Absolute Value Equations Research Articles

On Picard-SHSS iteration method for absolute value equation

Picard-type methods are efficient methods for solving the absolute value equation $ Ax-|x| = b $. To further improve the performance of Picard iteration method, a new inexact Picard iteration method, named Picard-SHSS iteration method, is proposed to solve the absolute value equation. The sufficient condition for the convergence of the proposed method for solving the absolute value equation is given. A numerical example is given to demonstrate the effectiveness of the new method.

The Picard-HSS-SOR iteration method for absolute value equations

In this paper, we present the Picard-HSS-SOR iteration method for finding the solution of the absolute value equation (AVE), which is more efficient than the Picard-HSS iteration method for AVE. The convergence results of the Picard-HSS-SOR iteration method are proved under certain assumptions imposed on the involved parameter. Numerical experiments demonstrate that the Picard-HSS-SOR iteration method for solving absolute value equations is feasible and effective.

Modified SOR-Like Method for Absolute Value Equations

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like method are presented. The computational efficiency of the modified SOR-like method is better than that of the SOR-like method by some numerical experiments.

On the unique solution of the generalized absolute value equation

In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) $$Ax-B|x|=b$$ with $$A, B\in \mathbb {R}^{n\times n}$$ from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system $$Ax=b$$ with $$A\in \mathbb {R}^{n\times n}$$ . Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.

On matrix characterizations for P-property of the linear transformation in second-order cone linear complementarity problems

The P-property of the linear transformation in second-order cone linear complementarity problems (SOCLCP) plays an important role in checking the globally uniquely solvable (GUS) property due to the work of Gowda et al. However, it is not easy to verify the P-property of the linear transformation, in general. In this paper, we provide matrix characterizations for checking the P-property, which is a new approach different from those in the literature. This is a do-able manipulation, which helps verifications of the P-property and globally uniquely solvable (GUS) property in second-order cone linear complementarity problems. Moreover, using an equivalence relation to the second-order cone linear complementarity problem, we study some sufficient and necessary conditions for the unique solution of the absolute value equations associated with second-order cone (SOCAVE).

A relaxed generalized Newton iteration method for generalized absolute value equations

To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations <i>Ax</i> - <i>B</i>|<i>x</i>| = <i>b</i>, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix <i>A</i> is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.

Optimal correction of the absolute value equations

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.

A cluster-based technique for solving the GAVE

Generalized absolute value equation (abbreviated as GAVE) has wide applications in scientific computing and engineering. In this paper, we propose a cluster-based technique for solving the GAVE. First, the GAVE is shown to be equivalent to a linear equation which is related to the signs of the solution. In order to avoid the disadvantage of the existence of the sign index sets which is problem depended, by the framework of a fixed-point iteration, the analysis of the signs of the iteration vectors is given. Some necessary conditions of the nonzero and zero entries of the iteration vector are presented. Then, the signs of the solution of the GAVE can be partitioned by a clustering process. Numerical examples show that the proposed technique can work and significantly accelerate the computational efficiency comparing to some existing methods.

An Analysis of Student Difficulties in Solving Problems of Reasoning Absolute Value Material

This study aims to determine the types and factors that cause student difficulties in solving reasoning problems on the absolute value material. This research is a type of qualitative descriptive. The subjects of this study were students of class X MIPA 5 SMAN 3 Surakarta in the 2019/2020 school year. Subjects were taken using a purposive sampling technique. The data collection technique used tests and interviews. Before the test is used, it is validated by an expert. Data analysis was carried out using data reduction, data presentation, and concluding. The results of this study indicate that the types of student difficulties are 1) fact difficulties, where the subject has difficulty in analyzing and linking the information from the questions with the questions in the questions. The difficulty of the subject is because the subject does not understand the problem and rarely does the problem in the form of a story; 2) Conceptual difficulty, where the subject has difficulty determining the limits of the two absolute value equations. In determining the limits on the absolute value equation, it is necessary to define absolute value. The difficulty of the subject is because the subject still does not understand the absolute value material; 3) skill difficulty, where the subject has difficulty performing algebraic operations in determining the value of x. The difficulty of the subject is because the subject is not careful in doing and checking the written answers again.

Preconditioned conjugate gradient methods for absolute value equations

We investigate the NP-hard absolute value equations (AVE), \(Ax-B|x| =b\), where \(A,B\) are given symmetric matrices in \(\mathbb{R}^{n\times n}, \ b\in \mathbb{R}^{n}\).By reformulating the AVE as an equivalent unconstrained convex quadratic optimization, we prove that the unique solution of the AVE is the unique minimum of the corresponding quadratic optimization. Then across the latter, we adopt the preconditioned conjugate gradient methods to determining an approximate solution of the AVE.The computational results show the efficiency of these approaches in dealing with the AVE.

A modified fixed point iteration method for solving the system of absolute value equations

ABSTRACT The fixed point iteration (FPI) method proposed by Ke [Appl Math Lett. 2020;99:105990] for solving the absolute value equations (AVE) with the form is interesting for its simplicity and efficiency. However, its convergence is only guaranteed for the case that , excluding the possible case that . To complete the gap, we develop a modified FPI (MFPI) method for solving the AVE with , which, besides keeping the simplicity of FPI, improves its efficiency by judiciously choosing the involving parameter. Under mild conditions, we prove its linear convergence. We present some preliminary numerical results for , demonstrating its convergence; and compare it with FPI when , illustrating its superiority.

Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation

The absolute value equation appears in various fields of applied mathematics such as operational research. Here we consider its generalized versionAX+B|X|=C, where A,B,C∈Cn×n are given, |X|=(|xi,j|) and X∈Cn×n is an unknown matrix that must be determined. In this investigation, based on the Picard matrix splitting iteration method, we applied a matrix splitting method for solving it. We will see that under the condition σmin(A)>nσmax(|B|), this method is convergent, where σmax(|B|) denotes the largest singular value of matrix |B| and σmin(A) denotes the smallest singular value of matrix A. Then we give some convergence theorems for our new method and analyze this procedure in detail. Then we consider a p-step iteration method for solving this equation and analyze this procedure. Numerical experiment results show the efficiency of the method.

The proximal methods for solving absolute value equation

<p style='text-indent:20px;'>In this paper, by considering that the objective function of the least squares NP-hard absolute value equations (AVE) <inline-formula><tex-math id="M1">\begin{document}$ Ax-\vert x\vert = b $\end{document}</tex-math></inline-formula>, is non-convex and non-smooth, two types of proximal algorithms are proposed to solve it. One of them is the proximal difference-of-convex algorithm with extrapolation and another is the proximal subgradient method. The convergence results of the proposed methods are proved under certain assumptions. Moreover, a numerical comparison is presented to demonstrate the effectiveness of the suggested methods.

A Three-Step Iterative Method for Solving Absolute Value Equations

In this paper, we transform the problem of solving the absolute value equations (AVEs)Ax−x=bwith singular values ofAgreater than 1 into the problem of finding the root of the system of nonlinear equation and propose a three-step algorithm for solving the system of nonlinear equation. The proposed method has the global linear convergence and the local quadratic convergence. Numerical examples show that this algorithm has high accuracy and fast convergence speed for solving the system of nonlinear equations.

The tensor splitting methods for solving tensor absolute value equation

Recently, D.-D. Liu et al. (2018) presented the tensor splitting methods for solving multilinear systems and S.-Q. Du et al. (2018) generalized tensor absolute value equations. In this paper, we verify the existence of solutions of tensor absolute value equations and propose the tensor splitting methods for solving this class of equation. Furthermore, the convergence analysis of the tensor splitting method is also studied under suitable conditions. Finally, numerical examples show that our algorithm is an efficient iterative method.

A new SOR-like method for solving absolute value equations

In this paper, by constructing equivalently the absolute value equation (AVE) as the new block equation of two-by-two, we present a new SOR-like (NSOR) method for solving the AVE. The convergence and optimal parameters of NSOR method for solving the AVE are studied in detail. Numerical experiments are presented to show that the NSOR method is implementable and efficient.

On the solution of general absolute value equations

In this note, we provide necessary and sufficient conditions that ensure the existence and uniqueness of solution of the general form of absolute value equations (AVEs), Ax−B|x|=b. The performed analysis is based on the equivalence between AVEs and horizontal linear complementarity problems (HLCPs). New sufficient conditions are proposed as well. We then compare the proposed conditions with recent results in the literature and we detail how efficient solution methods for HLCPs can be easily applied to the solution of general AVEs. Finally, we provide comments on the solvability of general AVEs under conditions larger than uniqueness of solution.

Efficiency of Dynamic Computer Environment in Learning Absolute Value Equation

The presented study analyzes the usage of the didactic efficiency of multiple representations in a computer environment in learning absolute value functions and equations. It is known that the axis of symmetry of the graph of the absolute value function is the y-axis. The research was applied at the University of Novi Sad, Serbia. The data were collected by testing a group of 226 students: major chemistry and physics students at the beginning of their common calculus course. The students worked individually in two groups: the experimental and control group. The experimental group of students practiced using GeoGebra software, and the control group of students practiced using paper and pencil. At the end of the experiment, which lasted for two weeks (six school classes), both groups were tested with a post-test of knowledge without using a computer. It can be concluded that GeoGebra software had a positive influence on the students’ achievements in solving absolute value equations.

Smoothing Secant Line Slope Using Aggregation Fischer Burmeister Function

As some of the objective functions are piecewise, so they are non-differentiable at specific points which have a significant impact on deep network rate and computational time. The non-differentiability issue increases the computational time dramatically. This issue is solved by the reformulation of the absolute value equation (AVE) through a parametrized single smooth equation. However, utilizing a single smoothing function is less effective to produce a better curve at the breaking points. Therefore, this work formulates a new smoothing function of Aggregation Fischer Burmister (AFB) via amalgamating of two popular smoothing functions: Aggregation (AGG) and Fischer-Burmeister (FB). These functions are having the ability of minimum estimation from both sides of the canonical piecewise function (CPF). If an amalgamation of smoothing functions can affect the differentiability of the piecewise objective function, then amalgamating the AGG and FB smoothing functions will produce a smooth secant line slope on both sides with less computational time. To evaluate the proposed technique, we implement a Newton algorithm using MATLAB, with random initial values. A new smoothing function is formulated by firstly converting the piecewise objective function to CPF. Then, we applied it to the Newton algorithm. Finally, to validate the AVE difficulty of the new piecewise function, we perform one run for each initial value, and 30 runs for time evaluation. The experimental analysis verified that the proposed technique outperformed other techniques of AGG and FB individually in terms of the natural logarithm, exponential, and square root. Hence, this novel technique yields promising smooth approximation for AVE with less computational time.

A special shift splitting iteration method for absolute value equation

In this paper, based on the shift splitting (SS) technique, we propose a special SS iteration method for solving the absolute value equation (AVE), which is obtained by reformulating equivalently the AVE as a two-by-two block nonlinear equation. Theoretical analysis shows that the special SS method is absolutely convergent under proper conditions. Numerical experiments are given to demonstrate the feasibility, robustness and effectiveness of the special SS method.