A verification method for enclosing solutions of absolute value equations

The formats of references [3], [6], and [12] need to be modified, that is "The unique solution of a class of the new generalized absolute value equation"--> "The unique solution of a class of the new generalized absolute value equation"; "A class of maximum-based iteration methods for the generalized absolute value equation" --> "A class of maximum-based iteration methods for the generalized absolute value equation"; "Bounds for the solutions of absolute value equations" --> "Bounds for the solutions of absolute value equations".2. The mathematical formula in reference [14] is garbled and needs to be modified as: "|Ax| -|B||x|=b".3. In the lower part of page 5, that is line 27, "It easy to see....." should be modified as "It is easy to see....."4. On the penultimate line of page 11, the quotation marks on both sides of "diag" need to be modified.5. In the middle and lower parts of page 5, that is, line 18 and line 27, does the beginning of the paragraph need to be moved backward?

We investigate the NP-hard absolute value equation (AVE) Ax − |x|=b, where A is an arbitrary square matrix whose singular values exceed one. The significance of the absolute value equations arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear complementarity problem that in turn is equivalent to the absolute value equation. In this paper, after a lower uniform smooth approximation function of absolute value function is established, absolute value equation is transformed into smooth optimization problem, and solved by smoothing Newton method. The convergence of method is shown, and the preliminary numerical experiments are given.

A direct generalized Newton method is proposed for solving the NP-hard absolute value equation (AVE) Ax − |x| = b when the singular values of A exceed 1. A simple MATLAB implementation of the method solved 100 randomly generated 1,000-dimensional AVEs to an accuracy of 10−6 in less than 10 s each. Similarly, AVEs corresponding to 100 randomly generated linear complementarity problems with 1,000 × 1,000 nonsymmetric positive definite matrices were also solved to the same accuracy in less than 29 s each.

We propose a hybrid algorithm for solving the NP-hard absolute value equation (AVE): \(Ax-|x|=b\), where \(A\) is an \(n\times n\) square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving iteratively a linear system of equations followed by a linear program. The algorithm was tested on 100 consecutively generated random solvable instances of the AVE with \(n=\) 50, 100, 200, 500 and 1000. The algorithm solved \(100\,\%\) of the test problems to an accuracy of \(10^{-8}\) by solving an average of 2.77 systems of linear equations and linear programs per AVE.

In this paper, an upper uniform smooth approximation function of absolute value function is proposed, and some properties of uniform smooth approximation function are studied. Then, absolute value equation (AVE), Ax - |x| = b, where A is a square matrix whose singular values exceed one, is transformed into smooth optimization problem by using the upper uniform smooth approximation function, and solved by quasi-Newton method. Numerical results in solving given AVE problems demonstrated that our algorithm is valid and superior to lower uniform smooth approximation function.

In this article, we take the piecewise linear equation system \(x-W|x|=b\), which is also known by absolute value equation, where \(W\in {\mathbb R}^ {n\times n}\), \(b\in {\mathbb R}^{n}\) are given and to undetermined the value of \(x\in {\mathbb R}^{n}\). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future. (small corrections operated in the pdf file on January 7, 2023)

In this paper, a class of absolute value equations (AVE) $ Ax-B|Cx| = d $ with $ A, B, C\in \mathbb{R}^{n\times n} $ is considered, which is a generalized form of the published works by Wu ^{[<xref ref-type="bibr" rid="b1">1</xref>]}, Wu and Shen ^{[<xref ref-type="bibr" rid="b2">2</xref>]} and Mezzadri ^{[<xref ref-type="bibr" rid="b3">3</xref>]}. Some conditions to guarantee the unique solution of the above AVE are gained. The corresponding results of the above published works are generalized.

The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), \(Ax-B\vert x\vert =b\), when the condition \(\Vert A^{-1}\Vert &lt;\frac{1}{\left\Vert B\right\Vert }\) holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where \(B=I\).&#x0D; Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition \(\Vert A^{-1}\Vert &lt;\frac{1}{4\Vert B\Vert }\), that the algorithm converges linearly global to the unique solution of the AVE.&#x0D; Numerical results are reported to show the efficiency of the proposed method and to compare with an available method.

The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.

The relaxed nonlinear PHSS-like iteration method for absolute value equations

We develop several methods (including two direct methods and an iterative method) for computing an enclosure of the solutions of the so-called Sylvester-like absolute value equations. The proposed direct methods are modifications of the Bauer-Skeel and Hansen-Bliek-Rohn bounds, which were introduced for outer approximation of the solutions of the standard and generalized absolute value equations. These approaches have the advantage of considerably reducing computational costs, in contrast to simple Kroneckerization, i.e., the direct application of the aforementioned bounds to the Kronecker form of the Sylvester-like absolute value equations. We also propose an iterative approach, which refines some initial bounds and produces highly efficient enclosures for the solutions. Moreover, the iterative method can be terminated at any time and provides a numerically guaranteed distance to the unique solution.

The system of absolute value equation, denoted by AVE, is a non-differentiable NP-hard problem. Many approaches have been proposed during the past decade and most of them focus on reformulating it as complementarity problem and then solve it accordingly. Another approach is to recast the AVE as a system of nonsmooth equations and then tackle with the nonsmooth equations. In this paper, we follow this path. In particular, we rewrite it as a system of smooth equations and propose four new smoothing functions along with a smoothing-type algorithm to solve the system of equations. The main contribution of this paper focuses on numerical comparisons which suggest a better choice of smoothing function along with the smoothing-type algorithm.

We investigate the optimum correction of an absolute value equation by minimally changing the coefficient matrix and right-hand side vector using Tikhonov regularization. Solving this problem is equivalent to minimizing the sum of fractional quadratic and quadratic functions. The primary difficulty with this problem is its nonconvexity. Nonetheless, we show that a global optimal solution to this problem can be found by solving an equation on a closed interval using the subgradient method. Some examples are provided to illustrate the efficiency and validity of the proposed method.

The main objective of this study is to discuss the optimum correction of linear inequality systems and absolute value equations (AVE). In this work, a simple and efficient feasible direction method will be provided for solving two fractional nonconvex minimization problems that result from the optimal correction of a linear system. We will show that, in some special-but frequently encountered-cases, we can solve convex optimization problems instead of not-necessarily-convex fractional problems. And, by using the method of feasible directions, we solve the optimal correction problem. Some examples are provided to illustrate the efficiency and validity of the proposed method.

Aptitude treatment interaction (ATI) studies essentially deal with two related questions: Can we adapt instruction to patterns of individual differences among students? If so, for which students is a particular method of instruction most effective? The general ATI problem as advanced by Cronbach (1957) is well known (e.g., Becker, 1970), and no further discussion of it is presented here.