Two-step nonlinear modulus-based matrix splitting iteration method for implicit complementarity problems

The modulus-based matrix splitting (MMS) iteration method has been proved to be a useful tool for solving different kinds of complementarity problems. However, directly extending the MMS iteration method to solve the implicit complementarity problem (ICP) suffers from the difficulty of selecting the initial vector. The recently proposed modified MMS (MMMS) iteration method does not need to select the initial vector but requires solving an additional linear system of equations at each iteration. To further improve the computing efficiency of the MMMS iteration method, based on the idea of the two-step matrix splitting iteration methodology and the nonlinear characteristics of the ICP, a two-step nonlinear MMS (TNMMS) iteration method is proposed in this paper. The convergence conditions of the proposed TNMMS iteration method are analyzed when the system matrix is either a positive definite matrix or an \(\varvec{H}_{+}\) ${\mathit{H}}_{+}$ -matrix. Finally, the effectiveness of the proposed TNMMS method and its superiority over the existing MMMS iteration method are illustrated by two numerical experiments.