Method of Alternating Projection for the Absolute Value Equation (original) (raw)
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Optimization Letters, 2011
We suggest an iterative method for solving absolute value equation Ax − |x| = b, where A ∈ R n×n is symmetric matrix and b ∈ R n , coupled with the minimization technique. We also discuss the convergence of the proposed method. Some examples are given to illustrate the implementation and efficiency of the method.
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Applied Mathematics and Computation, 2015
In this paper, we introduce and analyze two new methods for solving the NP-hard absolute value equations (AVE) Ax − |x| = b, where A is an arbitrary n × n real matrix and b ∈ R n , in the case, singular value of A exceeds 1. The comparison with other known methods is carried to show the effectiveness of the proposed methods for a variety of randomly generated problems. The ideas and techniques of this paper may stimulate further research.
An improved generalized Newton method for absolute value equations
SpringerPlus, 2016
Background We consider the absolute value equations (AVEs): where A ∈ R n×n , b ∈ R n , and |x| denotes a vector in R n , whose i-th component is |x i |. A more general form of the AVEs, Ax + B|x| = b, was introduced by Rohn (2004) and researched in a more general context in Mangasarian (2007a). Hu et al. (2011) proposed a generalized Newton method for solving absolute value equation Ax + B|x| = b associated with second order cones, and showed that the method is globally linearly and locally quadratically convergent under suitable assumptions. As was shown in Mangasarian and Meyer (2006) by Mangasarian, the general NP-hard linear complementarity problems (LCPs) (Cottle and Dantzing 1968; Chung 1989; Cottle et al. 1992) subsume many mathematical programming problems such as absolute value equations (AVEs) (1), which own much simpler structure than any LCP. Hence it has inspired many scholars to study AVEs. And in Mangasarian and Meyer (2006) the AVEs (1) was investigated in detail theoretically, the bilinear program and the generalized LCP were prescribed there for the special case when the singular values of A are not less than 1. Based on the LCP reformulation, sufficient conditions for the existence and nonexistence of solutions are given in this paper. Mangasarian also has used concave minimization model (Mangasarian 2007b), dual complementarity (Mangasarian 2013), linear complementarity (Mangasarian 2014a), linear programming (Mangasarian 2014b) and a hybrid algorithm (Mangasarian 2015) to solve AVEs (1). Hu and Huang reformulated a system of absolute value equations as a standard linear complementarity problem without any
On developing a stable and quadratic convergent method for solving absolute value equation
Journal of Computational and Applied Mathematics, 2017
We modify the generalized Newton method, proposed by Mangasarian [11], for solving NP-complete absolute value equation, so that it is numerically stable and has convergence order two. Moreover, the convergence conditions are weaker than already iterative methods, hence this method can be applied to a broad range of problems. Applicability of the proposed method is tested for various examples..
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Abstract and Applied Analysis, 2012
We suggest and analyze a residual iterative method for solving absolute value equationsAx-x=bwhereA∈Rn×n,b∈Rnare given andx∈Rnis unknown, using the projection technique. We also discuss the convergence of the proposed method. Several examples are given to illustrate the implementation and efficiency of the method. Comparison with other methods is also given. Results proved in this paper may stimulate further research in this fascinating field.
A new concave minimization algorithm for the absolute value equation solution
Optimization Letters, 2021
In this paper, we study the absolute value equation (AVE) Ax − b = |x|. One effective approach to handle AVE is by using concave minimization methods. We propose a new method based on concave minimization methods. We establish its finite convergence under mild conditions. We also study some classes of AVEs which are polynomial time solvable. Keywords Absolute value equation • Concave minimization algorithms • Linear complementarity problem where A ∈ R n×n , b ∈ R n and | • | denotes absolute value. In general, (AVE) is an NP-hard problem [16]. Since a general linear complementarity problem can be formulated as an absolute value equation, several methods, such as Newton-like methods [3,15,31] or concave optimization methods [20,21], have been proposed for solving (AVE).
Optimization Letters, 2014
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Properties of the solution set of absolute value equations and the related matrix classes
Cornell University - arXiv, 2022
The absolute value equations (AVE) problem is an algebraic problem of solving Ax + |x| = b. So far, most of the research focused on methods for solving AVEs, but we address the problem itself by analysing properties of AVE and the corresponding solution set. In particular, we investigate topological properties of the solution set, such as convexity, boundedness, connectedness, or whether it consists of finitely many solutions. Further, we address problems related to nonnegativity of solutions such as solvability or unique solvability. AVE can be formulated by means of different optimization problems, and in this regard we are interested in how the solutions of AVE are related with optima, Karush-Kuhn-Tucker points and feasible solutions of these optimization problems. We characterize the matrix classes associated with the above mentioned properties and inspect the computational complexity of the recognition problem; some of the classes are polynomially recognizable, but some others are proved to be NP-hard. For the intractable cases, we propose various sufficient conditions. We also post new challenging problems that raised during the investigation of the problem.
Mixed Integer Linear Programming Method for Absolute Value Equations
Proceedings of the 2009 International Symposium …, 2009
We formulate the NP-hard absolute value equation as linear complementary problem when the singular values of A exceed one, and we proposed a mixed integer linear programming method to absolute value equation problem. The effectiveness of the method is demonstrated by its ability to solve random problems.
Error bounds and a condition number for the absolute value equations
Mathematical Programming, 2022
Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations.