A hybrid algorithm for solving the absolute value equation

Abstract

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.