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 th e nonlinear complementarity problem (NCP) and the box constrained variation al inequalities (BVI). Instead of using an infinite sequence of smoothing a pproximation functions, we use a single smoothing approximation function an d Robinson's normal equation to reformulate NCP and BVI as an equivalent no nsmoorh equation H(u, x) = 0, where H : H-2n --> H-2n, u is an element of H -n is a parameter variable and x is an element of H-n is the original varia ble. 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-More smoothing functions can generate strongly semismooth functions, which play a fundame ntal 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 sol ve a linear system of equations and if we choose a certain smoothing functi on only a reduced form needs to be solved. Preliminary numerical results sh ow that the proposed methods for particularly chosen smoothing functions ar e very promising.