Finite algorithms for the numerical solutions of a class of nonlinear complementarity problems

In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a nonlinear source term or some contact problems. Based on the reformulated systems of the piecewise almost linear equations, we propose a class of semi-iterative algorithms to find the exact solution of the problems. We prove that the semi-iterative algorithms enjoy a nice monotone convergence property in the sense that subsets of the indices consisting of the indices, for which the corresponding components of the iterates violate the constraints, become smaller and smaller. Then the algorithms converge monotonically to the exact solutions of the problems in a finite number of steps. Some numerical experiments are presented to show the effectiveness of the proposed algorithms.