On the unique solvability and numerical study of absolute value equations

Abstract

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 \(\Vert A^{-1}\Vert <\frac{1}{\left\Vert B\right\Vert }\) holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where \(B=I\). Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition \(\Vert A^{-1}\Vert <\frac{1}{4\Vert B\Vert }\), that the algorithm converges linearly global to the unique solution of the AVE. Numerical results are reported to show the efficiency of the proposed method and to compare with an available method.

Keywords

• absolute value equations
• complementarity
• generalized Newton method
• global convergence
• optimization