Optimal correction of the absolute value equations
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Optimal correction of the absolute value equations Hossein Moosaei1,3 · Saeed Ketabchi2 · Milan Hladík1 Received: 27 February 2020 / Accepted: 31 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords Absolute value equation · Infeasible system · Non-convex optimization · Non-differentiable problem · Regularization technique
1 Introduction The absolute value equations (AVE) can be stated as follows: Ax − |x| = b,
(1)
where A and b are a real n × n matrix and an n-dimensional vector, respectively, and |x| denotes the component-wise absolute value of vector x ∈ Rn .
B
Milan Hladík [email protected] Hossein Moosaei [email protected]; [email protected]; [email protected] Saeed Ketabchi [email protected]
1
Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic
2
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
3
Department of Mathematics, Faculty of Science, University of Bojnord, Bojnord, Iran
123
Journal of Global Optimization
This system of equations, in a particular form suitable for solving interval linear equations, was first studied in [35] (and even earlier in report form in [34]). The general form of AVE was first introduced in [36], but this equation was named as an absolute value equation (AVE) by Mangasarian and Meyer in [25], and after that many researchers concentrated to analyze this problem in different aspects. The motivation of this problem comes from the point that some mathematical programming problems can be modeled by AVE and also the AVE is equivalent to the NP-hard linear complementary problem [8,9,16,25,26,32]. In the recent decade, attention of researchers was focused on several aspects of AVE. In fact, the AVE can be considered from four perspectives: (i) considering solvability, unsolvability, robustness or finding bounds of solution of AVE [14,25,33,37,42,43], (ii) the AVE has a unique solution, in this case finding the solution is the essential matter [17,23,24,27,29,38], (iii) computing the minimum norm solution in case AVE has more than one solution [19,28], (iv) the
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