Related papers: An Overview of Absolute Value Equations: From Theo…
The last two decades witnessed the increasing of the interests on the absolute value equations (AVE) of finding $x\in\mathbb{R}^n$ such that $Ax-|x|-b=0$, where $A\in \mathbb{R}^{n\times n}$ and $b\in \mathbb{R}^n$. In this paper, we pay…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches…
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
A solution of linear systems of equations Ax=b and Ax=0 is a vital part of many computational packages. This paper presents a novel formulation based on the projective extension of the Euclidean space using the outer product (extended…
Many systems of interest to control engineering can be modeled by linear complementarity problems. We introduce a new notion of equivalence between linear complementarity problems that sets the basis to translate the powerful tools of…
In this paper, we investigate global convergence properties of the inexact nonsmooth Newton method for solving the system of absolute value equations (AVE). Global $Q$-linear convergence is established under suitable assumptions. Moreover,…
An underdetermined generalized absolute value equation (GAVE) may have no solution, one solution, finitely many or infinitely many solutions. This paper is concerned with sufficient conditions that guarantee the existence of solutions to an…
One common type of symmetry is when values are symmetric. For example, if we are assigning colours (values) to nodes (variables) in a graph colouring problem then we can uniformly interchange the colours throughout a colouring. For a…
In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial…
Nonlinear eigenvalue problems (NEPs) present significant challenges due to their inherent complexity and the limitations of traditional linear eigenvalue theory. This paper addresses these challenges by introducing a nonlinear…
An inverse-free neural network model with mixed delays is proposed for solving the absolute value equation (AVE) $Ax -|x| - b =0$, which includes an inverse-free neural network model with discrete delay as a special case. By using the…
We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(\lambda)\mathbf{u}= 0$ on compact continua $\Omega \subset \mathbb{C}$. The method combines two key components: (1) a rational minimax…
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…
We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection…
We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…
$p$-adic linear regression is the problem of finding coefficients $\beta$ that minimise $\sum_i |y_i - x_i^\top\beta|_p$. We prove that computing an optimal solution is NP-hard via a polynomial-time reduction from Max Cut using a…
Optimization problems pervade essentially every scientific discipline and industry. Many such problems require finding a solution that maximizes the number of constraints satisfied. Often, these problems are particularly difficult to solve…
We present new convolution based smooth approximations to the absolute value function and apply them to construct gradient based algorithms such as the nonlinear conjugate gradient scheme to obtain sparse, regularized solutions of linear…
Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems…