A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson's normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u, x) = 0, where H : ℜ²ⁿ → ℜ²ⁿ, u ∈ ℜⁿ is a parameter variable and x ∈ ℜⁿ is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z^k = (u^k, x^k)} such that the mapping H( · ) is continuously differentiable at each z^k and may be non-differentiable at the limiting point of {z^k}. We prove that three most often used Gabriel–Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising.