A relaxed scalable modulus-based matrix splitting method for vertical nonlinear complementarity problems

A scalable modulus-based matrix splitting (SMMS) method is extended for solving the vertical nonlinear complementarity problem (VNCP) which shuns any auxiliary variables. With the aim to further enhance the efficiency, by introducing the relaxation matrix based on SMMS method, the relaxed scalable modulus-based matrix splitting (RSMMS) method is presented. When s=2, a comparison theorem between SMMS and RSMMS methods is presented to theoretically demonstrate the effectiveness of introducing the relaxation matrix. Under mild conditions, the convergence of the RSMMS method for arbitrary s is established and the RSMMS method is proved to converge to the unique solution of the VNCP. As a by-product, sufficient conditions are established for a VNCP to have a unique solution. Numerical results are given to demonstrate the efficiency of SMMS and RSMMS methods.