One-step smoothing Newton method for solving the mixed complementarity problem with a function

The mixed complementarity problem (denote by MCP(F)) can be reformulated as the solution of a smooth system of equations. In the paper, based on a perturbed mid function, we propose a new smoothing function, which has an important property, not satisfied by many other smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P₀ function are discussed. Then we presented a one-step smoothing Newton algorithm to solve the MCP with a P₀ function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the rate of convergence of the method is proved under some suitable assumptions.