A relaxed generalized Newton iteration method for generalized absolute value equations

To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations Ax - B vertical bar x vertical bar = b, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix A is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.

Automated summary

This paper proposes a new relaxed generalized Newton (RGN) iteration method to avoid singular generalized Jacobian matrix and accelerate the convergence of solving generalized absolute value equations. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions, with a specific sufficient condition when the coefficient matrix A is symmetric positive definite. Numerical experiments demonstrate the effectiveness of the new RGN iteration method for linear complementarity problems.