Lq. Qi et al., A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, MATH PROGR, 87(1), 2000, pp. 1-35
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.