating the Gaussian integrals (expectation, moments, etc.) involving the absolute value function has been playing important roles in various contents. For example, in [KN08] and [LW09], the expected number of zeros of random harmonic functions, which is also the average number of images of certain gravitational lensing system, was associated with the expectation of absolute value of certain...

A Symmetry Preserving Singular Value Decomposition

We consider the problem of classifying completely unlabeled data by using convex inequalities that contain absolute values of the data. This allows each data point to belong to either one of two classes by entering the inequality with a plus or minus value. By using such absolute value inequalities (AVIs) in support vector machine classifiers, unlabeled data can be successfully partitioned into...