Newton-type methods for solving quasi-complementarity problems via sign-based equation

In this paper, for solving quasi-complementarity problems, the discussion of the sign patterns of the solution is presented. Under some assumptions, the proposed theorem shows that the sign patterns of the solution of the quasi-complementarity problem can be obtained by solving a linear system. Then, a Newton's iteration can be applied to a differentiable nonlinear sign-based equation. The quadratic convergence conditions of the Newton iteration are given by discussing the singularity of the Jacobian. Moreover, a hybrid method is established by the existing modulus-based matrix splitting iteration method to obtain global convergence. By numerical examples, the proposed methods are shown to have higher precision and faster convergence rate than some existing methods. Meanwhile, the hybrid method is a practical efficient method in application.